Metaptuqiak Anˆlush II. Prìqeirec Shmei seic

Metaptuqiak Anˆlush II Prìqeirec Shmei seic Aj na, 2007

Perieqìmena 1 Basikèc 'Ennoiec 1 1.1 Q roi Banach 1 1.2 Fragmènoi grammikoð telestèc 14 1.3 Q roi peperasmènhc diˆstashc 20 1.4 Diaqwrisimìthta 25 1.5 Q roc phlðko 28 1.6 Stoiqei dhc jewrða q rwn Hilbert 31 2 Je rhma Hahn-Banach 43 2.1 Grammikˆ sunarthsoeid kai uperepðpeda 43 2.2 To L mma tou Zorn 45 2.3 To Je rhma Hahn - Banach 46 2.4 Diaqwristikˆ jewr mata 55 2.5 KlasikoÐ duðkoð q roi 59 2.6 DeÔteroc duðkìc kai autopˆjeia 63 3 Basikˆ jewr mata gia telestèc se q rouc Banach 67 3.1 H arq tou omoiìmorfou frˆgmatoc 67 3.2 Jewr mata anoikt c apeikìnishc kai kleistoô graf matoc 71 4 Bˆseic Schauder 77 4.1 Bˆseic Schauder 77 4.2 Basikèc akoloujðec 81 4.3 ParadeÐgmata bˆsewn Schauder 84 5 AsjeneÐc topologðec 89 5.1 Sunarthsoeidèc tou Minkowski 89 5.2 Topikˆ kurtoð q roi 91 5.3 Diaqwristikˆ jewr mata se topikˆ kurtoôc q rouc 95 5.4 H asjen c topologða 98 5.5 H asjen c- topologða 102 5.6 Metrikopoihsimìthta kai diaqwrisimìthta 106

iv Perieqomena 6 Je rhma Krein Milman 111 6.1 AkraÐa shmeða 111 6.2 Je rhma Krein Milman 113 6.3 Je rhma anaparˆstashc tou Riesz 115 6.4 Efarmogèc 120 6.4aþ Je rhma Stone Weierstrass 120 6.4bþ Oloklhrwtikèc anaparastˆseic 122 7 Jewr mata stajeroô shmeðou 127 7.1 Sustolèc se pl reic metrikoôc q rouc 127 7.2 Jewr mata stajeroô shmeðou se q rouc me nìrma 128

Kefˆlaio 1 Basikèc 'Ennoiec 1.1 Q roi Banach (a) OrismoÐ sumbolismìc 'Estw X ènac grammikìc q roc pˆnw apì to K (R C). Mia sunˆrthsh : X R lègetai nìrma an ikanopoieð ta ex c: (a) x 0 gia kˆje x X kai x = 0 an kai mìno an x = 0. (b) ax = a x gia kˆje x X, a K. (g) x + y x + y gia kˆje x, y X. O arijmìc x lègetai nìrma tou dianôsmatoc x X. To zeugˆri (X, ) eðnai ènac q roc me nìrma. Ston Ðdio grammikì q ro X mporoôme na jewr soume diˆforec nìrmec. 'Otan meletˆme mia sugkekrimènh nìrma pˆnw ston X, ja grˆfoume X antð tou (X, ). H nìrma epˆgei me fusiologikì trìpo mia metrik d ston X. An x, y X, orðzoume d(x, y) = x y. EÔkola elègqoume ìti h d ikanopoieð ta axi mata thc metrik c. Epiplèon, h d eðnai sumbibast me th grammik dom tou q rou: (a) h d eðnai analloðwth wc proc metaforèc, dhlad d(x + z, y + z) = d(x, y) gia kˆje x, y, z X. (b) h d eðnai omogen c, dhlad d(ax, ay) = a d(x, y) gia kˆje x, y X kai a K. JewroÔme gnwst th stoiqei dh jewrða twn metrik n q rwn. DÐnoume mìno kˆpoiouc orismoôc gia na sunennohjoôme gia ton sumbolismì. H anoikt mpˆla me kèntro to x X kai aktðna r > 0 eðnai to sônolo D(x, r) = {y X : x y < r}. H kleist mpˆla me kèntro to x X kai aktðna r > 0 eðnai to sônolo B(x, r) = {y X : x y r}.

2 Basikec Ennoiec H sfaðra me kèntro to x X kai aktðna r > 0 eðnai to sônolo S(x, r) = {y X : x y = r}. Parat rhsh: Apì to gegonìc ìti h d eðnai analloðwth wc proc metaforèc èpetai ìti D(x, r) = x + D(0, r) := {x + z : z < r} dhlad oi perioqèc tou x prokôptoun me metaforˆ twn perioq n tou 0 katˆ x. EpÐshc, an s, r > 0 tìte D(0, sr) = sd(0, r) := {sz : z D(0, r)}. Autì prokôptei eôkola apì to ìti h d eðnai omogen c. Me ˆlla lìgia, an gnwrðzoume thn anoikt ( thn kleist ) mpˆla me kèntro to 0 kai aktðna 1, tìte gnwrðzoume tic perioqèc kˆje shmeðou tou X. Grˆfoume B X gia thn kleist monadiaða mpˆla tou X: B X = {x X : x 1}. Entel c anˆloga orðzoume tic D X kai S X. 'Opwc se kˆje metrikì q ro, èna sônolo A X lègetai anoiktì an gia kˆje x A upˆrqei r > 0 ste D(x, r) A. To B X lègetai kleistì an to X\B eðnai anoiktì. An (x n ) eðnai mia akoloujða ston X kai x X, lème ìti h (x n ) sugklðnei sto x wc proc th nìrma (sugklðnei isqurˆ sto x) an x n x 0. Aut eðnai apl c h sôgklish wc proc thn d, epomènwc isqôei otid pote gnwrðzoume gia th sôgklish se metrikoôc q rouc. Gia parˆdeigma, èna sônolo B X eðnai kleistì an kai mìno an gia kˆje akoloujða (x n ) sto B me x n x X èpetai ìti x B. An A X, grˆfoume int(a) gia to eswterikì, A gia thn kleist j kh, kai A gia to sônoro tou A. Oi orismoð eðnai oi gnwstoð. Mia akoloujða (x n ) ston X lègetai akoloujða Cauchy an gia kˆje ε > 0 upˆrqei n 0 = n 0 (ε) N me thn idiìthta m, n n 0 = x n x m < ε. O X lègetai pl rhc an kˆje akoloujða Cauchy (x n ) ston X sugklðnei isqurˆ se kˆpoio x X. Orismìc. Q roc Banach eðnai ènac pl rhc q roc me nìrma. Kˆje grammikìc upìqwroc Y enìc q rou me nìrma (X, ) eðnai q roc me nìrma. JewroÔme apl c ton periorismì thc ston Y. Ja lème ìti o Y eðnai upìqwroc tou X, kai an eðnai kleistì uposônolo tou X ja lème ìti o Y eðnai kleistìc upìqwroc tou X. Den eðnai dôskolo na elègxete ìti ènac upìqwroc Y enìc q rou Banach X eðnai q roc Banach an kai mìno an eðnai kleistìc (ˆskhsh).

(b) Anisìthtec 1.1 Qwroi Banach 3 'Estw X grammikìc q roc kai èstw C èna kurtì uposônolo tou X. Mia sunˆrthsh f : C R lègetai kurt an (1) f(tx + (1 t)y) tf(x) + (1 t)f(y) gia kˆje x, y C kai t (0, 1). H f lègetai gnhsðwc kurt an opoted pote èqoume isìthta sthn (1) èpetai ìti x = y. H f : C R lègetai koðlh (antðstoiqa, gnhsðwc koðlh) an h f eðnai kurt (antðstoiqa, gnhsðwc kurt ). (i) Anisìthta tou Jensen. 'Estw f : C R koðlh sunˆrthsh. An x 1,..., x m C kai t j (0, 1) me t 1 + + t m = 1, tìte (2) m t i f(x i ) f(t 1 x 1 + + t m x m ). An epiplèon h f eðnai gnhsðwc koðlh, tìte isìthta èqoume an kai mìno an x 1 = = x m. Apìdeixh. Me epagwg wc proc m deðqnoume tautìqrona ìti t 1 x 1 + +t m x m C kai ìti isqôei h (2). Gia m = 2 autì eðnai ˆmeso apì ton orismì tou kurtoô sunìlou kai thc koðlhc sunˆrthshc. 'Estw ìti m 3 kai ac upojèsoume ìti h prìtash isqôei an k < m. Grˆfoume t 1 x 1 + + t m x m = t 1 x 1 + s m t j s x j, ìpou s = t 2 + + t m. Apì thn epagwgik upìjesh, m t j j=2 s x j C giatð m j=2 (t j/s) = 1. EpÐshc, t 1 + s = 1 ˆra m j=1 t jx j C. AfoÔ h f eðnai koðlh, j=2 m m f t j x j t 1 f(x 1 ) + sf t j s x j j=1 kai apì thn epagwgik upìjesh, j=2 j=2 m f t j m s x t j j s f(x j). Sunduˆzontac tic dôo anisìthtec, paðrnoume j=1 j=2 m m f t j x j t j t 1 f(x 1 ) + s s f(x j), dhlad to zhtoômeno. Elègxte mìnoi sac thn perðptwsh thc gnhsðwc koðlhc sunˆrthshc. H sunˆrthsh f : (0, + ) R me f(x) = ln x eðnai gnhsðwc koðlh. An loipìn a 1,..., a m > 0 kai t j (0, 1) me t 1 + + t m = 1, tìte (3) j=2 m t j ln a j ln(t 1 a 1 + + t m a m ). j=1

4 Basikec Ennoiec 'Epetai ìti (4) a t1 1 at2 2 at m m t 1 a 1 + + t m a m me isìthta mìno an a 1 = = a m. H anisìthta aut genikeôei thn anisìthta arijmhtikoô-gewmetrikoô mèsou. An t 1 = = t m = 1/m, paðrnoume m a 1 a m a 1 + + a m. m Ja qrhsimopoi soume mia ˆmesh sunèpeia thc (4). (ii) Anisìthta tou Young. An x, y 0 kai p, q > 1 me 1 p + 1 q = 1, tìte xy xp p + yq q me isìthta mìno an x p = y q. Apìdeixh. Efarmìzoume thn anisìthta (4) me a = x p, b = y q. AfoÔ 1 p + 1 q = 1, a 1/p b 1/q a p + b q me isìthta mìno an a = b. Orismìc. An p, q > 1 kai 1 p + 1 q = 1, lème ìti oi p kai q eðnai suzugeðc ekjètec. SumfwnoÔme ìti o suzug c ekjèthc tou p = 1 eðnai o q =. (iii) Anisìthta tou Hölder. 'Estw a 1,..., a m kai b 1,..., b m K kai p, q suzugeðc ekjètec. Tìte, m m a j b j a j p j=1 j=1 1/p m b j q Apìdeixh. Upojètoume ìti to dexiì mèloc den mhdenðzetai, alli c h anisìthta eðnai profan c. Upojètoume epðshc arqikˆ ìti j=1 j=1 m m a j p = b j q = 1. j=1 Tìte, qrhsimopoi ntac thn anisìthta tou Young paðrnoume m a j b j j=1 = 1 p m m ( aj p a j b j p j=1 m a j p + 1 q j=1 j=1 1/q + b j q q. ) m b j q = 1 p + 1 q = 1. Gia th genik perðptwsh, jètoume t = ( a j p ) 1/p, s = ( b j q ) 1/q. Parathr - ste ìti m j=1 a j /t p = j=1 m b j /s q = 1 j=1

1.1 Qwroi Banach 5 kai efarmìste thn prohgoômenh anisìthta. Exetˆste pìte isqôei isìthta. Sthn perðptwsh p = q = 2 paðrnoume thn anisìthta Cauchy-Schwarz: m m a j b j a j 2 j=1 j=1 1/2 m b j 2 j=1 1/2. (iv) Anisìthta tou Minkowski. 'Estw a 1,..., a m, b 1,..., b m K kai èstw 1 p < +. Tìte, m a j + b j p j=1 1/p m a j p j=1 1/p m + b j p Apìdeixh. An p = 1 h anisìthta eðnai ˆmesh sunèpeia thc trigwnik c anisìthtac. Upojètoume ìti p > 1 kai ìti a j + b j p > 0 (alli c, h anisìthta eðnai profan c). 'Estw q o suzug c ekjèthc tou p. Grˆfoume j=1 j=1 j=1 m m m a j + b j p a j + b j p 1 a j + a j + b j p 1 b j kai efarmìzoume thn anisìthta tou Hölder me ekjètec p kai q gia kajèna apì ta dôo ajroðsmata. Parathr ntac ìti q(p 1) = p, èqoume dhlad m a j + b j p j=1 m a j + b j p j=1 m + a j + b j p j=1 j=1 1/q m a j p j=1 1/q 1/q m m a j + b j p a j + b j p m a j p j=1 j=1 j=1 1/p m b j p 1/p j=1 1/p 1/p 1/p m + b j p. Diair ntac me ( a j + b j p ) 1/q kai qrhsimopoi ntac thn 1 (1/q) = 1/p paðrnoume to zhtoômeno. Parathr seic. (a) H anisìthta tou Minkowski epekteðnetai kai se ˆpeirec a- koloujðec. An (a j ), (b j ) eðnai dôo akoloujðec sto K kai j=1 a j p < +, j=1 b j p < +, tìte h seirˆ j=1 a j + b j p sugklðnei kai a j + b j p j=1 1/p a j p j=1 1/p j=1 + b j p j=1 1/p.,.

6 Basikec Ennoiec (b) Oi anisìthtec Hölder kai Minkowski isqôoun kai gia oloklhr simec sunart seic. 'Estw (Ω, A, µ) q roc mètrou, f, g : Ω K metr simec sunart seic kai p, q suzugeðc ekjètec. Anisìthta Hölder: An oi f p kai g q eðnai oloklhr simec, tìte h fg eðnai oloklhr simh kai Ω ( fg dµ Ω ) 1/p ( 1/q f p dµ g dµ) q. Ω Anisìthta Minkowski: An oi f p kai g p eðnai oloklhr simec, tìte h f +g p eðnai oloklhr simh kai ( 1/p f + g dµ) p Ω ( 1/p f dµ) p + Ω ( 1/p g dµ) p. Ω H apìdeixh thc anisìthtac Hölder eðnai entel c anˆlogh me aut n thc antðstoiqhc anisìthtac gia peperasmènec akoloujðec. Gia na deðxete ìti h f g eðnai oloklhr simh arkeð to olokl rwma thc fg na eðnai peperasmèno, kˆti pou eðnai sunèpeia thc anisìthtac pou ja deðxete. Gia na deðxete ìti h f + g p eðnai oloklhr simh sthn anisìthta Minkowski, parathr ste ìti f(x) + g(x) p ( f(x) + g(x) ) p [2 max{ f(x), g(x) }] p = 2 p max{ f(x) p, g(x) p } 2 p ( f(x) p + g(x) p ), x Ω. Katìpin, akolouj ste thn apìdeixh thc anisìthtac Minkowski gia peperasmènec akoloujðec. (g) KlasikoÐ q roi Banach 1. Nìrmec ston K n. (a) 'Estw 1 p <. An x = (x 1,..., x n ) K n, orðzoume n x p = x j p j=1 kai sumbolðzoume ton (K n, p ) me l n p. EÔkola elègqoume ìti h p eðnai nìrma h trigwnik anisìthta eðnai sunèpeia thc anisìthtac tou Minkowski. Sthn perðptwsh p = 2 paðrnoume ton EukleÐdeio q ro l n 2. (b) An p =, orðzoume x = max 1 j n x j kai sumbolðzoume ton (K n, ) me l n. Elègxte ìti h eðnai nìrma. 'Opwc ja doôme sth sunèqeia, kˆje q roc peperasmènhc diˆstashc me nìrma eðnai pl rhc. Sthn perðptwsh tou l n p, 1 p, o èlegqoc thc plhrìthtac mporeð na gðnei kai ˆmesa. 2. Q roi akolouji n 1/p

1.1 Qwroi Banach 7 (a) 'Estw 1 p <. JewroÔme to grammikì q ro ìlwn twn ˆpeirwn akolouji n x = (x n ) gia tic opoðec n=1 x n p < +. OrÐzoume ( ) 1/p x p = x n p. n=1 H p eðnai nìrma (h trigwnik anisìthta eðnai akrib c h anisìthta tou Minkowski). O q roc pou prokôptei me autìn ton trìpo ja sumbolðzetai me l p. Prìtash 1.1.1. O l p, 1 p < eðnai q roc Banach. Apìdeixh. 'Estw (x (k) ) k N akoloujða Cauchy ston l p. An x (k) = (x (k) n ), gia kˆje n N èqoume x (k) n x (l) n x (k) x (l) p, ˆra h akoloujða (x (k) n ) k N eðnai Cauchy sto K. Epomènwc, upˆrqei x n K ste x (k) n x n kaj c k. OrÐzoume x = (x n ). 'Estw ε > 0. Upˆrqei k 0 N ste x (k) x (l) p < ε gia kˆje k, l k 0. Eidikìtera, gia kˆje N N kai k, l k 0 èqoume N n=1 x (k) n x (l) n p < ε p. Af nontac to l, blèpoume ìti gia kˆje N N kai kˆje k k 0, N n=1 x (k) n x n p ε p. Af nontac t ra to N blèpoume ìti x (k) x p ε gia kˆje k k 0. Autì deðqnei tautìqrona ìti x (k) x l p = x l p kai x (k) x isqurˆ ston l p. (b) Sthn perðptwsh p = mporoôme na orðsoume tic ex c apeirodiˆstatec genikeôseic tou l n. (b1) Ton q ro l ìlwn twn fragmènwn akolouji n, me nìrma thn x = sup{ x n : n N}. (b2) Ton q ro c 0 ìlwn twn mhdenik n akolouji n, me nìrma pˆli thn x 0 = sup{ x n : n N}. ApodeiknÔetai eôkola ìti h eðnai nìrma ston l. EpÐshc, o c 0 eðnai kleistìc upìqwroc tou l (ˆskhsh). 'Opwc ja doôme sthn epìmenh parˆgrafo, o l eðnai pl rhc (ja deðxoume kˆti polô genikìtero). 'Ara, oi l, c 0 eðnai q roi Banach. 3. Q roi fragmènwn sunart sewn 'Estw A tuqìn mh kenì sônolo. JewroÔme ton grammikì q ro B(A) ìlwn twn fragmènwn sunart sewn f : A K me nìrma thn f = sup{ f(t) : t A}.

8 Basikec Ennoiec Elègxte ìti h eðnai nìrma. Sthn perðptwsh A = N, o q roc B(A) sumpðptei me ton l. Parathr ste ìti f n f 0 an gia kˆje ε > 0 upˆrqei n 0 N ste gia kˆje n n 0 kai kˆje t A na isqôei f n (t) f(t) < ε. Dhlad, f n f ston B(A) an kai mìno an f n f omoiìmorfa. Prìtash 1.1.2. O B(A) eðnai q roc Banach. Apìdeixh. 'Estw (f n ) akoloujða Cauchy ston B(A). Gia kˆje t A èqoume f n (t) f m (t) f n f m ˆra h (f n (t)) eðnai Cauchy sto K. Epomènwc, upˆrqei to lim n f n (t). OrÐzoume f : A K me f(t) = lim n f n (t). 'Estw ε > 0. Upˆrqei n 0 N ste gia kˆje n, m N kai kˆje t A na isqôei Af nontac to m blèpoume ìti f n (t) f m (t) f n f m < ε. f n f = sup{ f n (t) f(t) : t A} ε gia kˆje n n 0. Autì apodeiknôei tautìqrona ìti f B(A) kai ìti f n f 0. Ac upojèsoume t ra ìti K eðnai ènac sumpag c metrikìc q roc kai C(K) eðnai o grammikìc q roc ìlwn twn suneq n sunart sewn f : K K. JewroÔme ton C(K) san upìqwro tou B(K). Prìtash 1.1.3. O C(K) eðnai kleistìc upìqwroc tou B(K). Apìdeixh. Upojètoume ìti f n C(K) kai f n f omoiìmorfa. Ja deðxoume ìti h f eðnai suneq c. 'Estw ε > 0. Upˆrqei n N ste f n f < ε/3. H f n eðnai suneq c sto sumpagèc K, ˆra omoiìmorfa suneq c. Epomènwc, upˆrqei δ > 0 me thn idiìthta: d(x, y) < δ = f n (x) f n (y) < ε/3. An loipìn x, y K kai d(x, y) < δ, tìte f(x) f(y) f(x) f n (x) + f n (x) f n (y) + f n (y) f(y) f n f + f n (x) f n (y) + f n f < ε. Dhlad, h f eðnai (omoiìmorfa) suneq c sto K. San pìrisma paðrnoume ìti o C(K) eðnai q roc Banach gia kˆje sumpag metrikì q ro K. Eidikìtera, o C[a, b] eðnai q roc Banach. 4. Q roi L p 'Estw (Ω, A, µ) q roc mètrou kai èstw 1 p <. JewroÔme ton grammikì q ro L p (µ) ìlwn twn metr simwn sunart sewn f : Ω K gia tic opoðec Ω f p dµ <.

1.1 Qwroi Banach 9 OrÐzoume sqèsh isodunamðac ston L p (µ) jètontac f g an f = g µ-sqedìn pantoô. To sônolo L p (µ) twn klˆsewn isodunamðac [f], f L p (µ) gðnetai grammikìc q roc me prˆxeic tic [f] + [g] = [f + g], a[f] = [af]. Ja suneqðsoume na qrhsimopoioôme to sômbolo f gia thn klˆsh [f], enno ntac ìti h [f] L p (µ) antiproswpeôetai apì opoiad pote sunˆrthsh stoiqeðo thc. An loipìn f L p (µ), orðzoume ( 1/p f p = f dµ) p. Ω H p eðnai nìrma. H trigwnik anisìthta eðnai sunèpeia thc anisìthtac tou Minkowski gia oloklhr simec sunart seic. H taôtish sunart sewn pou sumpðptoun µ-sqedìn pantoô gðnetai gia na ikanopoieðtai h f p = 0 = f = 0. Prˆgmati, an Ω f p dµ = 0 tìte f = 0 µ-sqedìn pantoô, dhlad [f] = [0]. Prìtash 1.1.4. O L p (µ), 1 p < eðnai q roc Banach. Gia thn apìdeixh ja qrhsimopoi soume èna genikì krit rio. DÐnoume pr ta kˆpoiouc orismoôc. Orismìc. 'Estw (x n ) akoloujða se ènan q ro me nìrma X. Lème ìti h seirˆ n=1 x n sugklðnei an upˆrqei x X ste s n := n x k x. k=1 Lème ìti h seirˆ n=1 x n sugklðnei apolôtwc an n=1 x n < +. L mma 1.1.5. 'Estw X ènac q roc me nìrma. Ta ex c eðnai isodônama: (a) O X eðnai pl rhc. (b) An (x n ) eðnai akoloujða ston X me n=1 x n < +, tìte h seirˆ n=1 x n sugklðnei. Apìdeixh. Upojètoume pr ta ìti o X eðnai pl rhc. 'Estw (x k ) akoloujða ston X, me thn idiìthta k=1 x k <. Gia tuqìn ε > 0, upˆrqei n 0 (ε) N ste, gia kˆje n > m n 0, Tìte, an n > m n 0, x m+1 + + x n < ε. s n s m = x m+1 + + x n x m+1 + + x n < ε. To ε > 0 tan tuqìn, ˆra h (s n ) eðnai Cauchy. O X eðnai pl rhc, ˆra h s n sugklðnei se kˆpoio x X. AntÐstrofa, èstw (x n ) akoloujða Cauchy ston X. Gia ε = 1, k = 1, 2,..., 2 mporoôme na broôme k n 1 < n 2 < < n k < ste, gia kˆje n > m n k, x n x m < 1 2 k.

10 Basikec Ennoiec Eidikìtera, gia kˆje k N. 'Ara, n k+1 > n k n k = x nk+1 x nk < 1 2 k x nk+1 x nk < 1 < +. k=1 H k=1 (x n k+1 x nk ) sugklðnei apolôtwc, opìte (apì thn upìjes mac) sugklðnei: upˆrqei x X ste m (x nk+1 x nk ) x, k=1 dhlad, x nm+1 x n1 x. 'Ara, x nk x + x n1. DeÐxame ìti h (x n ) èqei sugklðnousa upakoloujða. EÐnai ìmwc kai akoloujða Cauchy, ˆra sugklðnei ston X. 'Epetai ìti o X eðnai pl rhc. Apìdeixh thc Prìtashc 1.1.4. 'Estw (f k ) akoloujða ston L p (µ) me thn idiìthta k=1 f k p = M < +. Gia kˆje n N orðzoume g n (x) = n k=1 f k(x), x Ω. Tìte, g n p n f k p M, k=1 dhlad g n L p (µ) kai Ω gp ndµ M p. H (g n ) eðnai aôxousa, ˆra orðzetai h g(x) = lim g n (x) [0, ]. Apì to L mma tou Fatou, Ω g p dµ lim inf n Ω g p ndµ M p. Sunep c, h g p eðnai oloklhr simh. 'Epetai ìti g(x) = k=1 f k(x) < + sqedìn pantoô. OrÐzoume s n (x) = n k=1 f k(x). Apì thn g(x) < + èqoume ìti h s(x) = lim s n (x) = k=1 f k(x) sugklðnei sqedìn pantoô. H s eðnai metr simh kai apì thn s n (x) g n (x) g(x) sumperaðnoume ìti s(x) g(x) sqedìn pantoô. 'Epetai ìti Ω s p dµ dhlad s L p (µ). Tèloc, parathroôme ìti Ω g p dµ M p <, s n (x) s(x) p 2 p max{ s n (x) p, s(x) p } 2 p g(x) p sqedìn pantoô. AfoÔ s n (x) s(x) p 0 sqedìn pantoô, qrhsimopoi ntac to je rhma kuriarqhmènhc sôgklishc blèpoume ìti Ω s n s p dµ 0. Autì deðqnei ìti s n s p 0. Apì to L mma 1.1.5 èpetai ìti o L p (µ) eðnai q roc Banach.

1.1 Qwroi Banach 11 O L (Ω, A, µ) orðzetai wc ex c: an f : Ω K eðnai mia metr simh sunˆrthsh me thn idiìthta to sônolo A(f) = {t 0 : µ({x Ω : f(x) > t}) = 0} na eðnai mh kenì, orðzoume to ousi dec ˆnw frˆgma thc f jètontac esssup(f) = inf{t 0 : µ({x Ω : f(x) > t}) = 0}. Parathr ste ìti to sônolo twn metr simwn sunart sewn f gia tic opoðec A(f) eðnai grammikìc q roc kai ìti esssup(f) = min A(f) (ˆskhsh). 'Opwc prðn, tautðzoume tic f kai g an f = g sqedìn pantoô kai jewroôme ton q ro L (Ω, A, µ) twn klˆsewn isodunamðac, me nìrma thn f = esssup(f). H eðnai nìrma kai o L (µ) := L (Ω, A, µ) eðnai q roc Banach (ˆskhsh). (d) Pl rwsh Kˆje q roc me nìrma X {emfuteôetai} isometrikˆ kai puknˆ se ènan q ro Banach. H diadikasða aut lègetai pl rwsh. Orismìc 'Estw (X, ) q roc me nìrma. O q roc Banach ( X, ) lègetai pl rwsh tou X an upˆrqei grammik apeikìnish τ : X X me tic ex c idiìthtec: (a) h τ eðnai isometrða, dhlad τ(x) = x gia kˆje x X. (b) o τ(x) eðnai puknìc upìqwroc tou X. Prìtash 1.1.6. Kˆje q roc me nìrma èqei mia pl rwsh. Apìdeixh. JewroÔme to sônolo [X] ìlwn twn akolouji n Cauchy (x n ) ston X. OrÐzoume mia sqèsh isodunamðac sto [X], jètontac (x n ) (y n ) x n y n 0. To sônolo X twn klˆsewn isodunamðac gðnetai grammikìc q roc wc ex c: an x, y eðnai oi klˆseic stic opoðec an koun oi akoloujðec Cauchy (x n ), (y n ) antðstoiqa, orðzoume x + y thn klˆsh thc akoloujðac Cauchy (x n + y n ) kai a x, a K, thn klˆsh thc (ax n ). Elègxte ìti oi prˆxeic orðzontai kalˆ kai ìti o ( X, +, ) eðnai grammikìc q roc. OrÐzoume nìrma ston X wc ex c: an x X kai (x n ) mia akoloujða Cauchy sthn klˆsh x, jètoume x = lim x n. n To ìrio autì upˆrqei giatð h akoloujða ( x n ) eðnai Cauchy sto R, kai eðnai anexˆrthto apì thn epilog thc akoloujðac Cauchy (x n ) sthn x. Elègxte ta parapˆnw, kaj c kai to ìti h eðnai nìrma. Tèloc, jewroôme thn apeikìnish τ : X X, ìpou τ(x) eðnai h klˆsh thc stajer c akoloujðac (x, x,..., x,...). H τ eðnai grammik apeikìnish kai τ(x) = lim n x = x gia kˆje x X. Dhlad, o X emfuteôetai {isometrikˆ} ston X. Isqurismìc: O τ(x) eðnai puknìc ston X.

12 Basikec Ennoiec Apìdeixh. 'Estw x X kai (x n ) akoloujða Cauchy sthn klˆsh x. Tìte, lim n x τ(x n ) = lim n lim m x m x n = 0, dhlad τ(x n ) x. AfoÔ to x tan tuqìn kai τ(x n ) τ(x), o τ(x) eðnai puknìc ston X. Mènei na deðxoume ìti o X eðnai pl rhc. Ja qrhsimopoi soume èna genikì epiqeðrhma. L mma 1.1.7. 'Estw (X, d) metrikìc q roc kai M puknì uposônolo tou X me thn idiìthta: kˆje akoloujða Cauchy stoiqeðwn tou M sugklðnei se stoiqeðo tou X. Tìte, o X eðnai pl rhc. Apìdeixh. 'Estw (x n ) akoloujða Cauchy ston X. MporoÔme na broôme m n M me d(x n, m n ) < 1/n. Tìte, h (m n ) eðnai Cauchy ˆra upˆrqei x X ste m n x. AfoÔ d(x n, x) d(x n, m n ) + d(m n, x) < 1/n + d(m n, x), blèpoume ìti x n x. 'Eqoume dh deð ìti o τ(x) eðnai puknìc ston X. EpÐshc, h (τ(x n )) eðnai akoloujða Cauchy an kai mìno an h (x n ) eðnai Cauchy, kai tìte τ(x n ) x X, ìpou x h klˆsh thc (x n ). SÔmfwna me to L mma, o X eðnai pl rhc. Ask seic 1. 'Estw X q roc me nìrma, x X kai r > 0. DeÐxte ìti int(b(x, r)) = D(x, r), D(x, r) = B(x, r), D(x, r) = B(x, r) = S(x, r). 2. 'Estw X q roc me nìrma. DeÐxte ìti oi apeikonðseic + : X X X me (x, y) x + y : K X X me (a, x) ax : X R + me x x eðnai suneqeðc (wc proc tic fusiologikèc metrikèc se kˆje perðptwsh). 3. 'Estw X q roc Banach kai Y upìqwroc tou X. DeÐxte ìti o Y eðnai q roc Banach an kai mìno an eðnai kleistìc. 4. 'Estw X grammikìc q roc kai 1, 2 dôo nìrmec ston X. DeÐxte ìti x 1 x 2 gia kˆje x X an kai mìno an B (X, 2 ) B (X, 1 ). 5. 'Estw X q roc me nìrma kai Y grammikìc upìqwroc tou X. DeÐxte ìti an int(y ), tìte Y = X. 6. 'Estw B(x n, r n) mia fjðnousa akoloujða apì kleistèc mpˆlec se ènan q ro Banach X. DeÐxte ìti n=1 B(xn, rn). [Upìdeixh: DeÐxte ìti xn+1 xn rn rn+1 gia kˆje n.] 7. 'Estw X n-diˆstatoc pragmatikìc grammikìc q roc, kai x 1,..., x m dianôsmata pou parˆgoun ton X. Tìte, gia kˆje x X upˆrqoun λ 1,..., λ m R (ìqi anagkastikˆ monadikˆ), ste x = m λixi. OrÐzoume { m x = inf λ i : λ i R, x = m λ ix i }.

1.1 Qwroi Banach 13 DeÐxte ìti o (X, ) eðnai q roc me nìrma. 8. 'Estw K R n kurtì, sumpagèc, summetrikì wc proc to 0. Upojètoume ìti upˆrqei δ > 0 ste h EukleÐdeia mpˆla me kèntro to 0 kai aktðna δ na perièqetai sto K. OrÐzoume : R n R me x = min{λ 0 : x λk}. DeÐxte ìti h orðzetai kalˆ kai eðnai nìrma ston R n. DeÐxte ìti K = {x R n : x 1}. 9. JewroÔme ton R n me tic nìrmec p, 1 p. DeÐxte ìti an 1 p < q kai x R n, tìte x q x p n (1/p) (1/q) x q. DeÐxte ìti gia kˆje ε > 0 upˆrqei N N ste, gia kˆje p > N kai kˆje x R n, x x p (1 + ε) x. 10. DeÐxte ìti o c 0 eðnai kleistìc upìqwroc tou l. 11. Ja grˆfoume L p[0, 1] gia ton L p(λ), ìpou λ to mètro Lebesgue sto [0, 1]. (a) DeÐxte ìti o L [0, 1] eðnai q roc Banach. (b) An f L [0, 1], deðxte ìti f p f kaj c p. 12. 'Estw 1 p q. DeÐxte ìti an x l p tìte x q x p kai an f L q [0, 1] tìte f p f q. Eidikìtera, l p l q kai L q[0, 1] L p[0, 1]. 13. 'Estw 1 p < kai f n, f L p [0, 1] me f n f sqedìn pantoô. DeÐxte ìti f n p f p an kai mìno an f n f p 0. 14. 'Estw 1 p < kai f n L p [0, 1] me f n f sqedìn pantoô. DeÐxte ìti ta ex c eðnai isodônama: (a) f L p[0, 1] kai f n f p 0. (b) Gia kˆje ε > 0 upˆrqei δ > 0 ste, gia kˆje metr simo A [0, 1] me λ(a) < δ kai kˆje n N, na isqôei A f n p dλ < ε. 15. 'Estw C k [0, 1] o q roc ìlwn twn f : [0, 1] R pou èqoun k suneqeðc parag gouc, me nìrma thn f = max 0 s k (max{ f s (t) : t [0, 1]}). DeÐxte ìti o C k [0, 1] eðnai q roc Banach. 16. 'Estw f : [0, 1] R. H kômansh thc f orðzetai apì thn V (f) = sup { n f(t i) f(t i 1) : n N, 0 = t 0 < t 1 < < t n = 1 }. An V (f) < +, tìte lème ìti h f èqei fragmènh kômansh. JewroÔme ton q ro BV [0, 1] ìlwn twn sunart sewn f : [0, 1] R pou èqoun fragmènh kômansh, eðnai suneqeðc apì dexiˆ kai ikanopoioôn thn f(0) = 0. DeÐxte ìti h f = V (f) eðnai nìrma ston BV [0, 1] kai ìti o (BV [0, 1], ) eðnai q roc Banach.

14 Basikec Ennoiec 17. 'Estw x = (x n ) l. DeÐxte ìti h apìstash tou x apì ton c 0 eðnai Ðsh me d(x, c 0 ) = lim sup x n. n 18. 'Estw 1 p < + kai K kleistì kai fragmèno uposônolo tou l p. DeÐxte ìti to K eðnai sumpagèc an kai mìno an gia kˆje ε > 0 upˆrqei n 0(ε) N ste gia kˆje n n 0 kai kˆje x = (ξ k ) K, ξ k p < ε. k=n 19. DeÐxte ìti o q roc C[0, 1] twn suneq n sunart sewn f : [0, 1] R me nìrma thn f 1 = 1 f(t) dt den eðnai pl rhc. 0 20. DeÐxte ìti o q roc c 0 twn mhdenik n akolouji n me nìrma thn x = n=1 den eðnai pl rhc. 21. DeÐxte ìti h pl rwsh enìc q rou me nìrma X eðnai monadik. An X eðnai mia ˆllh pl rwsh tou X kai τ h isometrik emfôteush tou X ston X, tìte upˆrqei isometrða epð Φ : X X ste Φ(τ(x)) = τ (x) gia kˆje x X. x n 2 n 1.2 Fragmènoi grammikoð telestèc (a) Telestèc kai sunarthsoeid 'Estw X kai Y dôo q roi me nìrma. Mia apeikìnish T : X Y lègetai grammikìc telest c an T (ax 1 + bx 2 ) = at (x 1 ) + bt (x 2 ) gia kˆje x 1, x 2 X kai a, b K. H eikìna tou T eðnai o upìqwroc Im(T ) = {T (x) : x X} kai o pur nac tou T eðnai o upìqwroc Ker(T ) = {x X : T (x) = 0}. O T eðnai grammikìc isomorfismìc an eðnai èna proc èna kai epð, dhlad an Im(T ) = Y kai Ker(T ) = {0}. 'Enac grammikìc telest c T : X Y lègetai fragmènoc an upˆrqei M 0 ste T (x) M x gia kˆje x X. Apì thn grammikìthta tou T kai tic idiìthtec thc nìrmac èpetai ìti o T eðnai fragmènoc an kai mìno an eðnai suneq c: Je rhma 1.2.1. 'Estw X, Y q roi me nìrma kai T : X Y grammikìc telest c. Ta ex c eðnai isodônama: (a) O T eðnai suneq c apeikìnish. (b) O T eðnai suneq c sto 0. (g) O T eðnai fragmènoc. Apìdeixh. An o T eðnai suneq c, tìte eðnai suneq c kai sto 0. Upojètoume ìti o T eðnai suneq c sto 0. Gia ε = 1 > 0, mporoôme na broôme δ > 0 ste x δ = T (x) 1.

1.2 Fragmenoi grammikoi telestec 15 'Estw x X, x 0. Tìte, (δ/2 x )x δ ˆra T ((δ/2 x )x) 1. Dhlad, T (x) M x gia kˆje x X, ìpou M = 2/δ. Tèloc, upojètoume ìti o T eðnai fragmènoc kai deðqnoume ìti eðnai suneq c. Upˆrqei M > 0 me thn idiìthta T (x) M x gia kˆje x X. 'Estw x 0 X kai ε > 0. Epilègoume δ = ε/m. Tìte, an x x 0 < δ èqoume T (x) T (x 0 ) = T (x x 0 ) M x x 0 Mδ = ε. SumbolÐzoume me B(X, Y ) to sônolo t n fragmènwn grammik n telest n T : X Y. O B(X, Y ) eðnai grammikìc q roc. Orismìc. An T B(X, Y ) jètoume T = inf{m 0 : x X, T (x) M x }. AfoÔ o T eðnai fragmènoc, to sônolo ston orismì eðnai mh kenì, ˆra h T orðzetai kalˆ. ParathroÔme epðshc ìti to inf eðnai sthn pragmatikìthta min. Dhlad, T (x) T x, x X. Autì faðnetai wc ex c: paðrnoume fjðnousa akoloujða M n T me thn idiìthta Af nontac to n blèpoume ìti T (x) M n x, x X. T (x) lim n M n x = T x. Prìtash 1.2.2. 'Estw T : X Y fragmènoc grammikìc telest c. Tìte, T = sup{ T (x) : x 1} = sup{ T (x) : x = 1}. Apìdeixh. DeÐqnoume mìno thn pr th isìthta. 'Estw A = sup{ T (x) : x 1}. An x 1, tìte T (x) T x T. 'Ara, A T. AntÐstrofa, an x 0 tìte (x/ x ) 1 ˆra T (x/ x ) A = T (x) A x. Apì ton orismì thc T paðrnoume T A. Prìtash 1.2.3. H apeikìnish : B(X, Y ) R + me T T eðnai nìrma. Apìdeixh. (a) Profan c T 0 gia kˆje T B(X, Y ). An T = 0, tìte T (x) 0 x = 0 gia kˆje x X, opìte T (x) = 0 gia kˆje x X. 'Ara, T = 0. (b) An a K kai T B(X, Y ), tìte at = sup{ at (x) : x = 1} = sup{ a T (x) : x = 1} = a sup{ T (x) : x = 1} = a T.

16 Basikec Ennoiec (g) An T, S B(X, Y ) kai x X, tìte (T + S)(x) = T (x) + S(x) T (x) + S(x) T x + S x = ( T + S ) x, ˆra T + S B(X, Y ) kai T + S T + S. Prìtash 1.2.4. 'Estw X q roc me nìrma kai èstw Y q roc Banach. Tìte, o B(X, Y ) eðnai q roc Banach. Apìdeixh. 'Estw (T n ) akoloujða Cauchy ston B(X, Y ). Gia kˆje ε > 0 upˆrqei n 0 (ε) N ste T n T m ε an n, m n 0. Tìte, an x X kai n, m n 0, èqoume T n (x) T m (x) ε x. Autì deðqnei ìti h (T n (x)) eðnai Cauchy ston Y kai afoô o Y eðnai pl rhc upˆrqei y x Y me T n (x) y x. OrÐzoume T : X Y me T (x) = y x = lim n T n (x). EÔkola elègqoume ìti o T eðnai grammikìc telest c. Ja deðxoume tautìqrona ìti T B(X, Y ) kai T T n 0. Gia kˆje x X kai n n 0, T (x) T n (x) = lim n (T m (x) T n (x)) = lim n T m (x) T n (x) lim sup T m T n x ε x. n Autì deðqnei ìti (T T n ) B(X, Y ), ˆra T = (T T n )+T n B(X, Y ). EpÐshc, T T n ε gia kˆje n n 0, kai afoô to ε > 0 tan tuqìn, T T n 0. Parat rhsh. An X, Y, Z eðnai q roi me nìrma kai an T B(X, Y ), S B(Y, Z), tìte (S T )(x) = S(T (x)) S T (x) ( S T ) x, ˆra S T B(X, Z) kai S T S T. Eidikìtera, an T B(X, X) tìte T m B(X, X) gia kˆje m kai T m T m. Orismìc. Kˆje grammikìc telest c f : X K lègetai grammikì sunarthsoeidèc. O q roc B(X, K) ìlwn twn fragmènwn grammik n sunarthsoeid n f : X K lègetai duðkìc q roc tou X kai sumbolðzetai me X. AfoÔ o (K, ) eðnai pl rhc, paðrnoume to ex c. Prìtash 1.2.5. 'Estw X q roc me nìrma. O duðkìc q roc X tou X eðnai q roc Banach me nìrma thn f = sup{ f(x) : x = 1}. An X eðnai ènac q roc me nìrma, tìte X. To sunarthsoeidèc f : X K me f(x) = 0, x K eðnai fragmèno. Apì thn ˆllh pleurˆ, se kˆje apeirodiˆstato q ro me nìrma X mporoôme na orðsoume èna mh fragmèno grammikì sunarthsoeidèc wc ex c: jewroôme akoloujða grammikˆ anexˆrthtwn dianusmˆtwn x n me x n = 1, thn opoða epekteðnoume se bˆsh tou X prosjètontac èna sônolo dianusmˆtwn {z i : i I}. OrÐzoume

1.2 Fragmenoi grammikoi telestec 17 f(x n ) = n kai f(z i ) = 0, i I, kai epekteðnoume grammikˆ. 'Etsi paðrnoume èna grammikì sunarthsoeidèc f gia to opoðo sup{ f(x) : x = 1} sup{ f(x n ) : n N} = +. To er thma {pìso ploôsioc eðnai o duðkìc q roc} ja mac apasqol sei argìtera. Ja doôme (je rhma Hahn-Banach) ìti o X perièqei pˆnta pollˆ sunarthsoeid. (b) ParadeÐgmata telest n kai sunarthsoeid n 1. 'Estw 1 p. OrÐzoume R, L : l p l p me R(x 1,..., x n,...) = (0, x 1,..., x n,...), L(x 1,..., x n,...) = (x 2,..., x n,...). O R (dexiˆ metatìpish) kai o L (arister metatìpish) eðnai grammikoð telestèc. O R eðnai èna proc èna allˆ ìqi epð, en o L eðnai epð allˆ ìqi èna proc èna. Akìma, R = L = 1 kai L R = Id, en R L Id kai Ker(R L) = {x l p : x n = 0, n 2}. 2. 'Estw 1 p kai q o suzug c ekjèthc tou p. Gia stajerì y l q orðzoume f y : l p K me f y (x) = x n y n. n=1 Apì thn anisìthta tou Hölder èpetai ìti gia kˆje x l p h seirˆ n x ny n sugklðnei apolôtwc kai f y (x) = x n y n y q x p. n=1 'Ara f y l p kai f y y q (h grammikìthta tou f y eðnai faner ). 3. TeleÐwc anˆloga, an p kai q eðnai suzugeðc ekjètec kai g L q (µ), orðzoume φ g : L p (µ) K me φ g (f) = Ω fgdµ. Tìte, φ g (L p (µ)) kai φ g g L q (µ). 4. 'Estw t [0, 1]. OrÐzoume δ t : C[0, 1] K me δ t (f) = f(t). Tìte, δ t (C[0, 1]) kai δ t = 1. 5. 'Estw X, Y sumpageðc metrikoð q roi kai τ : Y X suneq c. OrÐzoume A : C(X) C(Y ) me (Af)(y) = f(τ(y)). Tìte, A B(C(X), C(Y )) kai A = 1. 6. 'Estw κ : [0, 1] [0, 1] R metr simh sunˆrthsh me thn idiìthta 1 0 1 0 κ(x, y) dλ(y) c 1, κ(x, y) dλ(x) c 2, x (c.p.) y (c.p)

18 Basikec Ennoiec ìpou c 1, c 2 jetikèc stajerèc. Gia kˆje 1 < p < + orðzoume T : L p ([0, 1]) L p ([0, 1]) me (T f)(x) = 1 0 κ(x, y)f(y)dλ(y). Ja deðxoume tautìqrona ìti h (T f)(x) orðzetai kalˆ x (c.p) kai ìti o T eðnai fragmènoc telest c. 'Eqoume (T f)(x) = 1 0 1 0 ( 1 0 c 1/q 1 κ(x, y)f(y)dλ(y) κ(x, y) 1/q κ(x, y) 1/p f(y) dλ(y) ) 1/q ( 1 κ(x, y) dλ(y) ( 1 κ(x, y) f(y) p dλ(y) ìpou q o suzug c ekjèthc tou p. Tìte, T f p p = 0 1 0 c p/q 1 = c p/q 1 c p/q (T f)(x) p dx 1 1 0 1 0 0 0 ) 1/p κ(x, y) f(y) p dλ(y) ) 1/p κ(x, y) f(y) p dλ(y)dλ(x) f(y) p ( 1 1 1 c 2 f(y) p dλ(y). 0 0 ) κ(x, y) dλ(x) dλ(y) Dhlad, T f p c 1/q 1 c 1/p 2 f p. 7. 'Estw X o upìqwroc tou C[0, 1] pou apoteleðtai apì tic suneq c paragwgðsimec sunart seic. OrÐzoume D : X C[0, 1] me Df = f. O D eðnai ènac mh fragmènoc grammikìc telest c (ˆskhsh). (g) IsomorfismoÐ, isometrðec, isodônamec nìrmec 'Estw X, Y q roi me nìrma. 'Enac grammikìc telest c T : X Y lègetai isomorfismìc an eðnai isomorfismìc grammik n q rwn (dhl. èna proc èna kai epð) kai oi T, T 1 eðnai fragmènoi telestèc. EÔkola elègqoume ìti an o T : X Y eðnai isomorfismìc, upˆrqoun M 1, M 2 > 0 ste ( ) 1 M 2 x T (x) M 1 x, x X. AntÐstrofa an o T : X Y eðnai grammikìc isomorfismìc kai upˆrqoun M 1, M 2 > 0 ste na isqôei h ( ), tìte o T eðnai isomorfismìc q rwn me nìrma.

1.2 Fragmenoi grammikoi telestec 19 Lème ìti dôo q roi me nìrma X kai Y eðnai isìmorfoi an upˆrqei isomorfismìc T : X Y. Oi X kai Y lègontai isometrikˆ isìmorfoi an upˆrqei isomorfismìc T : X Y me thn epiplèon idiìthta T (x) = x, x X. 'Enac tètoioc isomorfismìc lègetai isometrða. Parathr ste ìti kˆje isometrða diathreð tic apostˆseic: an x 1, x 2 X, tìte T (x 1 ) T (x 2 ) = x 1 x 2. Epomènwc, dôo isometrikˆ isìmorfoi q roi {tautðzontai} tìso san grammikoð ìso kai san metrikoð q roi. DÔo nìrmec 1 kai 2 pˆnw ston Ðdio grammikì q ro X lègontai isodônamec an h tautotik apeikìnish Id : (X, 1 ) (X, 2 ) eðnai isomorfismìc. IsodÔnama, an upˆrqoun jetikèc stajerèc a, b ste a x 1 x 2 b x 1 gia kˆje x X. Parathr seic. 1. O isomorfismìc q rwn me nìrma kai h isodunamða norm n eðnai sqèseic isodunamðac. 2. An ènac q roc me nìrma eðnai pl rhc, tìte eðnai pl rhc kai wc proc kˆje isodônamh nìrma. 3. An 1 kai 2 eðnai dôo isodônamec nìrmec, tìte h anisìthta a x 1 x 2 b x 1, x X, eðnai isodônamh me thn ab 2 B 1 bb 2, ìpou B i h monadiaða mpˆla tou (X, i ). Ask seic 1. 'Estw X q roc Banach kai T B(X, X) me thn idiìthta n=1 T n < +. An y X orðzoume ton metasqhmatismì S y : X X me S y(x) = y + T (x). DeÐxte ìti o S y èqei monadikì stajerì shmeðo (S y(x 0) = x 0), to x 0 = y + n=1 T n (y). 2. DÐnontai g : [0, 1] R kai K : [0, 1] [0, 1] R suneqeðc. DeÐxte ìti upˆrqei suneq c sunˆrthsh f : [0, 1] R pou ikanopoieð thn exðswsh tou Volterra f(t) = g(t) + t 0 K(s, t)f(s)ds gia kˆje t [0, 1]. [Upìdeixh: An M = max{ K(s, t) : 0 s, t 1} kai T : C[0, 1] C[0, 1] o telest c pou orðzetai apì thn (T f)(t) = t 0 K(s, t)f(s)ds, deðxte ìti T n M n /n! gia kˆje n N.] 3. 'Estw X, Y q roi me nìrma kai T : X Y grammikìc telest c me thn idiìthta: an (x n ) akoloujða ston X me x n 0, tìte h (T (x n )) eðnai fragmènh akoloujða ston Y. DeÐxte ìti o T eðnai fragmènoc.

20 Basikec Ennoiec 4. DeÐxte ìti o l p eðnai isometrikˆ isìmorfoc me ènan upìqwro tou L p [0, 1] gia kˆje p 1. [Upìdeixh: Jewr ste ton upìqwro tou L p[0, 1] pou parˆgetai apì tic f n = (n(n + 1)) 1/p χ [ 1 n+1, 1 n ].] 5. Ston c 00 orðste nìrma me thn ex c idiìthta: h den eðnai isodônamh me thn, allˆ oi q roi (c 00, ) kai (c 00, ) eðnai isometrikˆ isìmorfoi. [Upìdeixh: An T : c 00 c 00 eðnai grammikìc isomorfismìc, h x = T x eðnai nìrma ston c 00.] 6. (Krit rio tou Schur) 'Estw (a ij) i,j=1 ènac ˆpeiroc pðnakac me a ij 0 gia kˆje i, j. Upojètoume akìma ìti upˆrqoun b, c > 0 kai p i > 0 ste, gia kˆje i, j, a ij p i bp j, a ij p j cp i. j=1 DeÐxte ìti o telest c T : l 2 l 2 pou orðzetai apì thn eðnai fragmènoc, kai T bc. 7. An x 0, x 1,..., x n R, tìte T ((ξ i ) i ) = ( ) ξ j a ij n i,j=0 j=1 n x i x j i + j + 1 π x 2 i. Aut eðnai h anisìthta tou Hilbert. Ja qreiasteðte to krit rio tou Schur, kai thn i=0 i 1 i + 1 + j + 1 2 2 i=0 1 < i + 1 2 0 dx (x + j + 1 ) x = π. 2 j + 1 2 1.3 Q roi peperasmènhc diˆstashc (a) Q roi peperasmènhc diˆstashc 'Estw X ènac grammikìc q roc diˆstashc n kai èstw {e 1,..., e n } mia bˆsh tou pˆnw apì to K. H l 1 -nìrma ston X orðzetai wc ex c: an x = n a ie i X, jètoume x 1 = n a i. L mma 1.3.1. H monadiaða mpˆla B 1 tou (X, 1 ) eðnai sumpag c. Apìdeixh. 'Estw (x (k) ) mia akoloujða sthn B 1. Kˆje x (k) grˆfetai monos manta sth morf x (k) = n a (k) i e i ìpou n a i 1. Eidikìtera, gia kˆje i n kai kˆje k, èqoume a (k) i 1. AfoÔ h (a (k) 1 ) eðnai fragmènh, èqei upakoloujða pou sugklðnei se kˆpoio a 1 K. Epilègontac diadoqikˆ upakoloujðec, mporoôme se n b mata na broôme

1.3 Qwroi peperasmenhc diastashc 21 aôxousa akoloujða deikt n m 1 < < m k < me thn idiìthta: gia kˆje i n, OrÐzoume x = n a ie i. Tìte, a (m k) i a i K. lim x k x(m k) 1 = lim k n a i a (m k) i = 0. Dhlad, x (mk) x. Profan c x B 1, ˆra deðxame ìti h B 1 eðnai akoloujiakˆ sumpag c. Je rhma 1.3.2. 'Estw X ènac grammikìc q roc diˆstashc n. Opoiesd pote dôo nìrmec ston X eðnai isodônamec. Apìdeixh. JewroÔme thn bˆsh {e 1,..., e n } kai th nìrma 1 tou L mmatoc 1.3.1. 'Estw tuqoôsa nìrma ston X. Ja deðxoume ìti oi kai 1 eðnai isodônamec. Autì apodeiknôei to je rhma, afoô h isodunamða norm n eðnai sqèsh isodunamðac. JewroÔme th monadiaða sfaðra S 1 = {x X : x 1 = 1} tou (X, 1 ), kai th sunˆrthsh f : S 1 R + me f(x) = x. H f eðnai suneq c sunˆrthsh: an x, y S 1, tìte n f(x) f(y) = x y x y = (x i y i )e i n ( ) x i y i e i max e n i x i y i i n ( ) = max e i x y 1. i n ParathroÔme akìma ìti f(x) > 0 gia kˆje x S 1 (giatð x = 0 = x = 0 = x / S 1 ) kai ìti h S 1 eðnai sumpag c wc kleistì uposônolo thc B 1. 'Ara h f paðrnei mia gnhsðwc jetik elˆqisth tim m kai mia mègisth tim M sthn S 1. Dhlad, an x S 1 èqoume 0 < m x M. AfoÔ oi dôo nìrmec eðnai jetikˆ omogeneðc, èpetai ìti m x 1 x M x 1 gia kˆje x X, dhlad oi kai 1 eðnai isodônamec. Je rhma 1.3.3. 'Estw X q roc peperasmènhc diˆstashc me nìrma kai èstw Y q roc me nìrma. Kˆje grammikìc telest c T : X Y eðnai fragmènoc. Apìdeixh. OrÐzoume ston X mia deôterh nìrma wc ex c: x = x X + T (x) Y.

22 Basikec Ennoiec (elègxte ìti eðnai nìrma). Oi X m, M > 0 ste gia kˆje x X. Eidikìtera, kai eðnai isodônamec, ˆra upˆrqoun m x X x X + T (x) Y M x X T (x) Y M x X gia kˆje x X, ˆra o T eðnai fragmènoc. Pìrisma 1.3.4. An X kai Y eðnai dôo n-diˆstatoi q roi me nìrma, tìte oi X kai Y eðnai isìmorfoi. Apìdeixh. 'Estw X kai Y dôo q roi me nìrma, diˆstashc dimx = dimy = n. AfoÔ oi dôo q roi èqoun thn Ðdia diˆstash, upˆrqei grammikìc isomorfismìc T : X Y. AfoÔ h diˆstash twn X kai Y eðnai peperasmènh, oi T kai T 1 eðnai fragmènoi telestèc. 'Ara o T eðnai isomorfismìc q rwn me nìrma. Pìrisma 1.3.5. Kˆje q roc peperasmènhc diˆstashc me nìrma eðnai pl rhc. Apìdeixh. An ènac q roc eðnai pl rhc wc proc kˆpoia nìrma, tìte eðnai pl rhc kai wc proc kˆje isodônamh nìrma. An dimx = n, tìte ìlec oi nìrmec ston X eðnai isodônamec. ArkeÐ loipìn na deðxoume ìti o X eðnai pl rhc wc proc mða apì autèc. DeÐxte ìti o X eðnai pl rhc wc proc thn n a i e i = max a i. i n Pìrisma 1.3.6. Se ènan q ro peperasmènhc diˆstashc, èna sônolo eðnai sumpagèc an kai mìno an eðnai kleistì kai fragmèno. Apìdeixh. 'Estw A kleistì kai fragmèno uposônolo tou X wc proc kˆpoia nìrma. Ja deðxoume ìti to A eðnai akoloujiakˆ sumpagèc. AfoÔ oi kai 1 eðnai isodônamec, upˆrqoun a, b > 0 ste a x x 1 b x gia kˆje x X. 'Estw (x n ) akoloujða sto A. AfoÔ to A eðnai fragmèno wc proc thn upˆrqei M > 0 ste x n M = x 1 bm gia kˆje n. To sônolo (bm)b 1 eðnai sumpagèc apì to L mma 1.3.1, ˆra upˆrqei upakoloujða (x nk ) pou sugklðnei se kˆpoio x X wc proc thn 1. 'Omwc tìte, x x nk 1 a x x n k 1 0, ˆra x nk x wc proc thn. Tèloc, x A giatð to A eðnai -kleistì. O antðstrofoc isqurismìc isqôei se kˆje metrikì q ro. Pìrisma 1.3.7. Kˆje upìqwroc peperasmènhc diˆstashc enìc q rou me nìrma eðnai kleistìc. Apìdeixh. 'Estw X q roc me nìrma kai èstw Y upìqwroc tou X me dimy <. Apì to Pìrisma 1.3.5, o Y eðnai pl rhc, ˆra eðnai kleistì uposônolo tou X.

(b) To L mma tou F. Riesz 1.3 Qwroi peperasmenhc diastashc 23 EÐdame ìti h monadiaða mpˆla enìc q rou peperasmènhc diˆstashc eðnai sumpag c. H sumpˆgeia thc monadiaðac mpˆlac qarakthrðzei touc q rouc peperasmènhc diˆstashc. H apìdeixh basðzetai sto ex c l mma. L mma tou Riesz. 'Estw X q roc me nìrma kai èstw Y ènac gn sioc kleistìc upìqwroc tou X. (a) Gia kˆje ε (0, 1) upˆrqei x ε S X tou opoðou h apìstash apì ton Y eðnai toulˆqiston 1 ε: d(x ε, Y ) := inf{ x ε y : y Y } 1 ε. (b) An o Y èqei peperasmènh diˆstash, tìte upˆrqei x S X tou opoðou h apìstash apì ton Y eðnai h mègisth dunat : d(x, Y ) = 1. Apìdeixh. (a) 'Estw ε (0, 1). O Y eðnai gn sioc upìqwroc tou X, epomènwc mporoôme na broôme x 0 X\Y. AfoÔ o Y eðnai kleistìc, d(x 0, Y ) = d > 0. AfoÔ d/(1 ε) > d, upˆrqei y 0 Y ste 0 x 0 y 0 < d 1 ε. JewroÔme to x ε = (x 0 y 0 )/ x 0 y 0 S X. Gia kˆje y Y èqoume y 0 + x 0 y 0 y Y, sunep c x ε y = x 0 y 0 x 0 y 0 y = 1 x 0 y 0 x 0 (y 0 + x 0 y 0 y) d x 0 y 0 > 1 ε. 'Ara, d(x ε, Y ) 1 ε. (b) Ac upojèsoume t ra ìti o Y èqei peperasmènh diˆstash. Gia to x 0 sto (a), brðskoume y n Y me d(x 0, y n ) d := d(x 0, Y ). H (y n ) eðnai fragmènh ston Y, epomènwc èqei sugklðnousa upakoloujða (y nk ) (ta kleistˆ kai fragmèna uposônola tou Y eðnai sumpag ). An lim k y nk = y 0, tìte y 0 Y kai x 0 y 0 = d. H apìdeixh suneqðzetai ìpwc sto (a). Pìrisma 1.3.8. 'Estw X 1 X 2 X n upìqwroi peperasmènhc diˆstashc enìc q rou me nìrma X (ìloi oi egkleismoð eðnai gn sioi). Tìte, mporoôme na broôme monadiaða dianôsmata x n X n ste d(x n, X n 1 ) = 1, n 2. Eidikìtera, se kˆje apeirodiˆstato q ro me nìrma X mporoôme na broôme akoloujða (x n ) apì monadiaða dianôsmata me thn idiìthta x n x m 1 an n m. Apìdeixh. BrÐskoume to x n efarmìzontac to deôtero mèroc tou L mmatoc tou Riesz gia to zeugˆri X n 1 X n, dimx n 1 <.

24 Basikec Ennoiec 'Estw t ra apeirodiˆstatoc q roc me nìrma X. Gia thn kataskeu thc akoloujðac (x n ), epilègoume tuqìn x 1 S X kai orðzoume X 1 = span{x 1 }. Apì to L mma tou Riesz upˆrqei x 2 S X ste d(x 2, X 1 ) = 1. OrÐzoume X 2 = span{x 1, x 2 } kai suneqðzoume me ton Ðdio trìpo: an ta x 1,..., x n èqoun oristeð, jètoume X n = span{x 1,..., x n } kai, qrhsimopoi ntac to gegonìc ìti o X eðnai apeirodiˆstatoc, brðskoume x n+1 S X ste d(x n+1, X n ) = 1. Apì thn kataskeu eðnai fanerì ìti an n < m tìte x n X m 1 ˆra x n x m d(x m, X m 1 ) = 1. MporoÔme t ra na qarakthrðsoume touc q rouc peperasmènhc diˆstashc wc ex c: Je rhma 1.3.9. 'Enac q roc me nìrma èqei peperasmènh diˆstash an kai mìno an h monadiaða mpˆla tou eðnai sumpag c. Apìdeixh. 'Eqoume deð ìti an dimx < tìte h B X eðnai sumpag c. Ac upojèsoume ìti o X eðnai apeirodiˆstatoc. SÔmfwna me to prohgoômeno pìrisma, mporoôme na broôme akoloujða (x n ) monadiaðwn dianusmˆtwn me x n x m 1 an n m. H (x n ) den èqei sugklðnousa upakoloujða, ˆra h B X den eðnai sumpag c. Ask seic 1. 'Estw X q roc me nìrma kai èstw 0 < θ < 1. 'Ena A B X lègetai θ-dðktuo gia thn B X an gia kˆje x B X upˆrqei a A me x a < θ. An to A eðnai θ-dðktuo gia thn B X, deðxte ìti gia kˆje x B X upˆrqoun a n A, n N, ste x = θ n a n. n=0 2. 'Estw X = (R n, ) kai èstw ε > 0. (a) 'Estw x 1,..., x k B X me thn idiìthta: x i x j ε an i j. DeÐxte ìti k (1 + 2/ε) n. (b) DeÐxte ìti upˆrqei ε-dðktuo gia thn B X me plhjˆrijmo N (1 + 2/ε) n. [Upìdeixh gia to (a): Oi mpˆlec B(x i, ε/2) perièqontai sthn B(0, 1 + ε/2) kai èqoun xèna eswterikˆ.] 3. 'Estw X apeirodiˆstatoc q roc me nìrma. (a) DeÐxte ìti upˆrqoun x 1, x 2,..., x n,... B X ste x n + 1 4 B X B X kai ta x n + 1 4 B X na eðnai xèna. (b) DeÐxte ìti den upˆrqei mètro Borel µ ston X pou na ikanopoieð ta ex c: 1. To µ eðnai analloðwto wc proc tic metaforèc, dhlad µ(x + A) = µ(a) gia kˆje sônolo Borel A kai kˆje x X. 2. µ(a) > 0 gia kˆje mh kenì anoiktì A X. 3. Upˆrqei mh kenì anoiktì A 0 X me µ(a 0) < +.

1.4 Diaqwrisimìthta 1.4 Diaqwrisimothta 25 'Estw X q roc me nìrma kai D X. To D lègetai puknì ston X an D = X. IsodÔnama, an gia kˆje x X kai kˆje ε > 0 upˆrqei z D me x z < ε. Orismìc. O X lègetai diaqwrðsimoc an upˆrqei arijm simo sônolo D X pou eðnai puknì ston X. (a) ParadeÐgmata 1. Oi q roi l p, 1 p < kai c 0 eðnai diaqwrðsimoi. DeÐqnoume autìn ton isqurismì gia ton l p (h perðptwsh tou c 0 af netai wc ˆskhsh). Prìtash 1.4.1. Gia kˆje 1 p <, o l p eðnai diaqwrðsimoc. Apìdeixh. Upojètoume ìti K = R (h migadik perðptwsh eðnai entel c anˆlogh). JewroÔme to sônolo D = {y = (y 1,..., y N, 0, 0,...) : N N, y n Q}. To D eðnai arijm simo. Ja deðxoume ìti D = l p. 'Estw x = (x n ) l p kai ε > 0. H seirˆ n x n p sugklðnei, ˆra upˆrqei N N ste n=n+1 x n p < εp 2. Gia kˆje n = 1,..., N mporoôme na broôme rhtì osod pote kontˆ ston x n. MporoÔme loipìn na broôme rhtoôc y n, n = 1,..., N pou na ikanopoioôn thn Prosjètontac, èqoume x n y n p < εp, n = 1,..., N. 2N N n=1 x n y n p < εp 2. OrÐzoume y = (y 1,..., y N, 0, 0,...). Tìte, y D kai x y p = < ( N x n y n p + n=1 ( ε p 2 + εp 2 ) 1/p = ε. n=n+1 x n p ) 1/p AfoÔ ta x l p kai ε > 0 tan tuqìnta, èpetai to zhtoômeno. 2. O l den eðnai diaqwrðsimoc. Genikˆ, gia na deðxoume ìti ènac metrikìc q roc den eðnai diaqwrðsimoc qrhsimopoioôme sun jwc ton ex c isqurismì. L mma 1.4.2. 'Estw (X, d) metrikìc q roc. Ac upojèsoume ìti mporoôme na broôme x i, i I ston X kai a > 0 pou ikanopoioôn to ex c: gia kˆje i j I d(x i, x j ) a. Tìte, gia kˆje puknì D X èqoume card(i) card(d).

26 Basikec Ennoiec Apìdeixh. Oi mpˆlec D(x i, a/2), i I eðnai xènec. An to D eðnai puknì, se kˆje D(x i, a/2) upˆrqei kˆpoio d i D. An i j, tìte d i d j afoô D(x i, a/2) D(x j, a/2) =. 'Ara, h f : I D me f(i) = d i eðnai èna proc èna. Dhlad, to D èqei toulˆqiston tìsa stoiqeða ìsa to I. Sthn perðptwsh tou l, jewroôme to sônolo A = {x = (x n ) : x n {0, 1}, n N}. Kˆje akoloujða me ìrouc 0 1 eðnai fragmènh, ˆra A l. ParathroÔme ìti an x = (x n ), y = (y n ) A kai x y, tìte x y = 1. SÔmfwna me to l mma, an D eðnai puknì uposônolo tou l, tìte to D èqei toulˆqiston tìsa stoiqeða ìsa to A. 'Omwc, to diag nio epiqeðrhma tou Cantor deðqnei ìti to A eðnai uperarijm simo. Sunep c, o l den eðnai diaqwrðsimoc. 3. O B[a, b] kai o L [a, b] den eðnai diaqwrðsimoi q roi (ˆskhsh). Apì to je rhma prosèggishc tou Weierstrass prokôptei ìti o C[a, b] eðnai diaqwrðsimoc (ˆskhsh: efarmìste to krit rio thc 'Askhshc 1 se sunduasmì me to gegonìc ìti o q roc P [a, b] twn poluwnômwn eðnai puknìc ston C[a, b]). Sthn epìmenh parˆgrafo ja deðxoume ìti, genikìtera, an K eðnai ènac sumpag c metrikìc q roc, tìte o C(K) eðnai diaqwrðsimoc. (b) Diaqwrisimìthta tou C(K) 'Estw K ènac sumpag c metrikìc q roc. Ja deðxoume ìti o C(K) eðnai diaqwrðsimoc. H apìdeixh ja basisteð sto epìmeno L mma. L mma 1.4.3 (diamerðseic thc monˆdac). 'Estw K ènac sumpag c metrikìc q roc. Upojètoume ìti K = V 1 V n, ìpou V 1,..., V n anoiktˆ sônola. Upˆrqoun suneqeðc sunart seic φ i : K [0, 1] me supp(φ i ) V i (i = 1,..., n), ste φ 1 (x) + + φ n (x) = 1 gia kˆje x K. Apìdeixh. Gia kˆje x K mporoôme na broôme i(x) n kai anoikt perioq W x tou x ste W x V i(x). Apì th sumpˆgeia tou K, upˆrqoun x 1,..., x m K ste K = W x1 W xm. Gia kˆje i = 1,..., n, orðzoume H i na eðnai h ènwsh ekeðnwn twn W xj, j m ta opoða perièqontai sto V i. Efarmìzontac to L mma tou Urysohn, brðskoume suneqeðc sunart seic g i : K [0, 1] me thn idiìthta: g i 1 sto H i kai supp(g i ) V i. T ra, orðzoume tic φ 1,..., φ n wc ex c: φ 1 = g 1 φ 2 = (1 g 1 )g 2 φ n = (1 g 1 )(1 g 2 ) (1 g n 1 )g n. Apì ton orismì twn φ i èqoume supp(φ i ) supp(g i ) V i. Epagwgikˆ elègqoume ìti φ 1 + + φ k = 1 k (1 g i). Sunep c, φ 1 + + φ n = 1 n (1 g i ) 1, afoô kˆje x K an kei se kˆpoio H i kai 1 g i 0 sto H i.

1.4 Diaqwrisimothta 27 Je rhma 1.4.4. 'Estw K ènac sumpag c metrikìc q roc. O C(K) eðnai diaqwrðsimoc. Apìdeixh. Gia kˆje f C(K) orðzoume ω f (δ) = sup{ f(x) f(y) : d(x, y) δ}. Parathr ste ìti to gegonìc ìti h f eðnai suneq c (isodônama, omoiìmorfa suneq c) perigrˆfetai isodônama apì thn lim ω f (δ) = 0. δ 0 StajeropoioÔme + δ > 0 kai kalôptoume ton K me peperasmènec to pl joc mpˆlec D(x j, δ), j = 1,..., N. Apì to L mma 1.4.3 upˆrqoun suneqeðc sunart seic φ i : K [0, 1] me supp(φ i ) D(x i, δ) (i = 1,..., N), ste φ 1 (x)+ +φ N (x) = 1 gia kˆje x K. Jètoume F δ = span{φ 1,..., φ N }. Isqurismìc. dist(f, F δ ) ω f (δ). Prˆgmati, an orðsoume g = N f(x i)φ i, tìte g F δ kai, gia kˆje y K, N N N f(y) g(y) = f(y)φ i (y) f(x i )φ i (y) φ i (y) f(y) f(x i ), kai, lìgw twn supp(φ i ) D(x i, δ), to teleutaðo ˆjroisma isoôtai me {i:y D(x i,δ)} φ i (y) f(y) f(x i ) ω f (δ) {i:y D(x i,δ)} φ i (y) ω f (δ). T ra, apì ton isqurismì kai apì thn lim ω f (δ) = 0 sumperaðnoume ìti C(K) = δ 0 + n=1 F n. AfoÔ kˆje 1/2 F 1/2 èqei peperasmènh diˆstash, o n C(K) eðnai diaqwrðsimoc ('Askhsh 1). (g) Diaqwrisimìthta tou L p (K, B, µ), 1 p < 'Estw K ènac sumpag c metrikìc q roc. Grˆfoume B gia thn σ-ˆlgebra twn Borel uposunìlwn tou K. Je rhma 1.4.5. 'Estw µ èna peperasmèno mètro ston (K, B). Gia kˆje 1 p <, o L p (K, B, µ) eðnai diaqwrðsimoc. Apìdeixh. Perigrˆfoume ta b mata thc apìdeixhc. JewroÔme thn oikogèneia A = {A B : upˆrqei f n C(K) : 0 f n 1 kai f n χ A p 0}. ApodeiknÔoume ìti h A eðnai σ-ˆlgebra, deðqnontac diadoqikˆ ìti eðnai kleist wc proc sumplhr mata, perièqei to K, eðnai kleist wc proc peperasmènec tomèc kai wc proc aôxousec en seic (ˆskhsh). Sth sunèqeia apodeiknôoume ìti kˆje anoiktì U K an kei sthn A. Prˆgmati, an U eðnai anoiktì uposônolo tou K kai an orðsoume f n : K [0, 1] me f n (x) = nd(x, K \ U) 1 + nd(x, K \ U), tìte f n χ U kai, apì to je rhma kuriarqhmènhc sôgklishc, f n χ U p 0.

28 Basikec Ennoiec 'Epetai ìti A = B kai autì deðqnei ìti oi aplèc metr simec sunart seic φ : K R proseggðzontai apì suneqeðc me thn p. AfoÔ o C(K) eðnai diaqwrðsimoc wc proc thn kai f p f (µ(x)) 1/p, parathroôme ìti o (C(K), p ) eðnai diaqwrðsimoc. Oi aplèc sunart seic eðnai puknèc ston L p (K, B, µ), sunep c o L p (K, B, µ) eðnai diaqwrðsimoc. MporoÔme epðshc na deðxoume ìti o L p (R), 1 p < eðnai diaqwrðsimoc. H basik parat rhsh eðnai ìti an f L p kai an jèsoume f n = χ [ n,n] f, tìte f n f p 0. Dhlad, o upìqwroc pou apoteleðtai apì tic g L p pou mhdenðzontai èxw apì kˆpoio diˆsthma [ n, n], n N eðnai puknìc ston L p. Me bˆsh aut n thn parat rhsh, mporoôme na anaqjoôme sto Je rhma 1.4.5. Anˆlogo apotèlesma isqôei gia ton L p (R d ), d N (ˆskhsh). Ask seic 1. 'Estw X q roc me nìrma. Upojètoume ìti upˆrqei arijm simo sônolo A X me thn idiìthta o upìqwroc span(a) na eðnai puknìc ston X. DeÐxte ìti o X eðnai diaqwrðsimoc. 2. DeÐxte ìti o c 0 eðnai diaqwrðsimoc. 3. DeÐxte ìti o C[a, b] eðnai diaqwrðsimoc, en o B[a, b] ìqi. Exetˆste an o L [0, 1] eðnai diaqwrðsimoc. 4. DeÐxte ìti, gia kˆje d N kai gia kˆje 1 p <, o L p(r d ) eðnai diaqwrðsimoc. 1.5 Q roc phlðko 'Estw X q roc me nìrma kai èstw Z ènac grammikìc upìqwroc tou X. OrÐzoume mia sqèsh isodunamðac ston X wc ex c: x y x y Z. O q roc phlðko X/Z eðnai to sônolo twn klˆsewn isodunamðac [x] = x + Z, to opoðo gðnetai grammikìc q roc me prˆxeic tic [x] + [y] = [x + y], a[x] = [ax]. To oudètero stoiqeðo thc prìsjeshc eðnai h klˆsh [0] = Z. Ac upojèsoume epiplèon ìti o Z eðnai kleistìc upìqwroc tou X. OrÐzoume mia sunˆrthsh 0 : X/Z R + mèsw thc [x] 0 = inf{ y : y x} = inf{ x z : z Z}. Prìtash 1.5.1. An o Z eðnai kleistìc upìqwroc tou X, tìte h 0 eðnai nìrma ston X/Z. Apìdeixh. (a) Profan c [x] 0 0, kai an [x] 0 = 0 tìte upˆrqoun z n Z ste x z n 0. Dhlad, x = lim n z n Z = Z

1.5 Qwroc phliko 29 afoô o Z eðnai kleistìc. 'Omwc autì shmaðnei ìti [x] = [0]. AntÐstrofa, [0] 0 = inf{ z : z Z} = 0, afoô o Z eðnai upìqwroc. (b) An a 0, qrhsimopoi ntac thn az = Z èqoume a[x] 0 = [ax] 0 = inf{ ax z : z Z} = inf{ ax az : z Z} (g) OmoÐwc, afoô Z + Z = Z, = inf{ a x z : z Z} = a [x] 0. [x] + [y] 0 = [x + y] 0 = inf{ x + y z : z Z} = inf{ x + y (z 1 + z 2 ) : z 1, z 2 Z} inf{ x z 1 + y z 2 : z 1, z 2 Z} = inf{ x z 1 : z 1 Z} + inf{ y z 2 : z 2 Z} = [x] 0 + [y] 0. O q roc (X/Z, 0 ) lègetai q roc phlðko tou X (me ton Z). Prìtash 1.5.2. H fusiologik apeikìnish Q : X X/Z me Q(x) = [x] eðnai fragmènoc grammikìc telest c kai Q 1. Apìdeixh. H grammikìthta tou Q elègqetai eôkola. EpÐshc, afoô 0 Z, 'Ara, Q 1. Q(x) 0 = [x] 0 = inf{ x z : z Z} x. Prìtash 1.5.3. 'Estw X, Y q roi me nìrma, T B(X, Y ) kai Z = Ker(T ). OrÐzoume T 0 : X/Z Y me T 0 ([x]) = T (x). Tìte, o T 0 eðnai èna proc èna, fragmènoc grammikìc telest c kai T 0 = T. Apìdeixh. O Z eðnai kleistìc upìqwroc tou X giatð o T eðnai suneq c kai grammik apeikìnish. JewroÔme ton q ro phlðko X/Z. AfoÔ Z = Ker(T ) èqoume [x] = [x 1 ] = x x 1 Ker(T ) = T (x) = T (x 1 ), dhlad o T 0 orðzetai kalˆ. EÔkola elègqoume ìti o T 0 eðnai èna proc èna, grammikìc telest c. EpÐshc, an x X tìte gia kˆje z Z, ˆra T 0 ([x]) = T 0 [x z] = T (x z) T x z T 0 ([x]) T [x] 0. Dhlad, o T 0 eðnai fragmènoc kai T 0 T. 'Estw 0 < ε < T kai èstw x X me x = 1 kai T (x) > T ε. Tìte, T 0 ([x]) = T (x) > T ε kai [x] 0 x = 1. 'Ara, T 0 T 0([x]) [x] 0 > T ε. AfoÔ to ε tan tuqìn, T 0 T. Dhlad, T 0 = T.

30 Basikec Ennoiec Prìtash 1.5.4. 'Estw X q roc Banach kai èstw Z kleistìc upìqwroc tou X. Tìte, o X/Z eðnai q roc Banach. Apìdeixh. 'Estw ([x n ]) akoloujða Cauchy ston X/Z. Ja deðxoume ìti h ([x n ]) èqei sugklðnousa upakoloujða. AfoÔ h ([x n ]) eðnai Cauchy, mporoôme na broôme aôxousa akoloujða deikt n n 1 < n 2 < < n k < gia thn opoða [x nk+1 x nk ] 0 = [x nk+1 ] [x nk ] 0 < 1 2 k+1. Epagwgikˆ, brðskoume z k Z ste (x nk+1 z k+1 ) (x nk z k ) < 1 2 k. H epilog twn z k gðnetai wc ex c: jètoume z 1 = 0. Ac upojèsoume ìti èqoun epilegeð ta z 1,..., z k. AfoÔ [x nk+1 x nk ] 0 < 1/2 k+1, upˆrqei y k+1 Z me thn idiìthta x nk+1 x nk y k+1 < 1/2 k. Jètoume z k+1 = z k + y k+1, opìte (x nk+1 z k+1 ) (x nk z k ) = x nk+1 x nk y k+1 < 1 2 k. H akoloujða (x nk z k ) eðnai Cauchy ston X, epomènwc upˆrqei x 0 X ste x nk z k x 0. Apì thn Prìtash 1.5.2 èpetai ìti Q(x nk z k ) Q(x 0 ), dhlad [x nk ] [x 0 ]. KleÐnoume aut n thn parˆgrafo me mia tupik efarmog twn q rwn phlðkwn. Prìtash 1.5.5. 'Estw X q roc me nìrma, Z kleistìc upìqwroc tou X kai Y upìqwroc tou X peperasmènhc diˆstashc. Tìte, o Z +Y eðnai kleistìc upìqwroc tou X. Apìdeixh. JewroÔme thn fusiologik apeikìnish Q : X X/Z. AfoÔ o Y èqei peperasmènh diˆstash, o Q(Y ) eðnai upìqwroc peperasmènhc diˆstashc tou X/Z, ˆra kleistìc upìqwroc tou X/Z. AfoÔ h Q eðnai suneq c apeikìnish, o Q 1 (Q(Y )) eðnai kleistìc upìqwroc tou X. 'Omwc, x Q 1 (Q(Y )) Q(x) Q(Y ) y Y : x y Z x Y + Z. Ask seic 1. 'Estw X q roc me nìrma kai Y kleistìc upìqwroc tou X. An oi Y kai X/Y eðnai q roi Banach, tìte o X eðnai q roc Banach. 2. 'Estw X q roc Banach kai Y, Z kleistoð upìqwroi tou X. Upojètoume ìti o Y eðnai isìmorfoc me ton Z. EÐnai oi X/Y kai X/Z isìmorfoi? [Upìdeixh: Jewr ste touc X = l 2, Y = {x l 2 : x 1 = 0} kai Z = {x l 2 : x 1 = x 2 = 0}.] 3. 'Estw X grammikìc q roc kai Y upìqwroc tou X. 'Enac grammikìc telest c P : X Y lègetai probol epð tou Y an, gia kˆje y Y, P (y) = y.

1.6 Stoiqeiwdhc jewria qwrwn Hilbert 31 Upojètoume ìti o X eðnai q roc me nìrma, o Y eðnai kleistìc upìqwroc tou X kai ìti upˆrqei suneq c probol P : X Y. Jètoume Z = Ker(P ) kai jewroôme ton Y Z = (Y Z, 1) ìpou (y, z) 1 = y + z, gia kˆje (y, z) Y Z. (a) DeÐxte ìti o Y Z eðnai isìmorfoc me ton X. (b) DeÐxte ìti o X/Y eðnai isìmorfoc me ton Z kai o X/Z eðnai isìmorfoc me ton Y. 1.6 Stoiqei dhc jewrða q rwn Hilbert (a) Q roi Hilbert Orismìc. 'Estw X grammikìc q roc pˆnw apì to K. Miˆ sunˆrthsh, : X X K lègetai eswterikì ginìmeno an ikanopoieð ta ex c: (a) x, x 0 gia kˆje x X, me isìthta an kai mìno an x = 0. (b) x, y = y, x, gia kˆje x, y X. (g) gia kˆje y X h sunˆrthsh x x, y eðnai grammik. Prìtash 1.6.1. (anisìthta Cauchy-Schwarz) 'Estw X q roc me eswterikì ginìmeno. An x, y X, tìte x, y x, x y, y. Apìdeixh. Exetˆzoume pr ta thn perðptwsh K = C. 'Estw x, y X kai èstw M = x, y. Upˆrqei θ R ste x, y = Me iθ. Gia kˆje migadikì arijmì λ = re it èqoume 0 λx + y, λx + y = λ 2 x, x + λ x, y + λ x, y + y, y = λ 2 x, x + 2Re(λ x, y ) + y, y = r 2 x, x + 2Re(rMe i(θ+t) ) + y, y. Epilègoume to t ètsi ste e i(θ+t) = 1. Tìte, èqoume r 2 x, x 2rM + y, y 0 gia kˆje r > 0. PaÐrnontac r = y, y / x, x èqoume to zhtoômeno (h perðptwsh x = 0 y = 0 eðnai profan c). Sthn perðptwsh pou K = R, parathroôme ìti gia kˆje x, y X kai gia kˆje t R isqôei 0 tx + y, tx + y = t 2 x, x + 2t x, y + y, y. H diakrðnousa tou triwnômou wc proc t prèpei na eðnai mikrìterh Ðsh apì mhdèn. 'Ara, 4 x, y 2 4 x, x y, y 0. Autì dðnei to zhtoômeno. OrÐzoume : X R me x = x, x. H anisìthta Cauchy-Schwarz mˆc epitrèpei na deðxoume ìti h eðnai nìrma: Prìtash 1.6.2. 'Estw X q roc me eswterikì ginìmeno. H sunˆrthsh : X R, me x = x, x eðnai nìrma.

32 Basikec Ennoiec Apìdeixh. ArkeÐ na elègxoume thn trigwnik anisìthta (oi ˆllec idiìthtec eðnai aplèc). 'Omwc, x + y 2 = x + y, x + y = x 2 + x, y + y, x + y 2 = x 2 + y 2 + 2Re( x, y ) x 2 + y 2 + 2 x, y x 2 + y 2 + 2 x y = ( x + y ) 2, apì tic idiìthtec tou eswterikoô ginomènou kai thn anisìthta Cauchy-Schwarz. O (X, ) eðnai q roc me nìrma, kai èqoume deð ìti oi (x, y) x + y, (λ, x) λx eðnai suneqeðc wc proc thn. Apì thn anisìthta Cauchy-Schwarz èpetai eôkola ìti to eswterikì ginìmeno eðnai ki autì suneqèc wc proc thn : Prìtash 1.6.3. 'Estw X q roc me eswterikì ginìmeno kai èstw h epagìmenh nìrma. An x n x kai y n y wc proc thn, tìte Apìdeixh. Grˆfoume x n, y n x, y. x n, y n x, y = x n, y n y + x n x, y x n, y n y + x n x, y x n y n y + x n x y. H (x n ) sugklðnei ˆra eðnai fragmènh, kai y n y 0, x n x 0. 'Ara, x n, y n x, y. Eidikìtera, gia kˆje y X h apeikìnish x x, y eðnai fragmèno grammikì sunarthsoeidèc ston X. Orismìc. 'Enac q roc Banach lègetai q roc Hilbert an upˆrqei eswterikì ginìmeno, ston X ste x = x, x gia kˆje x X. Sth sunèqeia sumbolðzoume touc q rouc Hilbert me H. Kˆje q roc Hilbert ikanopoieð ton kanìna tou parallhlogrˆmmou: gia kˆje x, y H, x + y 2 + x y 2 = 2 x 2 + 2 y 2. AntÐstrofa, an h nìrma enìc q rou Banach X ikanopoieð ton kanìna tou parallhlogrˆmmou, tìte proèrqetai apì eswterikì ginìmeno to opoðo orðzetai apì thn sthn perðptwsh K = R, kai apì thn x, y = 1 4 { x + y 2 x y 2 } x, y = 1 4 ( x + y 2 x y 2 + i x + iy 2 i x iy 2 ) sthn perðptwsh K = C.

(b) Kajetìthta 1.6 Stoiqeiwdhc jewria qwrwn Hilbert 33 Orismìc. 'Estw X ènac q roc me eswterikì ginìmeno. Lème ìti ta x, y X eðnai orjog nia ( kˆjeta) kai grˆfoume x y, an x, y = 0. An x X kai M eðnai èna mh kenì uposônolo tou X, lème ìti to x eðnai kˆjeto sto M kai grˆfoume x M an x y gia kˆje y M. Parathr seic. 1. To 0 eðnai kˆjeto se kˆje x X, kai eðnai to monadikì stoiqeðo tou X pou èqei aut n thn idiìthta. 2. An x y, isqôei to Pujagìreio je rhma: x + y 2 = x 2 + y 2. Orismìc. 'Estw X ènac q roc me eswterikì ginìmeno kai èstw M grammikìc upìqwroc tou X. OrÐzoume M = {x X : y M, x, y = 0}. O M eðnai kleistìc grammikìc upìqwroc tou X (ˆskhsh). Prìtash 1.6.4. 'Estw H q roc Hilbert, M kleistìc grammikìc upìqwroc tou H, kai x H. Upˆrqei monadikì y 0 M ste x y 0 = d(x, M) = inf{ x y : y M}. To monadikì autì y 0 M sumbolðzetai me P M (x), kai onomˆzetai probol tou x ston M. Apìdeixh. Jètoume δ = d(x, M). Upˆrqei akoloujða (y n ) ston M ste x y n δ. Apì ton kanìna tou parallhlogrˆmmou, y n y m 2 = (y n x) + (x y m ) 2 = 2 y n x 2 + 2 y m x 2 (y n + y m ) 2x 2 2 = 2 y n x 2 + 2 y m x 2 4 y n + y m x 2. 'Omwc, y n+y m 2 M, ˆra y n+y m 2 x δ. Epomènwc, y n y m 2 2 y n x 2 + 2 y m x 2 4δ 2 2δ 2 + 2δ 2 4δ 2 = 0 ìtan m, n. 'Ara, h (y n ) eðnai akoloujða Cauchy ston H. O H eðnai pl rhc, ˆra upˆrqei y 0 H ste y n y 0. 'Epetai ìti y 0 M (o M eðnai kleistìc) kai x y 0 = lim n x y n = δ. Gia th monadikìthta, qrhsimopoioôme kai pˆli ton kanìna tou parallhlogrˆmmou. An x y = δ = x y, tìte 'Ara, y = y. 0 y y 2 = 2 x y 2 + 2 x y 2 4 y + y x 2 2δ 2 + 2δ 2 4δ 2 = 0. 2

34 Basikec Ennoiec Prìtash 1.6.5. Me tic upojèseic thc Prìtashc 1.6.4, x P M (x) M. Apìdeixh. Jètoume w = x P M (x). 'Estw ìti to w den eðnai kˆjeto ston M. Tìte, upˆrqei z M ste w, z > 0. Gia ε > 0 arketˆ mikrì, èqoume 2 w, z ε z 2 > 0. 'Ara, x (P M (x) + εz) 2 = w εz 2 = w εz, w εz to opoðo eðnai ˆtopo giatð P M (x) + εz M. = w 2 2ε w, z + ε z 2 = δ 2 ε(2 w, z ε z 2 ) < δ 2, Pìrisma 1.6.6. An H q roc Hilbert kai M kleistìc gn sioc upìqwroc tou H, tìte upˆrqei z H, z 0, ste z M. Apìdeixh. 'Estw x H\M. PaÐrnoume z = x P M (x) 0. Pìrisma 1.6.7. 'Enac grammikìc upìqwroc F tou H eðnai puknìc an kai mìno an to monadikì diˆnusma tou H pou eðnai kˆjeto ston F eðnai to 0. Apìdeixh. ( ) Upojètoume ìti o F eðnai puknìc ston H, kai ìti z, x = 0 gia kˆje x F. 'Estw y H. AfoÔ o F eðnai puknìc, upˆrqei akoloujða (y n ) F me y n y. Tìte, 0 = z, y n z, y. 'Ara, z, y = 0. AfoÔ z, y = 0 gia kˆje y H, èqoume z = 0. ( ) Ac upojèsoume ìti o F den eðnai puknìc ston H. Tìte, o F eðnai gn sioc kleistìc upìqwroc tou H. 'Ara, upˆrqei z 0, z F. Eidikìtera, z F, ˆtopo. Je rhma 1.6.8. 'Estw H q roc Hilbert, kai M kleistìc grammikìc upìqwroc tou H. Tìte, H = M M. Dhlad, kˆje x H grˆfetai monos manta sth morf x = x 1 + x 2, x 1 M, x 2 M. Apìdeixh. 'Estw x H. Grˆfoume x = P M (x)+(x P M (x)). Tìte, P M (x) M kai x P M (x) M. An x 1 + x 2 = x 1 + x 2 kai x 1, x 1 M, x 2, x 2 M, tìte to y = x 1 x 1 = x 2 x 2 M M giatð oi M, M eðnai upìqwroi, ˆra y y, to opoðo shmaðnei ìti y = 0. 'Ara, x 1 = x 1 kai x 2 = x 2, ap ìpou èpetai h monadikìthta. Pìrisma 1.6.9. 'Estw M {0} kleistìc grammikìc upìqwroc tou q rou Hilbert H. OrÐzoume P M : H H me P M (x) = P M (x 1 + x 2 ) = x 1, ìpou x = x 1 + x 2 ìpwc sto Je rhma. O P M eðnai fragmènoc grammikìc telest c, kai P M = 1.

1.6 Stoiqeiwdhc jewria qwrwn Hilbert 35 Apìdeixh. EÔkola elègqoume ìti o P M eðnai grammikìc telest c. EpÐshc, P M (x) 2 = x 1 2 x 1 2 + x 2 2 = x 1 + x 2 2 = x 2, dhlad o P M eðnai fragmènoc, kai P M 1. An x 0 M, x 0 0, tìte P M (x 0 ) = x 0. 'Ara, P M P M (x 0 ) x 0 = 1. 'Estw H {0} q roc Hilbert. Ja doôme ìti o H perièqei {pollˆ} sunarthsoeid, ta opoða anaparðstantai me polô sugkekrimèno trìpo apì ta stoiqeða tou H. L mma 1.6.10. Gia kˆje a H, h f a : H R me f a (x) = x, a an kei ston H, kai f a H = a H. Apìdeixh. 'Eqoume kai f a (λx + µy) = λx + µy, a = λ x, a + µ y, a = λf a (x) + µf a (y), f a (x) = x, a a x. 'Ara, f a H kai f a a. Tèloc, an a 0, f a f a(a) a An a = 0, profan c f a = 0 (f a 0). = a, a a = a. AntÐstrofa, kˆje f H anaparðstatai sth morf f = f a gia kˆpoio a H: Je rhma 1.6.11. (Je rhma anaparˆstashc tou Riesz) 'Estw H q roc Hilbert, kai f H. Upˆrqei monadikì a H ste f = f a. Apìdeixh. OrÐzoume M = Kerf = {x H : f(x) = 0}. O M eðnai kleistìc grammikìc upìqwroc tou H. An M = H, tìte f 0 kai f = f 0. An M H, tìte upˆrqei z 0, z H pou eðnai kˆjeto ston M. Tìte, gia kˆje y H èqoume f(f(z)y f(y)z) = f(z)f(y) f(y)f(z) = 0. 'Ara f(z)y f(y)z M, kai afoô z M paðrnoume f(z)y f(y)z, z = 0 = f(z) y, z = f(y) z, z = f(y) = y, f(z)z z 2 = fa (y), ìpou a = f(z)z/ z 2. H monadikìthta tou a eðnai apl. An f(y) = y, a = y, a gia kˆje y H, tìte a a y gia kˆje y H. 'Ara, a = a.

36 Basikec Ennoiec Pìrisma 1.6.12. 'Estw H q roc Hilbert. H apeikìnish T : H H me T (a) = f a eðnai antigrammik isometrða kai epð. ShmeÐwsh. Lègontac ìti h T eðnai antigrammik, ennooôme ìti T (λa + µa ) = λt (a) + µt (a ) gia kˆje a, a H kai gia kˆje λ, µ K. Apìdeixh. (a) Gia thn antigrammikìthta thc T, parathroôme ìti ˆra f λa+µa (x) = x, λa + µa = λ x, a + µ x, a = λf a (x) + µf a (x), T (λa + µa ) = f λa+µa = λf a + µf a = λt (a) + µt (a ). (b) Apì to L mma 1.6.10 èqoume T (a) = f a = a. Dhlad, h T eðnai isometrða. (g) An f H, upˆrqei a H ste T (a) = f a = f, apì to Je rhma anaparˆstashc tou Riesz. Dhlad, h T eðnai epð. Je rhma 1.6.13. 'Estw M kleistìc upìqwroc tou q rou Hilbert H kai èstw f M. Upˆrqei monadikì f H ste f M = f kai f H = f M. Apìdeixh. O M eðnai q roc Hilbert, ˆra to je rhma anaparˆstashc tou Riesz mac dðnei monadikì w M ste f(x) = x, w, x M. AfoÔ (profan c) w H, mporoôme na orðsoume f : H R me f(x) = x, w, x H. Tìte, to f eðnai fragmèno grammikì sunarthsoeidèc ston H, epekteðnei to f, kai f M = w = f H. Mènei na deðxoume th monadikìthta: èstw ìti kˆpoio g H ikanopoieð ta parapˆnw. Tìte, apì to je rhma anaparˆstashc tou Riesz ston H, upˆrqei u H ste g(x) = x, u, x H. 'Omwc tìte, x, w u = 0 gia kˆje x M, opìte w u = z M. Tìte, u 2 = w 2 + z 2 apì to Pujagìreio je rhma, kai afoô u = g = f = f = w, prèpei na èqoume z = 0, to opoðo dðnei z = 0 = w = u. 'Epetai ìti g = f. (g) Orjokanonikèc bˆseic 'Estw X q roc me eswterikì ginìmeno. Mia oikogèneia {e i : i I} X lègetai orjokanonik, an e i, e j = δ ij (1 an i = j kai 0 an i j). An {e i : i I} eðnai mia orjokanonik oikogèneia ston X, tìte to {e i : i I} eðnai grammikˆ

1.6 Stoiqeiwdhc jewria qwrwn Hilbert 37 anexˆrthto sônolo. Prˆgmati, an n k=1 λ ke ik = 0, tìte gia kˆje j = 1,..., n èqoume 0 = n λ k e ik, e ij = k=1 n λ k e ik, e ij = λ j. Orismìc. 'Estw H q roc Hilbert. Miˆ megistik (me thn ènnoia tou egkleismoô) orjokanonik oikogèneia lègetai orjokanonik bˆsh tou H. Parathr ste ìti an miˆ orjokanonik oikogèneia {e i : i I} eðnai orjokanonik bˆsh tou H tìte H = span{e i : i I}. Autì eðnai sunèpeia tou PorÐsmatoc 1.6.7. EpÐshc, qrhsimopoi ntac to L mma tou Zorn mporeðte eôkola na elègxete ìti kˆje q roc Hilbert èqei orjokanonik bˆsh. Prìtash 1.6.14. 'Estw H ènac apeirodiˆstatoc diaqwrðsimoc q roc Hilbert. Upˆrqei orjokanonik bˆsh {e n : n N} tou H kai gia kˆje x H èqoume x = k=1 x, e n e n. n=1 Apìdeixh. ParathroÔme pr ta ìti kˆje orjokanonik bˆsh {e i : i I} tou H eðnai arijm simo sônolo: prˆgmati, an e i e j eðnai stoiqeða thc bˆshc, tìte e i e j = 2. Thn Ðdia stigm, afoô o q roc eðnai diaqwrðsimoc den gðnetai na upˆrqoun uperarijm sima to pl joc shmeða tou pou na apèqoun anˆ dôo apìstash Ðsh me 2. JewroÔme loipìn mia orjokanonik bˆsh {e n : n N} tou H (h diˆtaxh twn stoiqeðwn thc bˆshc eðnai tuqoôsa). Isqurismìc. Gia kˆje x H kai kˆje n N, n d(x, span{e 1,..., e n }) = x x, e i e i. Prˆgmati, èstw λ 1,..., λ n K kai y = n λ ie i. ParathroÔme ìti 'Ara, n x 2 λ i e i = x 2 + n λ i x, e i 2 n x 2 λ i e i x 2 n x, e i 2. n x, e i 2 kai isìthta mporeð na isqôei mìno an λ i = x, e i, i = 1,..., n, dhlad an y = n x, e i e i. 'Estw x X, kai ε > 0. AfoÔ H = span{e n : n N}, upˆrqoun N N kai λ 1,..., λ N R ste N x λ n e n < ε. n=1

38 Basikec Ennoiec 'Omwc tìte, gia kˆje M > N èqoume M x x, e i e i = d(x, span{e 1,..., e M }) d(x, span{e 1,..., e N }) N x λ n e n < ε. AfoÔ to ε > 0 tan tuqìn, autì shmaðnei ìti M n=1 x, e n e n x kaj c to M, dhlad x = x, e n e n. n=1 Prìtash 1.6.15. 'Estw {e i : i N} orjokanonik akoloujða se ènan q ro Hilbert H. Tìte, (a) IsqÔei h anisìthta tou Bessel n=1 x, e i ) 2 x 2, x H. (b) An h {e i : i N} eðnai orjokanonik bˆsh, tìte isqôei h isìthta tou Parseval x, e i ) 2 = x 2, x H. (g) An h isìthta tou Parseval isqôei gia kˆje x H, tìte h {e i : i N} eðnai orjokanonik bˆsh tou H. (d) An span{e i : i N} = H, tìte h {e i : i N} eðnai orjokanonik bˆsh tou H. Apìdeixh. (a) Sthn apìdeixh thc Prìtashc 1.6.14 eðdame ìti gia kˆje n N 0 d 2 n (x, span{e 1,..., e n }) = x 2 x, e i e i = x 2 n x, e i 2. Af nontac to n paðrnoume thn anisìthta tou Bessel. (b) An h {e i : i N} eðnai orjokanonik bˆsh, tìte h Prìtash 1.6.14 deðqnei ìti lim d(x, span{e 1,..., e n }) = 0. Sunep c, n x 2 n x, e i 2 = d 2 (x, span{e 1,..., e n }) 0, ap ìpou paðrnoume thn isìthta tou Parseval. (g) An h {e i : i N} den eðnai orjokanonik bˆsh tou H, tìte span{e i : i N} = H. Sunep c, upˆrqei y 0 ston H me thn idiìthta y e i gia kˆje i N. 'Omwc tìte, apì thn isìthta tou Parseval paðrnoume y 2 = y, e i 2 = 0,

1.6 Stoiqeiwdhc jewria qwrwn Hilbert 39 to opoðo eðnai ˆtopo. (d) 'Estw ìti span{e i : i N} = H kai ìti h {e i : i N} den eðnai orjokanonik bˆsh tou H. Epilègoume mh mhdenikì y me thn idiìthta y e i, i N. Upˆrqoun z m span{e i : i N} ste z m y. Tìte, èqoume y, z m = 0 gia kˆje m, ˆra dhlad y = 0, to opoðo eðnai ˆtopo. y, y = lim m z m, y = 0, Je rhma 1.6.16. (Riesz-Fisher) Kˆje diaqwrðsimoc q roc Hilbert H eðnai isometrikˆ isìmorfoc me ton l 2. Apìdeixh. O H èqei orjokanonik bˆsh {e n : n N}. OrÐzoume T : H l 2 me T (x) = ( x, e 1,..., x, e n,...). (a) O T eðnai kalˆ orismènoc, giatð n x, e n 2 = x 2 < +, ˆra T (x) l 2. (b) H grammikìthta tou T elègqetai eôkola. (g) T (x) 2 l 2 = n x, e n 2 = x 2, ˆra o T eðnai isometrða (eidikìtera, eðnai èna proc èna). (d) 'Estw (a 1,..., a n,...) l 2. OrÐzoume x N = N n=1 a ne n. Tìte, an N > M èqoume x N x M 2 = N n=m+1 a 2 n 0 kaj c N, M, kai autì deðqnei ìti h (x N ) eðnai akoloujða Cauchy ston H. O H eðnai pl rhc, ˆra upˆrqei x H ste x N x. 'Eqoume x N, e m x, e m kaj c N, kai an N > m, 'Ara, x, e m = a m, m N. Tèloc, N x N, e m = a n e n, e m = a m. n=1 T (x) = ( x, e m ) m N = (a m ) m N, ˆra o T eðnai epð. Ask seic 1. 'Estw X q roc me eswterikì ginìmeno, kai èstw x, y X. DeÐxte ìti (a) x y an kai mìno an x + ay = x ay gia kˆje a K. (b) x y an kai mìno an x + ay x gia kˆje a K. 2. 'Estw H q roc Hilbert kai èstw x n, y n sth monadiaða mpˆla tou H me thn idiìthta x n, y n 1. DeÐxte ìti x n y n 0. 3. 'Estw H q roc Hilbert, kai x n, x H me tic idiìthtec: x n x, kai, gia kˆje y H, x n, y x, y. DeÐxte ìti x n x 0.

40 Basikec Ennoiec 4. 'Estw X q roc me eswterikì ginìmeno, kai x 1,..., x n X. DeÐxte ìti n x i x j 2 = n x i 2 n 2 x i. i j An x i x j 2 gia i j, deðxte ìti an mia mpˆla perièqei ìla ta x i, prèpei na èqei aktðna toulˆqiston 2(n 1)/n. 5. 'Estw X q roc me eswterikì ginìmeno, kai èstw x 1,..., x n X. DeÐxte ìti ε i =±1 n 2 n ε i x i = 2 n x i 2, ìpou to exwterikì ˆjroisma eðnai pˆnw apì ìlec tic akoloujðec (ε 1,..., ε n) { 1, 1} n. 6. 'Estw X q roc me eswterikì ginìmeno, kai èstw A, B mh kenˆ uposônola tou X, me A B. DeÐxte ìti (a) A A, (b) B A, (g) A = A. 7. 'Estw H q roc Hilbert kai èstw Y upìqwroc tou H. DeÐxte ìti o Y eðnai kleistìc an kai mìno an Y = Y. 8. 'Estw M, N kleistoð upìqwroi enìc q rou Hilbert. DeÐxte ìti (M + N) = M N, (M N) = M + N. 9. D ste parˆdeigma q rou Hilbert H kai grammikoô upìqwrou F tou H me thn idiìthta H F + F. 10. 'Estw H q roc Hilbert kai èstw W, Z kleistoð upìqwroi tou H me thn idiìthta: an w W kai z Z, tìte w z (oi W kai Z eðnai kˆjetoi). DeÐxte ìti o W + Z eðnai kleistìc upìqwroc tou H. 11. Sto q ro C[ 1, 1] jewroôme to eswterikì ginìmeno f, g = 1 f(t)g(t)dt. 1 (a) An F = {f C[ 1, 1] : 1 f(t)dt = 0}, breðte ton F. 1 (b) An G = {f C[ 1, 1] : 1 f(t)dt = 0}, breðte ton 0 G kai apodeðxte ìti G G. 12. Se ènan q ro Hilbert H dðnetai ènac grammikìc telest c T : H H me tic idiìthtec: T 2 = T, T 1. An F = KerT, deðxte ìti: (a) T (H) F. (b) O T eðnai h orjog nia probol ston F. 13. 'Estw H q roc Hilbert kai èstw T : H H fragmènoc grammikìc telest c, tou opoðou h eikìna eðnai monodiˆstath. DeÐxte ìti upˆrqoun u, v H ste T (x) = x, u v, x H. Upìdeixh: Upˆrqei v H ste T (x) = λ xv, x H. DeÐxte ìti h x λ x eðnai fragmèno grammikì sunarthsoeidèc, kai qrhsimopoi ste to je rhma anaparˆstashc tou Riesz. 14. 'Estw X q roc me eswterikì ginìmeno kai èstw {e k } orjokanonik akìloujða ston X. An x, y X, deðxte ìti x, e k y, e k x y. k=1

1.6 Stoiqeiwdhc jewria qwrwn Hilbert 41 15. 'Estw Y kleistìc upìqwroc tou diaqwrðsimou q rou Hilbert H kai èstw {e n : n N} orjokanonik bˆsh tou Y. DeÐxte ìti an x H, tìte to plhsièstero shmeðo tou Y proc to x eðnai to n=1 x, en en. 16. 'Estw H diaqwrðsimoc q roc Hilbert, (e m) orjokanonik bˆsh tou H, kai (x n) akoloujða stoiqeðwn tou H. DeÐxte ìti ta ex c eðnai isodônama: (a) Gia kˆje x H, x, x n 0 kaj c n. (b) H (x n ) eðnai fragmènh kai, gia kˆje m N, e m, x n 0 kaj c n. 17. 'Estw H q roc Hilbert kai èstw (x n ) orjog nia akoloujða ston H (dhlad, an n m, tìte x n x m.) DeÐxte ìti h n x n sugklðnei an kai mìno an h n x n 2 sugklðnei.

Kefˆlaio 2 Je rhma Hahn-Banach 2.1 Grammikˆ sunarthsoeid kai uperepðpeda 'Estw X grammikìc q roc pˆnw apì to K. Lème ìti ènac grammikìc upìqwroc W tou X èqei peperasmènh sundiˆstash k N an dim(x/w ) = k. An o X/W èqei sundiˆstash 1 kai x 0 X, tìte to x 0 + W lègetai uperepðpedo tou X. Parat rhsh 'Eqoume dim(x/w ) = k an kai mìno an upˆrqoun x 1,..., x k grammikˆ anexˆrthta dianôsmata ston X me thn ex c idiìthta: gia kˆje x X upˆrqoun monadikˆ w W kai a 1,..., a k K ste ( ) x = w + a 1 x 1 + + a k x k. Prˆgmati, an dim(x/w ) = k kai {[x 1 ],..., [x k ]} eðnai mða bˆsh tou X/W, tìte gia kˆje x X upˆrqoun a 1,..., a k me [x] = a 1 [x 1 ] + + a k [x k ] = [a 1 x 1 + + a k x k ], dhlad x (a 1 x 1 + + a k x k ) =: w W. EpÐshc, a 1 x 1 + + a k x k = 0 = a 1 [x 1 ] + + a k [x k ] = [0] = a 1 = = a k = 0, dhlad ta x 1,..., x k eðnai grammikˆ anexˆrthta. Apì thn anexarthsða twn [x i ] èpetai kai h monadikìthta thc parˆstashc tou x sthn ( ). Sumplhr ste tic leptomèreiec, kaj c kai thn apìdeixh thc antðstrofhc kateôjunshc. Eidikìtera, an dim(x/w ) kai x 0 eðnai tuqìn stoiqeðo tou X\W, tìte kˆje x X grˆfetai monos manta sth morf x = w + ax 0 ìpou w W kai a K. Oi upìqwroi sundiˆstashc 1 eðnai akrib c oi pur nec twn mh tetrimmènwn grammik n sunarthsoeid n: Prìtash 2.1.1. 'Estw X grammikìc q roc kai èstw W upìqwroc tou X. O W èqei sundiˆstash 1 an kai mìno an upˆrqei grammikì sunarthsoeidèc f : X K, f 0, me Kerf = W. Apìdeixh. 'Estw f : X K mh mhdenikì grammikì sunarthsoeidèc. O pur nac W = Kerf eðnai grammikìc upìqwroc tou X. OrÐzoume f : X/W K me f([x]) = f(x). To f eðnai kalˆ orismèno, èna proc èna kai epð, dhlad grammikìc isomorfismìc. 'Ara, dim(x/w ) = dim(k) = 1.

44 Jewrhma Hahn-Banach AntÐstrofa, an dim(x/w ) = 1, upˆrqei grammikìc isomorfismìc T : X/W K. OrÐzoume f : X K me f(x) = T (Q(x)), ìpou Q(x) = [x] = x + W h fusiologik apeikìnish. To f eðnai mh mhdenikì grammikì sunarthsoeidèc kai Kerf = W. Sth sunèqeia upojètoume ìti èqoume kai mia nìrma ston grammikì q ro X. Prìtash 2.1.2. 'Estw X q roc me nìrma kai èstw W upìqwroc tou X sundiˆstashc 1. Tìte, o W eðnai kleistìc upìqwroc tou X puknìc upìqwroc tou X. Apìdeixh. O W eðnai kleistìc grammikìc upìqwroc tou X. An upojèsoume ìti upˆrqei x 0 W \W, tìte oi arqikèc mac parathr seic deðqnoun ìti kˆje x X grˆfetai sth morf x = w + ax 0 gia kˆpoia w W kai a K. AfoÔ x 0 W kai W W, autì shmaðnei ìti X W. Dhlad, o W eðnai puknìc. Oi kleistoð upìqwroi sundiˆstashc 1 eðnai akrib c oi pur nec twn fragmènwn mh mhdenik n grammik n sunarthsoeid n. Gia thn apìdeixh ja qreiastoôme èna l mma. L mma 2.1.3. 'Estw X grammikìc q roc. DÔo grammikˆ sunarthsoeid f, g : X K èqoun ton Ðdio pur na an kai mìno an upˆrqei b K\{0} tètoio ste g = bf. Apìdeixh. An g = bf, b 0, tìte profan c Kerf = Kerg. AntÐstrofa, èstw f, g : X K grammikˆ sunarthsoeid me Kerf = Kerg. An f 0, tìte g 0 kai g = 1 f. Upojètoume loipìn ìti f 0, opìte upˆrqei x 0 X me f(x 0 ) = 1. AfoÔ f(x 0 ) 0, èqoume g(x 0 ) 0. Gia kˆje x X, f(x f(x)x 0 ) = f(x) f(x)f(x 0 ) = 0 = g(x f(x)x 0 ) = 0 = g(x) = g(x 0 )f(x). Dhlad, g = bf ìpou b = g(x 0 ). Prìtash 2.1.4. 'Estw X q roc me nìrma kai f : X K mh mhdenikì grammikì sunarthsoeidèc. Tìte, f X an kai mìno an o Kerf eðnai kleistìc. Apìdeixh. An to f eðnai suneqèc, tìte o Kerf eðnai profan c kleistìc. Gia thn antðstrofh kateôjunsh, ac upojèsoume ìti f : X K eðnai èna mh mhdenikì grammikì sunarthsoeidèc kai ìti o Kerf eðnai kleistìc. JewroÔme ton q ro phlðko X/Kerf kai thn fusiologik apeikìnish Q : X X/Kerf. Apì thn Prìtash 2.1.1, o X/Kerf èqei diˆstash 1, ˆra upˆrqei grammikìc isomorfismìc T : X/Kerf K. Oi Q kai T eðnai suneqeðc, ˆra h g := T Q : X K eðnai fragmèno grammikì sunarthsoeidèc. ParathroÔme ìti g(x) = 0 an kai mìno an Q(x) = [x] = 0, dhlad an kai mìno an x Kerf. AfoÔ Kerg = Kerf, to L mma 2.1.3 deðqnei ìti f = bg gia kˆpoio b 0, b K. H sunèqeia tou f èpetai t ra apì th sunèqeia tou g.

2.2 To Lhmma tou Zorn 45 Ask seic 1. 'Estw X grammikìc q roc kai èstw W upìqwroc tou X. DeÐxte ìti dim(x/w ) = k an kai mìno an upˆrqoun x 1,..., x k grammikˆ anexˆrthta dianôsmata ston X me thn idiìthta: gia kˆje x X upˆrqoun monadikˆ w W kai a 1,..., a k K tètoia ste x = w + a 1x 1 + + a k x k. 2. 'Estw X grammikìc q roc kai èstw H X. DeÐxte ìti to H eðnai uperepðpedo an kai mìno an upˆrqoun t K kai mh mhdenikì grammikì sunarthsoeidèc f : X K tètoia ste H = {x X : f(x) = t}. 2.2 To L mma tou Zorn (a) 'Estw P èna mh kenì sônolo. MÐa sqèsh sto P lègetai merik diˆtaxh an ikanopoieð ta ex c: (1) gia kˆje a P, a a (anaklastik idiìthta) (2) an a b kai b a, tìte a = b (antisummetrik idiìthta) (3) an a b kai b c, tìte a c (metabatik idiìthta). To P lègetai merikˆ diatetagmèno wc proc thn. Apì ton orismì faðnetai ìti mporeð sto P na upˆrqoun a kai b gia ta opoða na mhn isqôei kamða apì tic a b kai b a (tìte, lème ìti ta a kai b den sugkrðnontai.) Ta a kai b sugkrðnontai an isqôei toulˆqiston mða apì tic a b b a. (b) 'Ena mh kenì uposônolo C tou P lègetai olikˆ diatetagmèno ( alusðda) an opoiad pote dôo stoiqeða tou sugkrðnontai. (g) An A, A P kai b P, lème ìti to b eðnai ˆnw frˆgma gia to A an: gia kˆje a A isqôei a b. 'Ena uposônolo A tou P mporeð na èqei na mhn èqei ˆnw frˆgma. (d) To m P lègetai megistikì stoiqeðo tou P an: gia kˆje a P me m a isqôei m = a. Dhlad, an den upˆrqei stoiqeðo tou P gn sia megalôtero apì to m. 'Ena merikˆ diatetagmèno sônolo P mporeð na èqei na mhn èqei megistikˆ stoiqeða. ParadeÐgmata (1) To sônolo R twn pragmatik n arijm n me th sun jh diˆtaxh eðnai èna olikˆ diatetagmèno sônolo pou den perièqei megistikˆ stoiqeða. (2) 'Estw X kai M = P(X) to sônolo ìlwn twn uposunìlwn tou X (to dunamosônolo tou X). OrÐzoume sto M wc ex c: A B A B. H eðnai merik diˆtaxh sto M (kai an to X èqei perissìtera apì èna stoiqeða, tìte upˆrqoun A, B M pou den sugkrðnontai: pˆrte p.q. A mh kenì, gn sio uposônolo tou X, kai B = X\A.) To M èqei èna (akrib c) megistikì stoiqeðo, to X. (3) JewroÔme to sônolo M = R n, n 2, twn diatetagmènwn n-ˆdwn pragmatik n arijm n. An x = (ξ 1,..., ξ n ), y = (η 1,..., η n ), lème ìti x y an ξ i η i

46 Jewrhma Hahn-Banach gia kˆje i = 1,..., n. To M eðnai merikˆ diatetagmèno wc proc thn, kai den perièqei megistikˆ stoiqeða. (4) JewroÔme to sônolo M = N twn fusik n arijm n, kai lème ìti m n an o m diaireð ton n. H eðnai merik diˆtaxh sto N (ta stoiqeða 3 kai 7 tou N den sugkrðnontai.) To A = {2 k : k = 0, 1, 2,...} eðnai olikˆ diatetagmèno uposônolo tou N. To N den perièqei megistikˆ stoiqeða wc proc thn. An jewr soume to sônolo P = {2, 3, 5, 7, 11,...} twn pr twn arijm n me thn Ðdia diˆtaxh, tìte kˆje stoiqeðo tou P eðnai megistikì. Pˆli me thn, to {2, 3, 4, 8} èqei dôo megistikˆ stoiqeða: to 3 kai to 8 (elègxte touc parapˆnw isqurismoôc.) L mma tou Zorn 'Estw P èna merikˆ diatetagmèno sônolo wc proc thn. Upojètoume ìti kˆje alusðda C P èqei ˆnw frˆgma sto P. Tìte, to P èqei toulˆqiston èna megistikì stoiqeðo. To L mma tou Zorn eðnai isodônamo me to axðwma thc epilog c kai ja to deqtoôme san axðwma. Ask seic 1. DeÐxte ìti kˆje grammikìc q roc X {0} èqei bˆsh. [Upìdeixh: Jewr ste to sônolo P ìlwn twn grammikˆ anexˆrthtwn uposunìlwn tou X me diˆtaxh thn A B A B. DeÐxte ìti to (P, ) ikanopoieð tic upojèseic tou L mmatoc tou Zorn.] 2. DeÐxte ìti kˆje q roc Hilbert èqei orjokanonik bˆsh. 2.3 To Je rhma Hahn - Banach Orismìc. 'Estw X grammikìc q roc pˆnw apì to R kai èstw p : X R. To p lègetai upogrammikì sunarthsoeidèc, an ikanopoieð ta ex c: (a) p(x + y) p(x) + p(y) gia kˆje x, y X, (b) p(λx) = λp(x) gia kˆje λ 0 kai kˆje x X, dhlad, an eðnai upoprosjetikì kai jetikˆ omogenèc. Parathr ste ìti: den a- paitoôme to p na paðrnei mh arnhtikèc timèc, oôte thn p(λx) = λ p(x), λ R. EÔkola elègqontai oi p(0) = 0 kai p( x) p(x). ParadeÐgmata (a) An f : X R eðnai grammikì sunarthsoeidèc, tìte ta f, f eðnai upogrammikˆ sunarthsoeid. (b) Kˆje nìrma : X R eðnai upogrammikì sunarthsoeidèc. (g) H p : l (R) R me p((ξ k )) = limsupξ k eðnai upogrammikì sunarthsoeidèc ston l. Je rhma 2.3.1 (Je rhma epèktashc tou Hahn). 'Estw X grammikìc q roc pˆnw apì to R kai èstw p : X R upogrammikì sunarthsoeidèc. 'Estw W grammikìc upìqwroc tou X, kai f : W R grammikì sunarthsoeidèc me thn idiìthta: gia kˆje x W, ( ) f(x) p(x).

Tìte, upˆrqei grammikì sunarthsoeidèc f : X R ste (a) f(x) = f(x) an x W (to f eðnai epèktash tou f), (b) f(x) p(x) gia kˆje x X. 2.3 To Jewrhma Hahn - Banach 47 Parathr ste ìti den upojètoume kamða topologik dom gia to q ro X (eðnai apl c ènac grammikìc q roc.) 'Opwc ja doôme, to ousiastikì b ma thc apìdeixhc eðnai p c ja epekteðnoume to f apì ènan upìqwro W 1 se ènan upìqwro W 2 pou èqei {mða diˆstash parapˆnw}, me grammikì trìpo kai qwrðc na qalˆsei h ( ). Apì th stigm pou autì eðnai dunatì, to L mma tou Zorn mˆc exasfalðzei miˆ {megistik epèktash} f, ki aut upoqreoôtai na èqei pedðo orismoô olìklhron ton X. L mma 2.3.2. Me tic upojèseic tou Jewr matoc, ac upojèsoume epiplèon ìti gia kˆpoion grammikì upìqwro W 1 tou X o opoðoc perièqei ton W, èqoume breð f 1 : W 1 R ste f 1 W = f kai f 1 (x) p(x) gia kˆje x W 1. 'Estw y X\W 1, kai W 2 = span{w 1, y}. Tìte, upˆrqei grammikì sunarthsoeidèc f 2 : W 2 R ste f 2 W1 = f 1 kai f 2 (x) p(x) gia kˆje x W 2. Apìdeixh. Kˆje z W 2 grˆfetai monos manta sth morf z = x + λy gia kˆpoia x W 1 kai λ R. H grammik epèktash f 2 pou zhtˆme prosdiorðzetai loipìn monos manta apì thn tim a R pou ja epilèxoume san f 2 (y). An jèsoume f 2 (y) = a, tìte prèpei na èqoume (1) f 2 (z) = f 2 (x) + λf 2 (y) = f 1 (x) + λa, afoô zhtˆme to f 2 na eðnai grammikì kai na epekteðnei to f 1. H ˆllh idiìthta pou zhtˆme apì to a eðnai h ex c: Gia kˆje x W 1 kai kˆje λ R, (2) f 1 (x) + λa p(x + λy). IsodÔnama, paðrnontac up ìyin tic (1) kai (2), zhtˆme a R me thn idiìthta: gia kˆje x W 1 kai kˆje λ > 0, (3) f 1 (x) + λa p(x + λy), f 1 (x) λa p(x λy). Epeid to p eðnai jetikˆ omogenèc kai to f 1 grammikì ston W 1, h (3) eðnai isodônamh me to ex c: gia kˆje x W 1 kai kˆje λ > 0, (4) f 1 ( x λ ) ( x ) + a p λ + y, f 1 ( x λ ) ( x ) a p λ y, kai epeid o W 1 eðnai upìqwroc, isodônama zhtˆme a R tètoio ste: gia kˆje x, x W 1, f 1 (x ) p(x y) a p(x + y) f 1 (x). Miˆ tètoia epilog tou a eðnai dunat an kai mìno an gia kˆje x, x W 1, f 1 (x) + f 1 (x ) p(x y) + p(x + y).

48 Jewrhma Hahn-Banach 'Omwc, f 1 (x) + f 1 (x ) = f 1 (x + x ) p(x + x ) = p((x y) + (x + y)) p(x y) + p(x + y), apì thn upogrammikìthta tou p, thn grammikìthta tou f 1 ston W 1, thn f 1 p ston W 1 kai to gegonìc ìti to p orðzetai se olìklhro ton X. Autì apodeiknôei to L mma. Apìdeixh tou Jewr matoc 2.3.1. 'Estw W h oikogèneia ìlwn twn zeugari n (W 1, f 1 ) pou ikanopoioôn ta ex c: (a) o W 1 eðnai grammikìc upìqwroc tou X, kai W W 1. (b) to f 1 : W 1 R eðnai grammikì kai f 1 W = f. (g) f 1 (x) p(x) gia kˆje x W 1. H W eðnai mh ken, afoô (W, f) W. OrÐzoume diˆtaxh sthn W jètontac (W 1, f 1 ) (W 2, f 2 ) an kai mìno an W 1 W 2 kai f 2 W1 = f 1. (1) To (W, ) eðnai merikˆ diatetagmèno sônolo. Ja deðxoume ìti ikanopoieð thn upìjesh tou L mmatoc tou Zorn: 'Estw C = {(W i, f i ) : i I} miˆ alusðda sto (W, ). OrÐzoume W = i I W i kai f : W R me f (x) = f i (x), x W i. ApodeiknÔoume eôkola ìti: (a) O W eðnai grammikìc upìqwroc tou X, kai W W i W gia kˆje i I (ja qreiasteðte to gegonìc ìti an i, j I tìte, eðte W i W j eðte W j W i, afoô h C eðnai alusðda.) (b) H f orðzetai kalˆ, eðnai grammik, kai f (x) p(x) gia kˆje x W (ki ed ja qreiasteðte to gegonìc ìti an i, j I tìte, eðte f j Wi = f i eðte f i Wj = f j afoô h C eðnai alusðda.) (g) Gia kˆje i I, f Wi = f i. 'Epetai ìti (W, f ) W, kai to (W, f ) eðnai ˆnw frˆgma thc C. (2) Apì to L mma tou Zorn, to (W, ) èqei megistikì stoiqeðo (W 0, f 0 ). Apì to L mma 2.3.2 blèpoume ìti W 0 = X: An ìqi, ja paðrname y X\W 0, kai orðzontac W 0 = span{w 0, y} ja epekteðname to f 0 se f 0 : W 0 R, opìte to (W 0, f 0) ja tan gn sia megalôtero apì to (W 0, f 0 ), ˆtopo. H f = f 0 : X R eðnai h zhtoômenh epèktash thc f ston X. Orismìc. 'Estw X grammikìc q roc pˆnw apì to K. H p : X R lègetai hminìrma an (a) p(x + y) p(x) + p(y) gia kˆje x, y X, (b) p(λx) = λ p(x) gia kˆje λ K kai kˆje x X. Parathr ste ìti kˆje hminìrma ikanopoieð ta ex c: (g) p(0) = 0 kai p( x) = p(x) gia kˆje x X. 'Epetai ìti p(x) 0 gia kˆje x X. (d) p(x) p(y) p(x y) gia kˆje x, y X. (e) To sônolo {x X : p(x) = 0} eðnai grammikìc upìqwroc tou X.

2.3 To Jewrhma Hahn - Banach 49 To Je rhma epèktashc tou Hahn paðrnei thn ex c morf an to p eðnai hminìrma. Je rhma 2.3.3. 'Estw X grammikìc q roc pˆnw apì to K kai èstw p : X R hminìrma. Upojètoume ìti W eðnai ènac grammikìc upìqwroc tou X, f : W K èna grammikì sunarthsoeidèc, kai f(x) p(x) gia kˆje x W. Tìte, upˆrqei grammikì sunarthsoeidèc f : X K ste f W = f kai f(x) p(x) gia kˆje x X. Apìdeixh. Exetˆzoume pr ta thn perðptwsh K = R. JewroÔme thn epèktash f pou mˆc exasfalðzei to Je rhma 2.3.1. Gia kˆje x X èqoume f(x) p(x), kai apì thn grammikìthta tou f kai thn p( x) = p(x) blèpoume ìti f(x) = f( x) p( x) = p(x). 'Ara, f(x) p(x) gia kˆje x X. Gia thn apìdeixh sth migadik perðptwsh, ja qreiastoôme kˆpoiec aplèc parathr seic. An X eðnai ènac grammikìc q roc pˆnw apì to C kai f : X C eðnai èna grammikì sunarthsoeidèc, tìte mporoôme na grˆyoume to f sth morf f = Ref + iimf, ìpou Ref, Imf : X R eðnai prosjetikˆ kai R-omogen sunarthsoeid (dhlad, an a R, tìte Ref(ax) = aref(x)). EpÐshc, to f prosdiorðzetai pl rwc apì to Ref. Prˆgmati, gia kˆje x X èqoume kai ˆra f(ix) = Ref(ix) + iimf(ix) f(ix) = if(x) = Imf(x) + iref(x), f(x) = Ref(x) + iimf(x) = Ref(x) iref(ix). AntÐstrofa, an h : X R eðnai èna prosjetikì kai R-omogenèc sunarthsoeidèc, tìte to g : X C me g(x) = h(x) ih(ix) eðnai grammikì sunarthsoeidèc kai Reg = h. Apìdeixh tou Jewr matoc 2.3.3 sthn perðptwsh K = C: Jètoume h = Ref. Tìte, to h : W R eðnai grammikì sunarthsoeidèc an doôme ton X san grammikì q ro pˆnw apì to R, kai h(x) = Ref(x) f(x) p(x) gia kˆje x W. Apì thn pragmatik perðptwsh tou Jewr matoc, upˆrqei h : X R prosjetikì kai R-omogenèc, me thn idiìthta h W = h kai h(x) p(x) gia kˆje x X. OrÐzoume f : X C me f(x) = h(x) i h(ix). Tìte, to f eðnai grammikì sunarthsoeidèc kai eôkola elègqoume ìti f W = f (giatð Re( f) = h kai h W = h = Ref).

50 Jewrhma Hahn-Banach Mènei na deðxoume ìti f(x) p(x) gia kˆje x X. Ac upojèsoume ìti f(x) = Re iθ. Apì th grammikìthta tou f èqoume f(e iθ x) = R R, ˆra h(e iθ x) = f(e iθ x). Tìte, f(x) = R = h(e iθ x) p(e iθ x) = e iθ p(x) = p(x), kai h apìdeixh eðnai pl rhc. Sto plaðsio twn q rwn me nìrma èqoume thn ex c {analutik morf } tou Jewr matoc Hahn-Banach. Je rhma 2.3.4 (Banach). 'Estw X q roc me nìrma, Y upìqwroc tou X, kai f : Y K fragmèno grammikì sunarthsoeidèc. Tìte, upˆrqei f : X R fragmèno grammikì sunarthsoeidèc me f Y = f kai f X = f Y. Dhlad, upˆrqei suneq c epèktash tou f ston X, me diat rhsh thc nìrmac. Apìdeixh. 'Eqoume f(x) f Y x gia kˆje x Y. OrÐzoume p : X R me p(x) = f Y x. To p eðnai hminìrma: (a) Gia thn upoprosjetikìthta, parathroôme ìti p(x + y) = f Y x + y f Y ( x + y ) = f Y x + f Y y = p(x) + p(y). (b) To p eðnai omogenèc: an λ K, tìte p(λx) = f Y λx = λ f Y x = λ p(x). Apì to Je rhma 2.3.3, upˆrqei grammik epèktash f : X K thc f, ste f(x) p(x) = f Y x gia kˆje x X. 'Ara, f X kai f X f Y. Apì thn ˆllh pleurˆ, afoô f Y = f, paðrnoume f X = sup x S X f(x) sup x S Y f(x) = f Y. Dhlad, f X = f Y. H analutik morf tou Jewr matoc Hahn-Banach mac epitrèpei na deðqnoume thn Ôparxh fragmènwn sunarthsoeid n me sugkekrimènec idiìthtec. Eidikìtera, deðqnei ìti o duðkìc q roc enìc mh tetrimmènou q rou me nìrma perièqei pollˆ sunarthsoeid. Qarakthristikˆ apotelèsmata autoô tou eðdouc eðnai ta ex c. Je rhma 2.3.5. 'Estw X {0} q roc me nìrma, kai x 0 X, x 0 0. Upˆrqei fragmèno grammikì sunarthsoeidèc f : X R ste f = 1 kai f(x 0 ) = x 0.

2.3 To Jewrhma Hahn - Banach 51 Apìdeixh. JewroÔme ton upìqwro W = span{x 0 } pou parˆgetai apì to x 0 kai orðzoume f : W R me f(λx 0 ) = λ x 0. To f eðnai grammikì, kai f W f(λx 0 ) = sup λ 0 λx 0 = 1. Apì to Je rhma 2.3.4, upˆrqei f : X R fragmèno grammikì sunarthsoeidèc, me f = f W = 1 kai f(x 0 ) = f(x 0 ) = x 0. Pìrisma 2.3.6. 'Estw X q roc me nìrma kai èstw x X. Tìte, x = max f =1 f(x). Apìdeixh. Apì to Je rhma 2.3.5, upˆrqei f X, f = 1 me f(x) = x. 'Ara, (1) sup f(x) f(x) = x. f =1 Apì thn ˆllh pleurˆ, an f = 1 tìte f(x) f x = x. 'Ara, (2) sup f(x) x. f =1 Apì tic (1) kai (2), x = sup f =1 f(x). To sup eðnai max lìgw tou f. Parat rhsh. GnwrÐzoume dh ìti f = sup x =1 f(x) gia kˆje f X (autì tan sunèpeia tou orismoô thc nìrmac telest.) To Je rhma Hahn-Banach, sth morf tou PorÐsmatoc 2.3.6, mac dðnei th duðk sqèsh x = max f =1 f(x). Dhlad, h nìrma tou x {piˆnetai} san tim kˆpoiou f apì th monadiaða sfaðra tou duðkoô q rou. Pìrisma 2.3.7. An f(x) = f(y) gia kˆje f X, tìte x = y. Apìdeixh. 'Eqoume x y = sup f =1 f(x y) = sup f =1 f(x) f(y) = 0, ˆra x = y. ShmeÐwsh. To Pìrisma 2.3.7 deðqnei ìti o X diaqwrðzei ta shmeða tou X: an x y, tìte upˆrqei f X tètoio ste f(x) f(y). To Je rhma pou akoloujeð deðqnei ìti o X diaqwrðzei shmeða apì kleistoôc upoq rouc: Je rhma 2.3.8. 'Estw X q roc me nìrma, Y gn sioc kleistìc upìqwroc tou X, kai x 0 X\Y. An δ = d(x 0, Y ) = inf{ x 0 y : y Y }, tìte upˆrqei f X ste f X = 1/δ, f(y) = 0 gia kˆje y Y, kai f(x 0 ) = 1. Apìdeixh. JewroÔme thn fusiologik apeikìnish Q : X X/Y. ParathroÔme ìti [x 0 ] 0 = inf{ x 0 y : y Y } = d(x 0, Y ) = δ.

52 Jewrhma Hahn-Banach Apì to Je rhma 2.3.5 upˆrqei g (X/Y ) me g = 1 kai g([x 0 ]) = δ. OrÐzoume f : X K me f(x) = 1 (g Q)(x). δ To f eðnai fragmèno grammikì sunarthsoeidèc kai f(y) = g([0])/δ = 0 an y Y. Apì ton orismì tou g, EpÐshc, f(x 0 ) = g([x 0]) δ = 1. f(x) = 1 g Q x g(q(x)) δ δ x δ, ˆra f 1/δ. Mènei na deðxoume ìti f 1/δ. ParathroÔme ìti f(x 0 y) = f(x 0 ) = 1 gia kˆje y Y. 'Ara, f sup y Y = f(x 0 y) x 0 y = 1 inf y Y x 0 y 1 [x 0 ] 0 = 1 δ. Autì oloklhr nei thn apìdeixh. KleÐnoume aut thn parˆgrafo me mia efarmog tou Jewr matoc 2.3.8. Je rhma 2.3.9. An o X eðnai diaqwrðsimoc, tìte o X eðnai diaqwrðsimoc. Apìdeixh. Ja mˆc qreiasteð èna L mma: L mma 2.3.10. H S X = {f X : f = 1} eðnai diaqwrðsimh wc proc thn epagìmenh metrik. Apìdeixh. Upˆrqei M arijm simo puknì uposônolo tou X. OrÐzoume M 1 = {g = f/ f : f M\{0}}. To M 1 eðnai arijm simo uposônolo thc S X, kai ja deðxoume ìti eðnai puknì sthn S X. 'Estw f S X. Upˆrqei akoloujða {h k } sto M me h k f. 'Ara, h k f = 1, dhlad, telikˆ h k 0. Oi l k = h k / h k an koun sto M 1, kai f l k = = f h k h k ( f h k + h k 1 1 h k ) f h k + h k 1 1 h k 0. 'Ara, M 1 = S X. Apìdeixh tou Jewr matoc. JewroÔme {g n : n N} arijm simo puknì uposônolo thc S X. Gia kˆje n N èqoume g n = sup x =1 g n (x) = 1, ˆra upˆrqei x n X, x = 1, ste g n (x n ) > 1 2.

2.3 To Jewrhma Hahn - Banach 53 OrÐzoume Y = span{x n : n N}. 'Estw ìti Y X. Tìte, apì to Je rhma 2.3.8, upˆrqei f X ste f = 1 kai f(y) = 0 gia kˆje y Y. Eidikìtera, f(x n ) = 0, n N. Tìte, gia kˆje n N, f g n ( f g n )(x n ) = f(x n ) g n (x n ) = g n (x n ) > 1 2, ˆtopo, giatð to {g n : n N} eðnai puknì sthn S X. 'Ara, Y = X. 'Omwc, o Y eðnai diaqwrðsimoc: oi peperasmènoi grammikoð sunduasmoð twn x n me rhtoôc suntelestèc eðnai puknoð ston Y. 'Ara, o X eðnai diaqwrðsimoc. O q roc L p [0, 1], 0 < p < 1. 'Estw 0 < p < 1. Me L p [0, 1] sumbolðzoume to grammikì q ro twn klˆsewn isodunamðac (tautðzoume dôo sunart seic an eðnai sqedìn pantoô Ðsec) twn Lebesgue metr simwn sunart sewn gia tic opoðec An f, g L p [0, 1], orðzoume [0,1] d(f, g) = f p dλ < +. [0,1] f g p dλ. H d eðnai metrik. H trigwnik anisìthta eðnai sunèpeia thc anisìthtac (a + b) p a p + b p, a, b 0, 0 < p < 1. EÔkola elègqoume ìti h d eðnai analloðwth wc proc metaforèc kai ìti oi prˆxeic tou grammikoô q rou eðnai suneqeðc wc proc thn d. EpÐshc, mimoômenoi thn a- pìdeixh thc plhrìthtac tou L p (µ), p 1, mporeðte na deðxete ìti o (L p [0, 1], d), 0 < p < 1 eðnai pl rhc. Ja exetˆsoume thn Ôparxh suneq n grammik n sunarthsoeid n F : L p [0, 1] R. L mma 2.3.11. 'Estw 0 < p < 1 kai èstw G mh kenì, anoiktì kai kurtì uposônolo tou L p [0, 1]. Tìte, G = L p [0, 1]. Apìdeixh. AfoÔ h metrik d eðnai analloðwth wc proc metaforèc, mporoôme epiplèon na upojèsoume ìti 0 G. AfoÔ to G eðnai anoiktì kai 0 G, upˆrqei R > 0 me thn idiìthta [0,1] f p dλ < R = f G. 'Estw g L p [0, 1]. Gia kˆje n N brðskoume diamèrish 0 = x 0 < x 1 <... < x n = 1 tètoia ste [x i,x i+1 ] g p dλ = 1 n [0,1] g p dλ, i = 0,..., n 1.

54 Jewrhma Hahn-Banach JewroÔme tic sunart seic Tìte, g = (g 1 +... + g n )/n kai [0,1] g i = ngχ [xi,x i+1 ], i = 0,..., n 1. g i p dλ = n p [x i,x i+1] g p dλ = 1 n 1 p g p dλ. [0,1] An to n eðnai arketˆ megˆlo petuqaðnoume g i G gia kˆje i: arkeð na epilèxoume n gia to opoðo [0,1] AfoÔ to G eðnai kurtì, blèpoume ìti g p dλ n 1 p R. g = g 1 +... + g n n kai afoô h g tan tuqoôsa, G = L p [0, 1]. G, Prìtash 2.3.12. An 0 < p < 1 kai F : L p [0, 1] R suneqèc grammikì sunarthsoeidèc, tìte F 0. Apìdeixh. Gia kˆje ε > 0, to sônolo G ε = {f L p [0, 1] : F (f) < ε} eðnai anoiktì, kurtì kai mh kenì giatð F (0) = 0. Apì to L mma, G ε = L p [0, 1]. Dhlad, f L p [0, 1] ε > 0, F (f) < ε. Af nontac to ε 0 + blèpoume ìti F (f) = 0 gia kˆje f L p [0, 1]. Dhlad, F 0. Ask seic 1. 'Estw X q roc me nìrma, x 1,..., x m grammikˆ anexˆrthta dianôsmata ston X, kai a 1,..., a m K. Upˆrqei f X ste f(x i) = a i, i = 1,..., m. 2. 'Estw X q roc me nìrma kai Y grammikìc upìqwroc tou X. DeÐxte ìti Y = {Kerf : f X, Y Kerf}. 3. 'Estw X grammikìc q roc pˆnw apì to K kai p 1, p 2 : X R hminìrmec. An f : X K eðnai èna grammikì sunarthsoeidèc me thn idiìthta f(x) p 1(x) + p 2(x) gia kˆje x X, deðxte ìti upˆrqoun grammikˆ sunarthsoeid f 1, f 2 : X K ste f = f 1 + f 2 kai f i(x) p i(x), i = 1, 2, x X. [Upìdeixh: BreÐte grammikì sunarthsoeidèc ψ : X X K me ψ(x 1, x 2 ) p 1 (x 1 ) + p 2(x 2) kai ψ(x, x) = f(x).]

2.4 Diaqwristika jewrhmata 55 4. 'Estw X kai Y q roi me nìrma. An o B(X, Y ) eðnai pl rhc kai X {0}, deðxte ìti o Y eðnai pl rhc. [Upìdeixh: Pˆrte x 0 S X kai f S X me f(x 0) = 1. Tìte, gia kˆje y Y o telest c T : X Y me T y(x) = f(x)y eðnai fragmènoc kai T y = y.] 5. 'Estw (X, ) q roc me nìrma kai èstw W grammikìc upìqwroc tou X. Upojètoume ìti eðnai mða ˆllh nìrma ston W pou eðnai isodônamh me ton periorismì thc ston W. DeÐxte ìti upˆrqei nìrma ston X pou eðnai isodônamh me thn ston X kai o periorismìc thc ston W eðnai h. 6. JewroÔme ton q ro l (R) twn fragmènwn pragmatik n akolouji n. DeÐxte ìti upˆrqei fragmèno grammikì sunarthsoeidèc f : l (R) R me thn ex c idiìthta: gia kˆje x = (x n ) l (R), lim inf n x n f(x) lim sup x n. n 7. 'Estw X q roc me nìrma kai èstw Y kleistìc upìqwroc tou X. O mhdenist c tou Y eðnai to N(Y ) = {f X : y Y f(y) = 0}. (a) DeÐxte ìti o N(Y ) eðnai kleistìc upìqwroc tou X kai o X /N(Y ) eðnai isometrikˆ isìmorfoc me ton Y. H isometrða eðnai o T : X /N(Y ) Y me T (f + N(Y )) = f Y. (b) DeÐxte ìti o (X/Y ) eðnai isometrikˆ isìmorfoc me ton N(Y ). H isometrða eðnai o S : (X/Y ) N(Y ) me S(g) = g Q, ìpou Q : X X/Y h fusiologik apeikìnish. 8. 'Estw X q roc me nìrma kai èstw Y kleistìc upìqwroc tou X peperasmènhc sundiˆstashc. An f : X K eðnai èna grammikì sunarthsoeidèc kai f Y Y, deðxte ìti f X. 2.4 Diaqwristikˆ jewr mata Orismìc. Se aut thn parˆgrafo upojètoume ìti X eðnai ènac q roc me nìrma pˆnw apì to R kai A, B eðnai mh kenˆ uposônola tou X. (a) Lème ìti ta A, B diaqwrðzontai, an upˆrqoun mh mhdenikì f X kai λ R ste: f(a) λ gia kˆje a A kai f(b) λ gia kˆje b B. (b) Lème ìti ta A, B diaqwrðzontai gn sia, an upˆrqoun f X kai λ R ste: f(a) > λ gia kˆje a A kai f(b) < λ gia kˆje b B. (g) Lème ìti ta A, B diaqwrðzontai austhrˆ, an upˆrqoun f X kai λ < µ sto R ste: f(a) µ gia kˆje a A kai f(b) λ gia kˆje b B. O ìroc {diaqwrðzontai} sto (a) dikaiologeðtai apì to gegonìc ìti to {x X : f(x) = λ} eðnai èna kleistì uperepðpedo, to opoðo qwrðzei ton X se dôo {hmiq rouc} ek twn opoðwn o ènac perièqei to A kai o ˆlloc to B. AnagkaÐa sunj kh gia ta (b), (g) eðnai h A B =. Ta diaqwristikˆ jewr mata pou ja suzht soume aforoôn kurtˆ sônola, kai h apìdeix touc basðzetai polô ousiastikˆ sto Je rhma Hahn-Banach. Gi autì kai anaferìmaste s autˆ me ton ìro {gewmetrik morf tou Jewr matoc Hahn- Banach}. L mma 2.4.1. 'Estw X q roc me nìrma kai èstw A X anoiktì kai kurtì me 0 A. OrÐzoume q A (x) = inf{t > 0 : x ta}.

56 Jewrhma Hahn-Banach Tìte, to q A eðnai èna mh arnhtikì upogrammikì sunarthsoeidèc, upˆrqei M > 0 ste ( ) q A (x) M x, x X kai ( ) A = {x X : q A (x) < 1}. Apìdeixh. H q A orðzetai kalˆ: to A eðnai anoiktì kai perièqei to 0, ˆra upˆrqei δ > 0 ste D(0, δ) A. 'Epetai ìti an 0 x X, tìte (δ/2 x )x A, ˆra 2 δ x {t > 0 : x ta} kai to sônolo autì eðnai kˆtw fragmèno apì to 0, ˆra èqei mègisto kˆtw frˆgma. EpÐshc, q A (x) 2 δ x, dhlad h ( ) isqôei me M = 2/δ (an x = 0, tìte 0 ta gia kˆje t > 0, ˆra q A (0) = 0.) DeÐqnoume t ra tic dôo idiìthtec tou upogrammikoô sunarthsoeidoôc: 'Estw λ > 0. Tìte, q A (x) = inf{t > 0 : λx ta} = inf{t > 0 : x (t/λ)a} = λ inf{(t/λ) : t > 0, x (t/λ)a} = λ inf{s > 0 : x sa} = λq A (x). Gia thn upoprosjetikìthta, èstw x, y X kai èstw ε > 0. Upˆrqoun t, s > 0 ste t < q A (x) + ε, s < q A (y) + ε, kai x ta, y sa. Apì thn kurtìthta tou A èqoume ta + sa = (t + s)a, ˆra x + y (t + s)a. 'Epetai ìti kai afoô to ε > 0 tan tuqìn, q A (x + y) t + s < q A (x) + q A (y) + 2ε, q A (x + y) q A (x) + q A (y). Tèloc, deðqnoume ìti A = {x X : q A (x) < 1}. An q A (x) < 1, tìte upˆrqei r ste q A (x) < r < 1 kai x ra A. AntÐstrofa, an x A, epeid to A eðnai anoiktì upˆrqei t > 0 ste x + tx A, opìte q A (x) 1 1+t < 1. Je rhma 2.4.2. 'Estw X q roc me nìrma kai èstw A mh kenì, anoiktì kurtì uposônolo tou X pou den perièqei to 0. Tìte, upˆrqei f X me thn idiìthta f(x) > 0 gia kˆje x A. Dhlad, to f diaqwrðzei to A apì to {0}. Apìdeixh. 'Estw x 0 A. To A = x 0 A eðnai anoiktì, kurtì kai perièqei to 0. SÔmfwna me to L mma, upˆrqei jetikì upogrammikì sunarthsoeidèc q : X R ste q(x) M x, x X,

2.4 Diaqwristika jewrhmata 57 kai q(x) < 1 an kai mìno an x A. Eidikìtera, q(x 0 ) 1 giatð x 0 / A. JewroÔme ton upìqwro W = span{x 0 } pou parˆgei to x 0, kai orðzoume f : W R me f(λx 0 ) = λq(x 0 ). H f frˆssetai apì to q: an λ 0, tìte f(λx 0 ) = q(λx 0 ), en an λ < 0, tìte f(λx 0 ) < 0 q(λx 0 ). Apì to Je rhma 2.3.1, h f epekteðnetai se grammikì sunarthsoeidèc f : X R, to opoðo ikanopoieð thn f(x) q(x) M x gia kˆje x X. EpÐshc, f(x) = f( x) q( x) M x = M x, ˆra f(x) M x gia kˆje x X, to opoðo apodeiknôei ìti to f eðnai fragmèno ( f X ). Tèloc, gia kˆje x A èqoume x 0 x A, ˆra q(x 0 x) < 1, ap ìpou blèpoume ìti f(x 0 ) f(x) = f(x 0 x) q(x 0 x) < 1. PaÐrnontac up ìyin kai thn q(x 0 ) 1, sumperaðnoume ìti x A, f(x) > f(x0 ) 1 = q(x 0 ) 1 0. Je rhma 2.4.3. 'Estw X q roc me nìrma kai èstw A, B xèna kurtˆ sônola, me to A anoiktì. Tìte, upˆrqoun f X kai λ R tètoia ste: f(a) < λ an a A, kai f(b) λ an b B. An to B eðnai ki autì anoiktì, tìte ta A, B diaqwrðzontai gn sia. Apìdeixh. Jètoume G = A B := {a b : a A, b B}. EÔkola elègqoume ìti to G eðnai kurtì, kai afoô G = b B (A b), to G eðnai anoiktì. Apì thn A B èpetai ìti 0 / G. Apì to prohgoômeno je rhma, upˆrqei f X tètoio ste f(x) > 0 gia kˆje x G. 'Estw a A, b B. Tìte, a b G ˆra f(a b) > 0. Dhlad, f(a) > f(b). Upˆrqei loipìn λ R me thn idiìthta ( ) sup{f(b) : b B} λ inf{f(a) : a A}. To A èqei upotejeð anoiktì kai kurtì, ˆra to f(a) eðnai èna anoiktì diˆsthma sto R. Tìte, h ( ) dðnei f(a) > λ gia kˆje a A kai f(b) λ gia kˆje b B. An to B eðnai ki autì anoiktì, tìte to f(b) eðnai epðshc anoiktì diˆsthma, ˆra f(b) < λ gia kˆje b B, dhlad ta A, B diaqwrðzontai gn sia. Tèloc, deðqnoume èna diaqwristikì je rhma gia xèna kleistˆ kai kurtˆ uposônola tou X, an èna apì autˆ eðnai sumpagèc. Ja qrhsimopoi soume to ex c l mma: L mma 2.4.4. 'Estw X q roc me nìrma, K sumpagèc uposônolo tou X, kai A anoiktì uposônolo tou X me K A. Tìte, upˆrqei r > 0 ste K + D(0, r) A, ìpou K + D(0, r) = {x + y : x K, y < r} = {x X : d(x, K) < r}.

58 Jewrhma Hahn-Banach Apìdeixh. Gia kˆje x K èqoume x A kai to A eðnai anoiktì, ˆra upˆrqei r x > 0 tètoio ste D(x, r x ) = x + D(0, r x ) A. Tìte, K x K D(x, r x /2), kai afoô to K eðnai sumpagèc, upˆrqoun x 1,..., x m K tètoia ste K D(x 1, r x1 /2)... D(x m, r xm /2). Jètoume r = min{r x1 /2,..., r xm /2}. Tìte, K + D(0, r) m (x i + D(0, r xi )) A. Je rhma 2.4.5. 'Estw X q roc me nìrma kai èstw A, B xèna kleistˆ kurtˆ uposônola tou X. An to B eðnai sumpagèc, tìte ta A, B diaqwrðzontai austhrˆ. Apìdeixh. AfoÔ ta A, B eðnai xèna, to sumpagèc B perièqetai sto anoiktì X\A, kai apì to L mma upˆrqei r > 0 tètoio ste B + D(0, r) X\A, ap ìpou paðrnoume (B + D(0, r/2)) (A + D(0, r/2)) =. Ta A + D(0, r/2), B + D(0, r/2) eðnai anoiktˆ kai kurtˆ: kurtˆ giatð ta A, B kai D(0, r/2) eðnai kurtˆ, kai anoiktˆ giatð h D(0, r/2) eðnai anoiktì sônolo. Apì to Je rhma 2.4.3 diaqwrðzontai gn sia, ˆra to Ðdio isqôei kai gia ta uposônolˆ touc A, B. Upˆrqoun loipìn f X kai λ R ste f(b) < λ < f(a) gia kˆje a A kai b B. To B eðnai sumpagèc, ˆra upˆrqei b 0 B ste f(b) f(b 0 ) gia kˆje b B. An µ = f(b 0 ) < λ, tìte B {x : f(x) µ}, A {x : f(x) λ}, ˆra ta A kai B diaqwrðzontai austhrˆ. Parat rhsh. An ta A, B upotejoôn apl c kleistˆ, tìte to Je rhma 2.4.5 paôei na isqôei. Gia parˆdeigma, sto R 2 jewroôme ta A = {(x, y) : y 0} kai B = {(x, y) : x > 0, xy 1}. Ta A, B eðnai kleistˆ, kurtˆ kai xèna, allˆ den diaqwrðzontai gn sia. Ask seic 1. JewroÔme ton L 2[ 1, 1] (to mètro eðnai to Lebesgue). Gia kˆje a R orðzoume E a = {f C[ 1, 1] : f(0) = a}. DeÐxte ìti kˆje E a eðnai kurtì kai puknì ston L 2 [ 1, 1]. An a b, deðxte ìti ta E a kai E b den diaqwrðzontai apì kanèna suneqèc grammikì sunarthsoeidèc F : L 2[ 1, 1] R.

2.5 Klasikoi duðkoi qwroi 59 2. 'Estw X q roc me nìrma pˆnw apì to R kai A, B mh kenˆ, xèna, kurtˆ uposônola tou X me thn idìthta 0 / A B. DeÐxte ìti upˆrqei suneqèc grammikì sunarthsoeidèc f : X R tètoio ste sup{f(x) : x B} < inf{f(x) : x A}. 2.5 KlasikoÐ duðkoð q roi Se aut n thn parˆgrafo perigrˆfoume touc duðkoôc q rouc orismènwn klasik n q rwn. Kˆpoiec apì tic apodeðxeic dðnontai leptomer c, oi upìloipec af nontai gia tic ask seic. Je rhma 2.5.1. O duðkìc q roc tou l p, 1 < p < eðnai isometrikˆ isìmorfoc me ton l q, ìpou q o suzug c ekjèthc tou p. Apìdeixh. 'Estw q o suzug c ekjèthc tou p. Gia kˆje y l q, h apeikìnish f y : l p K me f y (x) = n x ny n eðnai fragmèno grammikì sunarthsoeidèc kai f y y q. OrÐzoume T : l q l p me T (y) = f y. H grammikìthta tou T elègqetai eôkola. Ja deðxoume ìti o T eðnai isometrða. Gia kˆje y l q, stajeropoioôme N N kai jewroôme to x(n) = (x 1,..., x N, 0,...) ìpou x k = y k q 1 sign(y k ), k N. Tìte, kai opìte ( N ) 1/p ( N ) 1/p x(n) p = y k (q 1)p = y k q k=1 f y (x(n)) = N y k q, k=1 k=1 ( f y f N ) 1/q y(x(n)) = y k q x(n) p gia kˆje N N. 'Epetai ìti f y y q, ˆra k=1 T (y) l p = f y = y q. Mènei na deðxoume ìti o T eðnai epð. 'Estw f l p. Jètoume y = (y n ) ìpou y n = f(e n ). Ja deðxoume ìti y l q kai ìti f y = f (opìte T (y) = f). 'Estw N N. JewroÔme to x(n) = (x 1,..., x N, 0,...) ìpou x k = y k q 1 sign(y k ), k N. Tìte, kai N N f(x(n)) = y k q 1 sign(y k )f(e k ) = y k q k=1 k=1 ( N ) 1/p ( N ) 1/p x(n) p = y k (q 1)p = y k q k=1 k=1

60 Jewrhma Hahn-Banach ˆra dhlad, ( N N ) 1/p y k q = f(x(n)) f x(n) p = f y k q, k=1 N y k q f q k=1 gia kˆje N N. 'Epetai ìti y l q kai y q f. Gia na deðxoume ìti f y = f, parathroôme ìti ta dôo autˆ fragmèna sunarthsoeid sumfwnoôn se kˆje e n apì ton orismì tou y. Lìgw grammikìthtac sumfwnoôn ston span{e 1,..., e n,...} o opoðoc eðnai puknìc ston l p kai lìgw sunèqeiac sumfwnoôn se olìklhro ton l p. Je rhma 2.5.2. 'Estw (Ω, A, µ) q roc mètrou me µ(ω) < + kai 1 < p <. O duðkìc q roc tou L p (µ) eðnai isometrikˆ isìmorfoc me ton L q (µ), ìpou q o suzug c ekjèthc tou p. k=1 Apìdeixh. Gia kˆje g L q (µ), h apeikìnish φ g : L p (µ) K me φ g (f) = Ω fgdµ eðnai fragmèno grammikì sunarthsoeidèc kai φ g g q. OrÐzoume T : L q (µ) (L p (µ)) me T (g) = φ g. DeÐqnoume pr ta ìti o T eðnai isometrða. 'Estw g L q (µ), g 0. OrÐzoume f me f(ω) = g(ω) q 1 sign(g(ω)). Tìte, f L p (µ) kai Epomènwc, Ω f p dµ = Ω g q dµ. T (g) = φ g φ g(f) = g q q = g f p g q/p q. q Mènei na deðxoume ìti o T eðnai epð. 'Estw φ : L p (µ) K fragmèno grammikì sunarthsoeidèc. OrÐzoume ν : A K me ν(a) = φ(χ A ). H ν orðzetai kalˆ giatð χ A L p (µ) (afoô µ(ω) < ). EpÐshc, h ν eðnai mètro: apì thn grammikìthta tou φ èpetai ìti h ν eðnai peperasmèna prosjetik, kai an (A n ) eðnai mða fjðnousa akoloujða sthn A me n A n =, tìte ν(a n ) φ χ An p = φ (µ(a n )) 1/p 0. To Ðdio epiqeðrhma deðqnei ìti µ(a) = 0 = ν(a) = 0, dhlad to ν eðnai apìluta suneqèc wc proc to µ. Apì to je rhma Radon-Nikodym, upˆrqei µ-metr simh sunˆrthsh g ste φ(χ A ) = ν(a) = A gdµ = χ A gdµ. Ω

'Epetai ìti φ(f) = Ω fgdµ 2.5 Klasikoi duðkoi qwroi 61 gia kˆje apl metr simh sunˆrthsh f. Ja deðxoume ìti g L q (µ). JewroÔme mia akoloujða apì aplèc metr simec sunart seic h k me 0 h k g q kai jètoume sign(g). Tìte, g k p p = h k 1 kai g k = h 1/p k ˆra Ω g k g = φ(g k) φ g k p, Ω Apì thn ˆllh pleurˆ, g k g = h 1/p g k gdµ φ h k 1/p 1. k g h1/p k h1/q k h k 1 g k gdµ φ h k 1/p Ω = h k. 'Ara, to opoðo dðnei h k 1 φ q. PaÐrnontac k kai qrhsimopoi ntac to je rhma monìtonhc sôgklishc, blèpoume ìti Ω g q dµ = lim k Ω h k dµ φ q. Autì deðqnei ìti g L q (µ). T ra, ta φ g kai φ sumfwnoôn stic aplèc sunart seic oi opoðec eðnai puknèc ston L p (µ). Lìgw sunèqeiac, T (g) = φ g φ, dhlad o T eðnai isometrða epð. Sthn apìdeixh tou Jewr matoc 2.5.2 qrhsimopoi jhke to je rhma Radon Nikodym. DÐnoume mia apìdeixh pou qrhsimopoieð thn jewrða twn q rwn Hilbert (to epiqeðrhma eðnai tou von Neumann). Je rhma 2.5.3 (Radon Nikodym). 'Estw (Ω, A, µ) ènac q roc mètrou. Upojètoume ìti to µ eðnai σ-peperasmèno. 'Estw ν èna proshmasmèno mètro ston (Ω, A) to opoðo eðnai apolôtwc suneqèc wc proc to µ. Tìte, upˆrqei monadik h L 1 (Ω, A, µ) ste ν(a) = h dµ gia kˆje A A. A Apìdeixh. Ja exetˆsoume mìno thn perðptwsh pou to µ eðnai peperasmèno kai to ν eðnai mh arnhtikì (apì aut thn eidik perðptwsh paðrnoume to genikì sumpèrasma me basikˆ epiqeir mata thc jewrðac mètrou). JewroÔme to mètro λ = µ + ν ston (Ω, A). Apì tic upojèseic mac, to λ eðnai mh arnhtikì, peperasmèno mètro. JewroÔme ton q ro Hilbert L 2 (Ω, A, λ) kai orðzoume φ : L 2 (Ω, A, λ) R me φ(f) = Ω f dν. To φ eðnai grammikì sunarthsoeidèc ston L 2 (Ω, A, λ) kai φ(f) Ω 1, f dλ ( 1/2 λ(ω) f dλ) 2 = λ(ω) f 2, Ω

62 Jewrhma Hahn-Banach dhlad to φ eðnai fragmèno. Apì to je rhma anaparˆstashc tou Riesz, upˆrqei g L 2 (Ω, A, λ) me thn idiìthta dhlad, φ(g) = Ω Ω f dν = fg dλ, Ω fg dλ gia kˆje f L 2 (Ω, A, λ). DeÐqnoume pr ta ìti 0 g 1 sqedìn pantoô wc proc λ. Prˆgmati, an A n = {ω Ω : g(ω) 1 + 1/n} tìte, jewr ntac thn f = χ An blèpoume ìti λ(a n ) ν(a n ) ( 1 + 1 ) λ(a n ), n ˆra λ(a n ) = 0. AfoÔ A := {ω : g(ω) > 1} = n=1a n, èpetai ìti λ(a) = 0. Me anˆlogo trìpo, an B n = {ω Ω : g(ω) 1/n} tìte, jewr ntac thn f = χ Bn blèpoume ìti 0 ν(b n ) 1 n λ(b n), ˆra λ(b n ) = 0. AfoÔ B := {ω : g(ω) < 0} = n=1b n, èpetai ìti λ(b) = 0. Apì thn λ = µ + ν mporoôme t ra na grˆyoume ( ) Ω f(1 g) dν = Ω fg dµ gia kˆje f L 2 (Ω, A, λ), kai mporoôme na upojèsoume ìti oi g kai 1 g eðnai mh arnhtikèc pantoô sto Ω. Jètontac C = {ω : g(ω) = 1} kai jewr ntac thn f = χ C, blèpoume ìti µ(c) = 0. AfoÔ to ν eðnai apìluta suneqèc wc proc to µ, èpetai ìti ν(c) = 0. MporoÔme loipìn na upojèsoume ìti 0 g < 1 pantoô sto Ω. 'Estw A A. Gia kˆje n N jewroôme thn f n = (1 + g + + g n )χ A kai apì thn ( ) èqoume A (1 g n+1 ) dν = A g 1 gn+1 1 g Apì to je rhma monìtonhc sôgklishc sumperaðnoume ìti ν(a) = A dν = A g 1 g dµ. To A A tan tuqìn, opìte jètontac h = g 1 g èqoume to zhtoômeno (elègxte thn oloklhrwsimìthta kai th monadikìthta thc h). dµ.

Ask seic 2.6 Deuteroc duðkoc kai autopajeia 63 1. DeÐxte ìti o c 0 eðnai isometrikˆ isìmorfoc me ton l 1. 2. DeÐxte ìti o l 1 eðnai isometrikˆ isìmorfoc me ton l. 3. JewroÔme ton q ro c twn sugklinous n akolouji n me nìrma thn x = sup{ ξ k : k N}. (a) DeÐxte ìti oi q roi c kai c 0 eðnai isìmorfoi. [Upìdeixh: Jewr ste thn a- peikìnish T : c c 0 pou orðzetai wc ex c: an x = (a n ) me a n a, tìte T (x) = (a, a 1 a, a 2 a,...).] (b) DeÐxte ìti oi c kai c 0 den eðnai isometrikˆ isìmorfoi. [Upìdeixh: An S : c 0 c eðnai isometrða epð, parathr ste ìti upˆrqei n N gia to opoðo S(e n ) / c 0.] (g) DeÐxte ìti o c eðnai isometrikˆ isìmorfoc me ton l 1. 2.6 DeÔteroc duðkìc kai autopˆjeia 'Estw X ènac q roc me nìrma. 'Eqoume deð ìti o X eðnai q roc Banach me nìrma thn f = sup{ f(x) : x = 1}. MporoÔme loipìn na mil soume gia ton (X ), ton q ro ìlwn twn fragmènwn grammik n sunarthsoeid n F : X R, me nìrma thn F = sup F (f). f X =1 Gia eukolða grˆfoume X := (X ). O X lègetai deôteroc duðkìc tou X. 'Omoia mporoôme na orðsoume ton trðto duðkì X tou X kai oôtw kajex c. Sth sunèqeia, an X eðnai ènac q roc me nìrma ja grˆfoume x 1, x 2,... gia ta stoiqeða tou X, x 1, x 2,... gia ta stoiqeða tou X, x 1, x 2,... gia ta stoiqeða tou X klp. Kˆje x X orðzei me fusiologikì trìpo èna stoiqeðo τ(x) tou X wc ex c: orðzoume τ(x) : X R, me [τ(x)](x ) = x (x), x X. L mma 2.6.1. To τ(x) : X K eðnai fragmèno grammikì sunarthsoeidèc. Apìdeixh. Elègqoume pr ta th grammikìthta tou τ(x): An x 1, x 2 X kai λ, µ K, tìte EpÐshc, [τ(x)](λx 1 + µx 2) = (λx 1 + µx 2)(x) = λx 1(x) + µx 2(x) = λ[τ(x)](x 1) + µ[τ(x)](x 2). [τ(x)](x ) = x (x) x x = x x, x X. 'Ara, τ(x) X kai τ(x) X x X. L mma 2.6.2. H apeikìnish τ : X X me x τ(x) eðnai grammik isometrða. Eidikìtera, h τ eðnai èna proc èna.

64 Jewrhma Hahn-Banach Apìdeixh. (a) 'Estw x 1, x 2 X kai λ 1, λ 2 K. Gia kˆje x X èqoume [τ(λ 1 x 1 + λ 2 x 2 )](x ) = x (λ 1 x 1 + λ 2 x 2 ) = λ 1 x (x 1 ) + λ 2 x (x 2 ) = λ 1 [τ(x 1 )](x ) + λ 2 [τ(x 2 )](x ) = [λ 1 τ(x 1 ) + λ 2 τ(x 2 )](x ). 'Ara, τ(λ 1 x 1 + λ 2 x 2 ) = λ 1 τ(x 1 ) + λ 2 τ(x 2 ), dhlad h τ eðnai grammik. (b) 'Estw x X. Apì efarmog tou Jewr matoc Hahn-Banach, upˆrqei x 0 X ste x 0 = 1 kai x 0(x) = x. 'Ara, τ(x) X = sup [τ(x)](x ) [τ(x)](x 0) = x 0(x) = x. x =1 Dhlad, τ(x) X x. Sto prohgoômeno L mma eðdame ìti isqôei kai h antðstrofh anisìthta. Epomènwc, τ(x) X = x, kai h τ eðnai isometrða. (g) Profan c, τ(x) = 0 = τ(x) X = 0 = x = 0 = x = 0. Epeid h τ eðnai grammik, autì deðqnei ìti h τ eðnai èna proc èna. 'Amesh sunèpeia twn dôo lhmmˆtwn eðnai to ex c: Je rhma 2.6.3. Kˆje q roc X me nìrma emfuteôetai me fusiologikì trìpo isometrikˆ ston X mèsw thc τ : X X pou orðzetai apì thn [τ(x)](x ) = x (x). Parathr seic. (a) O X eðnai q roc Banach an kai mìno an o τ(x) eðnai kleistìc upìqwroc tou X. Prˆgmati, o X eðnai pl rhc kai o τ(x) grammikìc upìqwroc tou X. O τ(x) eðnai kleistìc an kai mìno an eðnai pl rhc, ìmwc o τ(x) eðnai isometrikˆ isìmorfoc me ton X. 'Ara, o τ(x) eðnai pl rhc an kai mìno an o X eðnai pl rhc. (b) Orismìc H apeikìnish τ lègetai kanonik emfôteush tou X ston X. O X lègetai autopaj c an τ(x) = X, dhlad an h τ eðnai epð. Tìte, o X eðnai isometrikˆ isìmorfoc me ton X. (g) H idiìthta thc autopˆjeiac den eðnai isodônamh me thn Ôparxh isometrikoô isomorfismoô anˆmesa stouc X kai X. An o X eðnai autopaj c, tìte eðnai isometrikˆ isìmorfoc me ton X, allˆ me polô isqurì trìpo: h kanonik emfôteush τ eðnai isometrða epð apì ton X ston X. O James (1951) èqei d sei parˆdeigma q rou pou eðnai isometrikˆ isìmorfoc me ton deôtero duðkì tou, qwrðc na eðnai autopaj c. (d) Upˆrqoun polloð mh autopajeðc q roi. Pr ta-pr ta, ènac q roc me nìrma pou den eðnai pl rhc den mporeð na eðnai autopaj c. 'Oloi oi q roi me nìrma pou èqoun ˆpeirh arijm simh diˆstash san grammikoð q roi, an koun se aut thn kathgorða: upojètontac ìti o q roc me nìrma X = span{x i : i N} eðnai pl rhc

2.6 Deuteroc duðkoc kai autopajeia 65 blèpoume ìti, apì to je rhma tou Baire, prèpei na upˆrqei n N ste o F n = span{x 1,... x n } na èqei mh kenì eswterikì (ˆskhsh: qrhsimopoi ste to gegonìc ìti kˆje upìqwroc F n eðnai kleistìc, afoô èqei peperasmènh diˆstash). 'Omwc tìte, X = F n (ˆskhsh) kai autì eðnai ˆtopo afoô o X eðnai apeirodiˆstatoc. O c 0 den eðnai autopaj c, giatð c 0 l 1 kai l 1 l. An o c 0 tan autopaj c, tìte oi c 0 kai l ja tan isometrikˆ isìmorfoi. Autì den mporeð na isqôei, afoô o c 0 eðnai diaqwrðsimoc en o l ìqi. O l 1 den eðnai autopaj c. An tan, ja tan isometrikˆ isìmorfoc me ton l, dhlad o l ja tan diaqwrðsimoc. 'Omwc autì ja s maine ìti kai o l eðnai diaqwrðsimoc, ˆtopo. Prìtash 2.6.4. O L p (µ), 1 < p <, eðnai autopaj c. Apìdeixh. 'Estw q o suzug c ekjèthc tou p. 'Eqoume deð ìti h apeikìnish T : L q (µ) (L p (µ)) pou orðzetai apì thn [T g](f) = Ω fg dµ, f L p (µ) eðnai isometrikìc isomorfismìc. 'Omoia, h apeikìnish S : L p (µ) (L q (µ)) pou orðzetai apì thn [Sf](g) = Ω fg dµ, g L q (µ) eðnai isometrikìc isomorfismìc. Jèloume na deðxoume ìti h τ : L p (µ) (L p (µ)) eðnai epð. 'Estw loipìn x (L p (µ)). Dhlad, h x : (L p (µ)) K eðnai fragmèno grammikì sunarthsoeidèc. Tìte, h x T : L q (µ) K eðnai fragmèno grammikì sunarthsoeidèc ston L q (µ), ˆra upˆrqei f L p (µ) ste x T = Sf. Ja deðxoume ìti τ(f) = x elègqontac ìti sumfwnoôn se kˆje x (L p (µ)). 'Estw x (L p (µ)). Upˆrqei g L q (µ) ste x = T g. Tìte, kai x (x ) = x (T g) = (x T )(g) = [Sf](g) = [τ(f)](x ) = x (f) = [T g](f) = Ω fg dµ. Ω fg dµ 'Ara x (x ) = [τ(f)](x ). Autì apodeiknôei ìti h τ eðnai epð, ˆra o L p (µ) eðnai autopaj c. Me teleðwc anˆlogo trìpo apodeiknôetai to ex c. Prìtash 2.6.5. O l p, 1 < p <, eðnai autopaj c. DeÐqnoume t ra èna akìma apotèlesma endeiktikì gia ton trìpo me ton opoðo h kanonik emfôteush τ : X X mporeð na mˆc d sei plhroforðec gia ton X. Prìtash 2.6.6. 'Estw X q roc Banach. O X eðnai autopaj c an kai mìno an o X eðnai autopaj c.

66 Jewrhma Hahn-Banach Apìdeixh. 'Estw τ : X X kai τ 1 : X X oi kanonikèc emfuteôseic. Upojètoume pr ta ìti τ(x) = X. 'Estw x 0 X. Zhtˆme x 0 X ste τ 1 (x 0) = x 0. OrÐzoume to x 0 wc ex c: an x X tìte τ(x) X, opìte jètoume x 0(x) := (x 0 )(τ(x)). EÔkola elègqoume ìti to x 0 eðnai grammikì sunarthsoeidèc, kai afoô x 0(x) = (x 0 )(τ(x)) x 0 τ(x) = x 0 x, blèpoume ìti x 0 X kai x 0 x 0. Ja deðxoume ìti τ 1 (x 0) = x 0. 'Estw x X. Tìte, upˆrqei x X ste τ(x) = x. 'Ara, [τ 1 (x 0)](x ) = x (x 0) = [τ(x)](x 0) = x 0(x) = x 0 (τ(x)) = x 0 (x ). AntÐstrofa, èstw ìti τ 1 (X ) = X kai ac upojèsoume ìti τ(x) X. Tìte, o τ(x) eðnai gn sioc kleistìc upìqwroc tou X ˆra upˆrqei mh mhdenikì fragmèno grammikì sunarthsoeidèc x 0 X ste x 0 (τ(x)) = 0 gia kˆje x X. Apì thn upìjes mac, upˆrqei x 0 X me τ 1 (x 0) = x 0. Tìte, gia kˆje x X èqoume x 0(x) = [τ(x)](x 0) = [τ 1 (x 0)](τ(x)) = x 0 (τ(x)) = 0, dhlad x 0 = 0. 'Omwc tìte, x 0 = τ 1 (x 0) = 0, to opoðo eðnai ˆtopo.

Kefˆlaio 3 Basikˆ jewr mata gia telestèc se q rouc Banach 3.1 H arq tou omoiìmorfou frˆgmatoc Prìgonoc thc arq c omoiìmorfou frˆgmatoc mporeð na jewrhjeð to ex c je rhma tou Osgood: an {f n } eðnai mia akoloujða suneq n sunart sewn sto [0, 1] me thn idiìthta h {f n (t)} na eðnai fragmènh gia kˆje t [0, 1], tìte upˆrqei upodiˆsthma [a, b] tou [0, 1] sto opoðo h {f n } eðnai omoiìmorfa fragmènh. Me to Ðdio ousiastikˆ epiqeðrhma apodeiknôetai h ex c genikìterh prìtash. Prìtash 3.1.1. 'Estw X pl rhc metrikìc q roc kai èstw F oikogèneia suneq n sunart sewn f : X R me thn ex c idiìthta: gia kˆje x X to sônolo {f(x) : f F} eðnai fragmèno. Tìte, upˆrqoun x 0 X kai r, M > 0 ste f(x) M gia kˆje x D(x 0, r) kai gia kˆje f F. Apìdeixh. Gia kˆje m N orðzoume A m = {x X : f F, f(x) m}. (i) Kˆje A m eðnai kleistì: an x k A m kai x k x, tìte f F, f(x k ) m kai, apì th sunèqeia twn f F paðrnoume f(x k ) f(x) kaj c k, ˆra f F, f(x) m, dhlad x A m. (ii) X = m=1 A m: 'Estw x X. Apì thn upìjesh, to {f(x) : f F} eðnai fragmèno, dhlad upˆrqei M x > 0 ste, gia kˆje f F, f(x) M x. Upˆrqei m = m(x) N me m M x. Tìte, x A m.

68 Basika jewrhmata gia telestec se qwrouc Banach (iii) O X eðnai pl rhc metrikìc q roc, opìte to Je rhma tou Baire mac exasfalðzei ìti kˆpoio A m0 èqei mh kenì eswterikì, dhlad upˆrqoun x 0 X kai r > 0 ste D(x 0, r) A m0. 'Omwc tìte, h F eðnai omoiìmorfa fragmènh sthn D(x 0, r): x D(x 0, r) f F, f(x) m 0. H arq tou omoiìmorfou frˆgmatoc diatup netai gia mia oikogèneia telest n F ston B(X, Y ) pou èqoun thn idiìthta to {T (x) : T F} na eðnai fragmèno ston Y gia kˆje x X. An o X eðnai pl rhc, h grammikìthta twn T F kai h apl idèa thc apìdeixhc thc Prìtashc 3.1.1 mac dðnoun ìti oi nìrmec T, T F eðnai omoiìmorfa fragmènec: Je rhma 3.1.2. 'Estw X q roc Banach, Y q roc me nìrma, kai èstw F mia oikogèneia apì fragmènouc grammikoôc telestèc T : X Y me thn idiìthta: gia kˆje x X, sup{ T x : T F} < +. Tìte, upˆrqei M > 0 ste T F, T M. Apìdeixh. Gia kˆje T F orðzoume f T : X R me f T (x) = T (x). Kˆje f T eðnai Lipschitz suneq c sunˆrthsh. An x, y X, tìte f T (x) f T (y) = T (x) T (y) T (x) T (y) T x y. Apì thn upìjes mac gia thn F blèpoume ìti gia kˆje x X sup{ f T (x) : T F} = sup{ T (x) : T F} < + dhlad to sônolo {f T (x) : T F} eðnai fragmèno. Apì thn Prìtash 3.1.1 upˆrqoun x 0 X kai r, M 1 > 0 ste gia kˆje x B(x 0, r) kai gia kˆje T F, f T (x) = T (x) M 1. 'Estw x B X. Tìte, gia kˆje T F èqoume T (x 0 +rx) M 1 kai T (x 0 ) M 1 (giatð x 0, x 0 + rx B(x 0, r)). 'Ara, gia kˆje T F T (x) = 1 r T (rx) = 1 r T (x 0 + rx) T (x 0 ) 1 r ( T (x 0 + rx) + T (x 0 ) ) 2M 1 r. AfoÔ to x B X tan tuqìn, èpetai ìti T M := 2M 1 /r gia kˆje T F. ShmeÐwsh. H upìjesh ìti o X eðnai q roc Banach eðnai aparaðthth ìpwc deðqnei to ex c parˆdeigma: JewroÔme to q ro c 00 twn telikˆ mhdenik n pragmatik n

3.1 H arqh tou omoiomorfou fragmatoc 69 akolouji n, me th nìrma x = sup k a k (ìpou x = (a k ) k N ). OrÐzoume mia akoloujða sunarthsoeid n f n : c 00 R me f n (x) = Kˆje f n eðnai grammikì sunarthsoeidèc kai n f n (x) = a k k=1 n k=1 n a k. k=1 a k n sup a k = n x, k dhlad f n n. IsqÔei mˆlista isìthta: pˆrte to x (n) = (1,..., 1, 0,...) me n monˆdec. Tìte, x (n) = 1 kai f n (x (n) ) = n. Gia kˆje x c 00, h akoloujða (f n (x)) eðnai telikˆ stajer. Epomènwc, to {f n (x) : n N} eðnai fragmèno. 'Omwc oi nìrmec twn f n den eðnai omoiìmorfa fragmènec: f n = n. To prìblhma eðnai bèbaia ìti o X den eðnai pl rhc. H arq tou omoiìmorfou frˆgmatoc efarmìzetai suqnˆ sthn ex c morf (je- rhma Banach-Steinhaus): Je rhma 3.1.3. 'Estw X q roc Banach, Y q roc me nìrma kai èstw (T n ) akoloujða fragmènwn telest n T n : X Y me thn idiìthta: gia kˆje x X upˆrqei to y x = lim n T n (x) Y. Tìte, an orðsoume T : X Y me T (x) = y x, o T eðnai fragmènoc telest c kai T lim inf T n. n Apìdeixh. EÔkola elègqoume ìti o T eðnai grammikìc telest c. Apì thn upìjesh, gia kˆje x X to sônolo {T n (x) : n N} eðnai fragmèno. Apì thn arq tou omoiìmorfou frˆgmatoc upˆrqei M > 0 ste T n M gia kˆje n N. Tìte, gia kˆje x X èqoume T (x) = lim T n (x) = lim T n (x) lim inf T n n n x M x. n Dhlad, o T eðnai fragmènoc kai T M. Gia thn akrðbeia, h apìdeixh deðqnei ìti T lim inf n T n. Upˆrqoun diˆforec parallagèc thc {arq c omoiìmorfou frˆgmatoc} gia oikogèneiec fragmènwn telest n. Sthn Prìtash pou akoloujeð, den upojètoume thn plhrìthta tou pedðou orismoô allˆ periorðzoume touc telestèc se èna {mikrì} sônolo. Prìtash 3.1.4. 'Estw X, Y q roi me nìrma, K sumpagèc kai kurtì uposônolo tou X, kai F oikogèneia fragmènwn telest n T : X Y me thn idiìthta: gia kˆje x K to sônolo { T (x) : T F} eðnai fragmèno. Tìte, upˆrqei a > 0 ste T (K) ab Y gia kˆje T F. Dhlad, oi eikìnec T (K), T F eðnai omoiìmorfa fragmènec.

70 Basika jewrhmata gia telestec se qwrouc Banach Apìdeixh. OrÐzoume A n = {x X : T F T (x) n}, n N. Kˆje A n eðnai kleistì sônolo kai apì thn upìjes mac, K = (K A n ). n=1 AfoÔ to K eðnai sumpagèc, upˆrqei m ste to K A m na èqei mh kenì eswterikì sto K. MporoÔme loipìn na broôme x 0 K kai r > 0 ste ( ) K (x 0 + rb X ) A m. EpÐshc, lìgw thc sumpˆgeiac tou K mporoôme na broôme M > 1 arketˆ megˆlo ste K x 0 + MrB X. 'Estw T F kai x K. JewroÔme to z = ( 1 1 M ) x 0 + 1 M x. To K eðnai kurtì, ˆra z K. EpÐshc, z x 0 = (x x 0 )/M rb X, dhlad z x 0 + rb X. Apì thn ( ) èqoume T (z) m kai T (x 0 ) m. Ara, grˆfontac to x sth morf x = Mz (M 1)x 0 paðrnoume T (x) M T (z) + (M 1) T (x 0 ) (2M 1)m. Autì apodeiknôei ìti T (K) ab Y gia kˆje T F, me a = (2M 1)m. H Prìtash pou akoloujeð eðnai eidik perðptwsh thc arq c tou omoiìmorfou frˆgmatoc kai dðnei krit rio gia to pìte èna sônolo fragmènwn grammik n sunarthsoeid n eðnai fragmèno. Prìtash 3.1.5. 'Estw X q roc Banach kai èstw A X. To A eðnai fragmèno an kai mìno an gia kˆje x X isqôei sup{ x (x) : x A} < +. Apìdeixh. An to A eðnai fragmèno, upˆrqei M > 0 ste x M gia kˆje x A. Tìte, gia kˆje x X èqoume sup{ x (x) : x A} M x < +. To antðstrofo eðnai eidik perðptwsh thc arq c tou omoiìmorfou frˆgmatoc (Je rhma 3.1.2). H duðk prìtash qrhsimopoieð thn kanonik emfôteush tou X ston X. Prìtash 3.1.6. 'Estw X q roc me nìrma kai èstw A X. To A eðnai fragmèno an kai mìno an gia kˆje x X isqôei sup{ x (x) : x A} < +. Apìdeixh: ( ) Upojètoume ìti upˆrqei M > 0 ste x M gia kˆje x A. Tìte, gia kˆje x X kai gia kˆje x A èqoume x (x) x x M x.

3.2 Jewrhmata anoikthc apeikonishc kai kleistou grafhmatoc 71 Dhlad, to {x (x) : x A} eðnai fragmèno. ( ) JewroÔme to sônolo τ(a) = {τ(x) : x A} X. To τ(a) eðnai uposônolo tou duðkoô tou X kai apì thn upìjes mac, sup{ (τ(x))(x ) : τ(x) τ(a)} = sup{ x (x) : x A} < +. AfoÔ o X eðnai q roc Banach, efarmìzetai h prohgoômenh Prìtash (me touc X, X, τ(a) sthn jèsh twn X, X, A antðstoiqa) kai èqoume ìti to τ(a) eðnai fragmèno ston X. AfoÔ h τ eðnai isometrða, to A eðnai fragmèno ston X. Ask seic 1. Qrhsimopoi ntac thn arq tou omoiìmorfou frˆgmatoc deðxte ìti an (a k ) eðnai mia akoloujða pragmatik n arijm n me thn idiìthta gia kˆje b = (b k ) c 0 h seirˆ k=1 a kb k na sugklðnei, tìte k=1 a k < +. 2. An x = (x i ) kai y = (y i ) eðnai dôo akoloujðec pragmatik n arijm n, orðzoume x, y := x i y i, an bèbaia h seirˆ sugklðnei. 'Estw 1 < p < +, q o suzug c ekjèthc tou p kai x (n) = (x (n) i ) mia akoloujða ston l p. DeÐxte ìti ta akìlouja eðnai isodônama: (a) Gia kˆje y l q, x (n), y 0. (b) Upˆrqei K > 0 ste x (n) p K gia kˆje n N, kai x (n) i 0 gia kˆje i N. 3. 'Estw (Ω, A, µ) q roc mètrou me µ(ω) < +. 'Estw (f n ) akoloujða ston L p (µ), 1 < p < +. An q eðnai o suzug c ekjèthc tou p, deðxte ìti ta ex c eðnai isodônama: (a) Gia kˆje g L q(µ), fngdµ 0. Ω (b) IsqÔei sup n f n p < + kai gia kˆje E A, fndµ 0. E 4. 'Estw X q roc Banach, (f n ) akoloujða ston X kai (ɛ n ) akoloujða jetik n pragmatik n arijm n me ɛ n 0. Upojètoume ìti gia kˆje x X upˆrqei K x > 0 me thn idiìthta f n(x) K xɛ n gia kˆje n N. DeÐxte ìti f n 0. 5. 'Estw X, Y, Z q roi Banach kai T : X Y Z apeikìnish me thn idiìthta: gia kˆje x X o T x : Y Z me T x (y) = T (x, y) eðnai fragmènoc grammikìc telest c, kai gia kˆje y Y o T y : X Z me T y(x) = T (x, y) eðnai fragmènoc grammikìc telest c. DeÐxte ìti upˆrqei M > 0 ste T (x, y) M x y, x X, y Y. 3.2 Jewr mata anoikt c apeikìnishc kai kleistoô graf matoc Orismìc 'Estw X kai Y metrikoð q roi, kai T : X Y sunˆrthsh. H T lègetai anoikt apeikìnish an gia kˆje A X anoiktì, to T (A) eðnai anoiktì uposônolo tou Y. An T : X Y suneq c, h T den eðnai aparaðthta anoikt : gia parˆdeigma, h T : (0, 2π) R me T (x) = sin x eðnai suneq c, allˆ T ((0, 2π)) = [ 1, 1].

72 Basika jewrhmata gia telestec se qwrouc Banach Je rhma anoikt c apeikìnishc (Schauder) 'Estw X kai Y q roi Banach, kai èstw T : X Y fragmènoc kai epð grammikìc telest c. Tìte, o T eðnai anoikt apeikìnish. To je rhma eðnai ousiastikˆ sunèpeia thc akìloujhc Prìtashc: Prìtash 3.2.1. An X, Y q roi Banach kai T : X Y fragmènoc kai epð grammikìc telest c, tìte to T (D X (0, 1)) perièqei anoikt mpˆla me kèntro to 0 ston Y. Apìdeixh. B ma 1 Parathr ste ìti AfoÔ o T eðnai epð, paðrnoume X = D X (0, k). k=1 'Ara, Y = T (X) = Y = T (D X (0, k)). k=1 T (D X (0, k)). k=1 O Y eðnai pl rhc, kai kˆje T (D X (0, k)) kleistì. Apì to je rhma tou Baire, u- pˆrqei m ste to T (D X (0, m)) na perièqei mia mpˆla ston Y. Dhlad, upˆrqoun y 0 Y kai r > 0 ste (1) D Y (y 0, ρ) T (D X (0, m)). B ma 2 To sônolo T (D X (0, m)) eðnai kurtì kai summetrikì wc proc to 0. Apì thn (1) èqoume D Y ( y 0, ρ) = D Y (y 0, ρ) T (D X (0, m)) kai 'Ara, D Y (0, ρ) = 1 2 D Y (y 0, ρ) + 1 2 D Y ( y 0, ρ) T (D X (0, m)). (2) T (D X (0, 1)) D Y (0, ρ/m) = D Y (0, δ), ìpou δ = ρ/m. B ma 3 EÐdame ìti h kleist j kh thc T (D X (0, 1)) perièqei miˆ anoikt mpˆla D Y (0, δ) ston Y. Mènei na doôme ìti to T (D X (0, 1)) èqei thn Ðdia idiìthta. JewroÔme thn D Y (0, δ/2). 'Estw y Y me y < δ/2. Tìte, ˆra upˆrqei x 1 D X (0, 1/2) ste y 1 2 T (D X(0, 1)) = T (D X (0, 1/2)), y T x 1 < δ 2 2.

3.2 Jewrhmata anoikthc apeikonishc kai kleistou grafhmatoc 73 Tìte, y T x 1 1 2 2 T (D X (0, 1)) = T (D X (0, 1/2 2 )), ˆra upˆrqei x 2 D X (0, 1/2 2 ) ste y T x 1 T x 2 < δ 2 3. Epagwgikˆ, brðskoume x n D X (0, 1/2 n ) me thn idiìthta (3) y T x 1 T x n < δ 2 n+1. H akoloujða z n = x 1 + + x n eðnai Cauchy ston X: an m > n, tìte z m z n = x m + + x n+1 x m + + x n+1 < 1 2 m + + 1 2 n+1 < 1 2 n. O X eðnai pl rhc, ˆra upˆrqei x X ste z n x. ParathroÔme ìti z n = x 1 + + x n x 1 + + x n < x 1 + 1 2 2 + + 1 2 n < x 1 + 1 2. 'Ara, x = lim n z n x 1 + 1 2 < 1. Dhlad, x D X (0, 1). Apì thn (3), T (z n ) = T (x 1 ) + + T (x n ) y. 'Omwc z n x kai o T eðnai suneq c, ˆra T (z n ) T (x). Dhlad, T (x) = y, ki autì shmaðnei ìti y T (D X (0, 1)). To y D Y (0, δ/2) tan tuqìn, ˆra T (D X (0, 1)) D Y (0, δ/2). Parat rhsh. Mia polô qr simh sunèpeia thc apìdeixhc tou L mmatoc eðnai h ex c. Prìtash 3.2.2. 'Estw X, Y q roi Banach kai èstw T : X Y fragmènoc kai epð grammikìc telest c. Upˆrqei stajerˆ M > 0 me thn ex c idiìthta: gia kˆje y Y upˆrqei x X ste T (x) = y kai x M y. Apìdeixh tou jewr matoc anoikt c apeikìnishc: 'Estw A X anoiktì. Ja deðxoume ìti to T (A) eðnai anoiktì: èstw y T (A). Upˆrqei x A ste T x = y. To A eðnai anoiktì, ˆra upˆrqei r > 0 ste D X (x, r) A. Apì thn Prìtash 3.2.1, upˆrqei δ > 0 ste T (D X (0, 1)) D Y (0, δ). Tìte, T (D X (0, r)) = T (rd X (0, 1)) = rt (D X (0, 1)) rd Y (0, δ) = D Y (0, δr). An y D Y (y, δr), tìte y y D Y (0, δr), ˆra upˆrqei z D X (0, r) ste T (z) = y y. 'Epetai ìti x + z D X (x, r), kai T (x + z) = y. Dhlad, T (A) T (D X (x, r)) D Y (y, δr). To y T (A) tan tuqìn, ˆra to T (A) eðnai anoiktì. Mia eidik perðptwsh tou jewr matoc anoikt c apeikìnishc eðnai to je rhma antðstrofhc apeikìnishc.

74 Basika jewrhmata gia telestec se qwrouc Banach Je rhma 3.2.3. 'Estw X, Y q roi Banach kai èstw T : X Y fragmènoc, èna proc èna kai epð, grammikìc telest c. Tìte, o T 1 : Y X eðnai fragmènoc grammikìc telest c. Apìdeixh. O T 1 orðzetai kalˆ kai eðnai grammikìc telest c. AfoÔ o T eðnai anoikt apeikìnish, gia kˆje A X anoiktì èqoume ìti to (T 1 ) 1 (A) = T (A) eðnai anoiktì uposônolo tou Y. 'Ara, o T 1 eðnai suneq c. To epìmeno apotèlesma aut c thc paragrˆfou, to je rhma kleistoô graf - matoc, mac dðnei èna polô qr simo krit rio gia na elègqoume an ènac grammikìc telest c eðnai fragmènoc. Orismìc 'Estw X, Y q roi me nìrma, kai T : X Y grammikìc telest c. To grˆfhma tou T eðnai to sônolo Γ(T ) = {(x, y) : y = T x} X Y. O T èqei kleistì grˆfhma an isqôei to ex c: An x n x ston X, y n y ston Y, kai y n = T (x n ), n N, tìte y = T (x). IsodÔnama, an to Γ(T ) eðnai kleistì uposônolo tou X Y, me nìrma p.q. thn (x, y) = x X + y Y (ˆskhsh). Je rhma kleistoô graf matoc 'Estw X, Y q roi Banach, kai èstw T : X Y grammikìc telest c. An to grˆfhma Γ(T ) tou T eðnai kleistì uposônolo tou X Y, tìte o T eðnai fragmènoc. Apìdeixh. JewroÔme ton X Y me nìrma thn (x, y) = x X + y Y. O X Y eðnai pl rhc: an z n = (x n, y n ) eðnai akoloujða Cauchy ston X Y kai ε > 0, upˆrqei n 0 N ste, gia kˆje n > m n 0, ε > z n z m = (x n x m, y n y m ) = x n x m X + y n y m Y. Tìte ìmwc, x n x m < ε kai y n y m < ε, dhlad oi (x n ), (y n ) eðnai akoloujðec Cauchy stouc X, Y antðstoiqa. Oi X, Y eðnai pl reic, ˆra upˆrqoun x X kai y Y ste x n x kai y n y. 'Omwc tìte, an z = (x, y), èqoume 'Ara, z n z. z z n = x x n + y y n 0. Apì thn upìjes mac, to Γ(T ) eðnai kleistì uposônolo tou X Y, kai eôkola elègqoume ìti eðnai grammikìc upìqwroc tou X Y. AfoÔ o X Y eðnai q roc Banach, to Γ(T ) eðnai q roc Banach. OrÐzoume P : Γ(T ) X me P (x, T x) = x. O P eðnai grammikìc telest c, kai P (x, T x) = 0 = x = 0 = T x = 0 = (x, T x) = (0, 0), ˆra o P eðnai èna proc èna. Profan c, o P eðnai epð. Tèloc, o P eðnai fragmènoc: P (x, T x) = x X x X + T x Y = (x, T x) X Y.

3.2 Jewrhmata anoikthc apeikonishc kai kleistou grafhmatoc 75 Dhlad, P 1. Apì to je rhma antðstrofhc apeikìnishc, o P 1 : X Γ(T ) me P 1 (x) = (x, T x) eðnai fragmènoc. Dhlad, upˆrqei M > 0 ste, gia kˆje x X T x Y x X + T x Y = (x, T x) X Y = P 1 (x) X Y M x X. Sunep c, o T eðnai fragmènoc. KleÐnoume aut thn parˆgrafo me dôo paradeðgmata pou deðqnoun ìti h upìjesh thc plhrìthtac twn X kai Y sta jewr mata antðstrofhc apeikìnishc kai kleistoô graf matoc eðnai aparaðthth. (a) JewroÔme touc q rouc X = C 1 [0, 1] kai Y = C[0, 1] twn suneq c paragwgðsimwn kai suneq n f : [0, 1] R antðstoiqa, me nìrma thn. OrÐzoume T : X Y me T (f) = f. O T eðnai grammikìc telest c kai èqei kleistì grˆfhma: an f n f 0 kai f n g 0, tìte f = g, dhlad g = T (f). 'Omwc o T den eðnai fragmènoc telest c. An f n (t) = t n, tìte f n = 1 en T (f n ) = nt n 1 = n. To je rhma kleistoô graf matoc den efarmìzetai kai o lìgoc eðnai (anagkastikˆ) ìti o X den eðnai pl rhc (sto parˆdeigma autì, o Y eðnai pl rhc). (b) 'Estw X apeirodiˆstatoc diaqwrðsimoc q roc Banach. JewroÔme mia bˆsh Hamel {e i : i I} tou X. To I eðnai uperarijm simo sônolo. MporoÔme epðshc na upojèsoume ìti e i = 1 gia kˆje i I. OrÐzoume mia deôterh nìrma ston X wc ex c: kˆje x X grˆfetai me monadikì trìpo sth morf x = i I a ie i me peperasmènouc to pl joc mh mhdenikoôc suntelestèc a i. Jètoume x = i I kai jewroôme ton q ro Y = (X, ). JewroÔme t ra thn tautotik apeikìnish I : (X, ) (X, ). To grˆfhma Γ(I) = {(x, x) : x X} thc I eðnai kleistì: èstw ìti x n x 0 kai x n y 0. Genikˆ, ( ) x = a i e i a i e i = a i = x, i I i I i I ˆra x n y 0 x n y 0, dhlad y = I(x) = x. 'Omwc o I den eðnai fragmènoc. Apì thn ( ) o I 1 eðnai fragmènoc, an loipìn o I tan fragmènoc, tìte ja tan isomorfismìc. Autì den mporeð na sumbaðnei, giatð o (X, ) èqei upotejeð diaqwrðsimoc en o (X, ) den eðnai diaqwrðsimoc: parathr ste ìti an i j I tìte e i e j = 2 kai to I eðnai uperarijm simo. To je rhma kleistoô graf matoc den efarmìzetai kai o lìgoc eðnai anagkastikˆ ìti o Y den eðnai pl rhc (sto parˆdeigma autì, o X eðnai pl rhc). To Ðdio parˆdeigma deðqnei ìti h upìjesh thc plhrìthtac den mporeð na afairejeð apì to je rhma antðstrofhc apeikìnishc: o I 1 : Y X eðnai fragmènoc, 1-1 kai epð grammikìc telest c allˆ den eðnai isomorfismìc. a i

76 Basika jewrhmata gia telestec se qwrouc Banach Ask seic 1. 'Estw X kleistìc upìqwroc tou L 1[0, 2]. Upojètoume ìti gia kˆje f L 1[0, 1] upˆrqei f X ste f [0,1] = f. DeÐxte ìti upˆrqei stajerˆ M > 0 me thn ex c idiìthta: an f L 1 [0, 1], upˆrqei f X me f [0,1] = f kai f 1 M f 1. 2. 'Estw (x n ) akoloujða se ènan q ro me nìrma X, me thn idiìthta n=1 f(x n) < + gia kˆje f X. DeÐxte ìti upˆrqei stajerˆ M > 0 ste f(x n) M f n=1 gia kˆje f X. 3. 'Estw X, Y q roi Banach kai T : X Y grammikìc telest c. DeÐxte ìti o T eðnai fragmènoc an kai mìno an gia kˆje g Y isqôei g T X. 4. 'Estw X, Y q roi Banach kai T : X Y fragmènoc kai epð grammikìc telest c. (a) DeÐxte ìti upˆrqei K > 0 ste: gia kˆje y Y upˆrqei x X me T (x) = y kai x K y. (b) An H : l 1 Y eðnai ènac fragmènoc grammikìc telest c, deðxte ìti upˆrqei fragmènoc grammikìc telest c G : l 1 X ste T G = H. 5. 'Estw X q roc Banach, (x n ) akoloujða ston X kai x 0 X me x n x 0. JewroÔme mia akoloujða (f n ) ston X kai f X pou ikanopoioôn to ex c: gia kˆje x X, f n(x) f(x). DeÐxte ìti f n(x n) f(x). 6. 'Estw X, Y q roi Banach kai T : X Y fragmènoc kai epð telest c. An x 0 X, y 0 = T (x 0) kai y n y 0 ston Y, deðxte ìti upˆrqoun x n X me T (x n) = y n kai x n x 0.

Kefˆlaio 4 Bˆseic Schauder 4.1 Bˆseic Schauder Orismìc 'Estw X q roc Banach kai èstw (x n ) akoloujða ston X. H (x n ) lègetai bˆsh Schauder tou X an gia kˆje x X upˆrqoun monadikoð a n = a n (x) K ste x = a n x n. n=1 An (x n ) eðnai mia bˆsh Schauder tou X, tìte mporoôme na blèpoume ton X san {q ro akolouji n} mèsw thc taôtishc x (a 1 (x), a 2 (x),...). H bˆsh mac parèqei èna {sôsthma suntetagmènwn} gia ton X. 'Estw ìti h (x n ) eðnai bˆsh Schauder tou X. Den eðnai dôskolo na elègxete ìti ta x n, n N eðnai grammikˆ anexˆrthta (eidikìtera, x n 0 gia kˆje n N). EpÐshc, o X eðnai diaqwrðsimoc (ˆskhsh). Je rhma 4.1.1. 'Estw X q roc Banach me bˆsh Schauder thn (x n ). Gia kˆje n N jewroôme thn fusiologik {probol } P n : X X pou orðzetai mèsw thc ( ) P n (x) = P n a i x i = n a i x i. Tìte, kˆje P n eðnai fragmènoc telest c kai sup n P n < +. Apìdeixh. Kˆje P n eðnai grammikìc telest c kai P n P n = P n, kˆti pou dikaiologeð ton ìro {probol }. Piì genikˆ, an n < m tìte P n P m = P m P n = P n. Autì pou den eðnai profanèc eðnai ìti oi telestèc P n eðnai fragmènoi. Gia kˆje x X èqoume P n (x) x, ˆra to { P n (x) : n N} eðnai fragmèno sto R. OrÐzoume mia nèa nìrma ston X mèsw thc x = sup P n (x). n

78 Baseic Schauder EÔkola elègqoume ìti h eðnai nìrma pou ikanopoieð thn x = lim P n(x) sup P n (x) = x, x X. n n Isqurismìc: O (X, ) eðnai pl rhc. Apìdeixh. 'Estw y k = ak i x i mia -Cauchy akoloujða ston X. Tìte, oi akoloujðec (P n (y k )) k eðnai -Cauchy ston X omoiìmorfa wc proc n: Gia kˆje ε > 0 upˆrqei k 0 (ε) N ste (1) k 1, k 2 k 0 (ε) n N, P n (y k1 ) P n (y k2 ) < ε. 'Omwc gia kˆje n N h akoloujða (P n (y k )) k brðsketai se ènan q ro peperasmènhc diˆstashc (ton X n = span{x 1,..., x n }), epomènwc sugklðnei se kˆpoio z n X n. EpÐshc, lìgw thc (1), h sôgklish eðnai omoiìmorfh wc proc n. Dhlad, gia kˆje ε > 0 upˆrqei k 0 (ε) N ste (2) k k 0 (ε) n N, P n (y k1 ) z n ε. H akoloujða (z n ) n eðnai -Cauchy: èstw ε > 0. Epilègoume k 0 N ste, gia kˆje k k 0 kai kˆje n N, na èqoume P n (y k ) z n < ε/3. Tìte, gia kˆje n, m N paðrnoume z n z m < 2ε 3 + P n(y k ) P m (y k ) an k k 0. AfoÔ lim n P n(y k ) = y k, an stajeropoi soume kˆpoio k k 0 kai pˆroume ta n, m arketˆ megˆla, blèpoume ìti dhlad P n (y k ) P m (y k ) < ε 3, z n z m < ε. O (X, ) eðnai pl rhc, ˆra upˆrqei z X tètoio ste z n z 0. ParathroÔme ìti an m > n, tìte (3) P n (z m ) = P n (lim k P m (y k )) = lim k P n (P m (y k )) = lim k P n (y k ) = z n. (h deôterh isìthta isqôei giatð ìla ta P m (y k ) an koun ston peperasmènhc diˆstashc q ro (X m, ) ston opoðo h P n eðnai suneq c). Apì thn (3) sumperaðnoume ìti upˆrqei akoloujða (a i ) i N sto K ste z n = gia kˆje n N (ˆskhsh). 'Epetai ìti n a i x i z = lim n z n = a i x i.

4.1 Baseic Schauder 79 Tèloc, apì thn (2) blèpoume ìti z y k = sup z n P n (y k ) 0 n ìtan k, to opoðo apodeiknôei ton isqurismì. JewroÔme t ra thn tautotik apeikìnish I : (X, ) (X, ). H I eðnai grammikìc, 1-1 kai epð telest c. AfoÔ x x gia kˆje x X, o I eðnai fragmènoc telest c. Oi dôo q roi eðnai pl reic, ˆra efarmìzetai to je rhma antðstrofhc apeikìnishc: o I 1 eðnai fragmènoc, dhlad upˆrqei K > 0 ste sup P n (x) = x K x n gia kˆje x X. 'Ara, kˆje P n eðnai fragmènoc telest c kai sup P n K < +. n Orismìc. O arijmìc M = sup n P n lègetai stajerˆ thc bˆshc (x i ). An M = 1 tìte h bˆsh (x i ) lègetai monìtonh. Apì thn P n P m = P n, n < m paðrnoume to ex c. Pìrisma 4.1.2. 'Estw X q roc Banach me bˆsh Schauder thn (x i ). Gia kˆje n < m kai kˆje a 1,..., a m K isqôei n a i x i M m a i x i ìpou M h stajerˆ thc bˆshc (x i ). Apìdeixh. Jètoume x = m a ix i. Tìte, P n (x) = n a ix i. To sumpèrasma prokôptei apì thn P n (x) P n x M x. 'Estw X q roc Banach me bˆsh Schauder thn (x i ). Gia kˆje i N orðzoume x i : X K me x i ( n=1 a nx n ) = a i. Kˆje x i eðnai grammikì sunarthsoeidèc, kai x j (x i) = δ ij, i, j N. Orismìc. H akoloujða {x i : i N} eðnai h diorjog nia akoloujða thc {x i : i N}. Kˆje x X grˆfetai sth morf Epiplèon, kˆje x i X : x = x i (x)x i. Prìtash 4.1.3. 'Estw X q roc Banach me bˆsh Schauder thn (x i ), kai èstw (x i ) h diorjog nia akoloujða thc (x i). Kˆje x i eðnai fragmèno grammikì sunarthsoeidèc kai x i 2M x i, ìpou M > 0 h stajerˆ thc bˆshc (x i ).

80 Baseic Schauder Apìdeixh. Jètoume P 0 0 kai jewroôme tuqìn x X. Apì thn x i (x)x i = P i (x) P i 1 (x) blèpoume ìti x i (x) = 1 x i P i(x) P i 1 (x) 2M x i x, ìpou M > 0 eðnai h stajerˆ thc bˆshc (x i ). Prìtash 4.1.4 (arq twn mikr n diataraq n). 'Estw X ènac q roc Banach kai èstw (x n ) bˆsh Schauder tou X, me stajerˆ bˆshc M > 0 kai δ := inf{ x n : n N} > 0. An (y n ) eðnai akoloujða ston X h opoða ikanopoieð thn x n y n < n=1 tìte h (y n ) eðnai epðshc bˆsh Schauder tou X. δ 2M, Apìdeixh. DeÐqnoume pr ta ìti an h seirˆ λ ix i sugklðnei, tìte h seirˆ λ iy i sugklðnei: èstw ìti x = λ ix i X. Gia kˆje n < m èqoume m i=n λ i y i max n i m λ i m m y i x i + λ i x i i=n i=n 2M m δ x m y i x i + λ i x i, i=n ìpou qrhsimopoi same thn trigwnik anisìthta kai thn λ i = x i (x) 2M x. δ Apì tic upojèseic mac, to dexiì mèloc teðnei sto mhdèn ìtan n, m. 'Ara, h n λ iy i eðnai akoloujða Cauchy ston X. 'Epetai ìti upˆrqei to y = λ iy i. OrÐzoume T : X X wc ex c: kˆje x X grˆfetai monos manta sth morf x = λ ix i. OrÐzoume ( ) T λ i x i = λ i y i. O T eðnai kalˆ orismènoc grammikìc telest c, kai gia kˆje x X èqoume dhlad i=n x T (x) = λ n (x n y n ) 2M δ x x n y n, n=1 I T 2M δ n=1 x n y n < 1. n=1 'Epetai ìti o T eðnai isomorfismìc: o antðstrofoc tou T eðnai o telest c S = k=0 (I T )k. H sôgklish thc seirˆc èpetai apì thn I T < 1: AfoÔ (I T ) k I T k gia kˆje k, paðrnoume (I T ) k I T k 1 = 1 I T. k=0 k=0

Qrhsimopoi ntac thn lim n (I T ) n = 0, blèpoume ìti ( ) T (I T ) k = lim k=0 n T Me ton Ðdio trìpo elègqoume ìti ( 4.2 Basikec akoloujiec 81 ( n ) (I T ) k = lim (I (I T n )n+1 ) = I. k=0 k=0 (I T ) k ) T = I. Den eðnai t ra dôskolo na elègxete ìti kˆje x X grˆfetai monos manta sth morf n=1 λ ny n : arkeð na grˆyete to T 1 (x) sth morf T 1 (x) = n=1 λ nx n. 4.2 Basikèc akoloujðec Orismìc. 'Estw X q roc Banach kai èstw (x n ) akoloujða ston X. H (x n ) lègetai basik akoloujða an eðnai bˆsh gia ton upìqwro Y = span{x n : n N} pou parˆgei. H epìmenh Prìtash mac dðnei èna krit rio gia na exetˆzoume an mia akoloujða eðnai basik. Prìtash 4.2.1. 'Estw X q roc Banach kai èstw (x n ) akoloujða mh mhdenik n dianusmˆtwn ston X. H (x n ) eðnai basik akoloujða an kai mìno an upˆrqei stajerˆ K > 0 me thn idiìthta (1) n m a i x i K a i x i gia kˆje n < m kai kˆje a 1,..., a m K. Apìdeixh. H mða kateôjunsh prokôptei ˆmesa an efarmìsoume to Pìrisma 3.3.2 gia ton Y = span{x n : n N}. Gia thn antðstrofh kateôjunsh, ac upojèsoume ìti h (x n ) ikanopoieð thn (1). ParathroÔme pr ta ìti ta dianôsmata x n eðnai grammikˆ anexˆrthta. An m a ix i = 0, tìte gia kˆje 2 j m èqoume a j x j j a i x i + j 1 a i x i 2K m a i x i = 0. H Ðdia anisìthta isqôei (me stajerˆ K antð gia 2K) ìtan j = 1. 'Epetai ìti a 1 = = a m = 0. JewroÔme ton F = span{x n : n N}. O F eðnai puknìc upìqwroc tou Y. Gia kˆje n orðzoume P n : F F me ( m ) P n a i x i = min{m,n} a i x i. O P n eðnai kalˆ orismènoc, lìgw thc grammik c anexarthsðac twn x i, kai apì thn (1) sumperaðnoume ìti P n K gia kˆje n. Qrhsimopoi ntac thn puknìthta

82 Baseic Schauder tou F ston Y mporoôme na epekteðnoume ton P n se olìklhro ton Y, me diat rhsh thc P n K. 'Estw x Y. Apì ton trìpo orismoô twn P n blèpoume ìti upˆrqei (monadik ) akoloujða (a i ) i N sto K ste P n (x) = n a i x i gia kˆje n N (qrhsimopoi ste thn P n P m = P n an n < m). Mènei na deðxoume ìti P n (x) x. 'Estw ε > 0. Upˆrqei z = n b ix i F ste x z < ε. Gia kˆje m > n èqoume P m (z) = z, ˆra x P m (x) x z + z P m (z) + P m (z) P m (x) (1 + P m ) x z < (1 + K) ε, to opoðo apodeiknôei to zhtoômeno. Kˆje apeirodiˆstatoc q roc Banach perièqei kleistì upìqwro me bˆsh. Me ˆlla lìgia, se kˆje apeirodiˆstato q ro Banach upˆrqei basik akoloujða. H apìdeixh basðzetai sto ex c L mma tou Mazur: Prìtash 4.2.2. 'Estw X ènac apeirodiˆstatoc q roc me nìrma kai èstw F upìqwroc tou X me peperasmènh diˆstash. Gia kˆje ε > 0 upˆrqei x X me x = 1, to opoðo ikanopoieð thn y (1 + ε) y + λx gia kˆje y F kai λ K. Apìdeixh. Yˆqnoume x S X to opoðo na eðnai {kˆjeto} ston F (sthn perðptwsh pou o X eðnai q roc Hilbert, opoiod pote x S X me x F ikanopoieð to zhtoômeno, me stajerˆ 1, apì to Pujagìreio Je rhma). MporoÔme na upojèsoume ìti 0 < ε < 1. AfoÔ dim(f ) <, h monadiaða sfaðra S F tou F eðnai sumpag c. 'Ara, mporoôme na broôme y 1,..., y k S F ta opoða na sqhmatðzoun ε/2 dðktuo: gia kˆje y S F upˆrqei j k ste y y j < ε/2. Apì to je rhma Hahn Banach, gia kˆje j = 1,..., k mporoôme na broôme yj X me yj = 1 kai y j (y j) = 1. O upìqwroc k Ker(yj ) èqei peperasmènh sundiˆstash, sunep c, afoô dim(x) =, upˆrqei x k Ker(yj ) me x = 1. j=1 Dhlad, y1(x) = = yk(x) = 0. 'Estw y S F kai λ K. Upˆrqei j k ste y y j < ε/2. MporoÔme loipìn na grˆyoume j=1 y + λx y j + λx y y j y j + λx ε 2 y j (y j + λx) ε 2 = y j (y j ) ε 2 = 1 ε 2 1 1 + ε,

4.2 Basikec akoloujiec 83 dhlad y = 1 (1 + ε) y + λx gia kˆje y S F kai kˆje λ K. 'Epetai to zhtoômeno (exhg ste giatð). Je rhma 4.2.3. Kˆje apeirodiˆstatoc q roc Banach X perièqei kleistì u- pìqwro me bˆsh. Apìdeixh. Ja deðxoume kˆti isqurìtero: gia kˆje ε > 0 upˆrqei basik akoloujða (x i ) ston X me stajerˆ M 1 + ε. 'Estw ε > 0. MporoÔme na broôme akoloujða (ε n ) jetik n arijm n, me n=1 (1 + ε) 1 + ε. JewroÔme tuqìn x 1 X me x 1 = 1 kai jètoume F 1 = span{x 1 }. Apì to L mma tou Mazur upˆrqei x 2 X me x 2 = 1, to opoðo ikanopoieð thn y (1 + ε 2 ) y + a 2 x 2 gia kˆje y F 1 kai kˆje a 2 K. Jètoume F 2 = span{x 1, x 2 } kai epilègoume x 3 X me x 3 = 1, to opoðo ikanopoieð thn y (1 + ε 3 ) y + a 3 x 3 gia kˆje y F 2 kai kˆje a 3 K. Sto n ostì b ma, jètoume F n = span{x 1,..., x n } kai epilègoume x n+1 X me x n+1 = 1, to opoðo ikanopoieð thn y (1 + ε n+1 ) y + a n+1 x n+1 gia kˆje y F n kai kˆje a n+1 K. Me autì ton trìpo orðzetai akoloujða (x i ) ston X. Ja deðxoume ìti: an n < m kai a 1,..., a m K, tìte ( ) n m a i x i (1 + ε) a i x i. Gia thn apìdeixh thc ( ) qrhsimopoioôme to gegonìc ìti gia kˆje n k < m èqoume y k = k a ix i F k kai thn epilog twn x i : èqoume n a i x i = y n (1 + ε n+1 ) y n + a n+1 x n+1 = (1 + ε n+1 ) y n+1, kai, epagwgikˆ, y n (1 + ε n+1 ) y n+1 dhlad, n a i x i m j=n+1 n+2 j=n+1 (1 + ε j ) y n+2 m j=n+1 m m (1 + ε j ) a i x i (1 + ε) a i x i. Autì apodeiknôei ìti h (x i ) eðnai basik, me stajerˆ M 1 + ε. (1 + ε j ) y m,

84 Baseic Schauder 4.3 ParadeÐgmata bˆsewn Schauder Kˆpoioi apì touc klasikoôc q rouc Banach èqoun mia polô fusiologik bˆsh Schauder. Gia parˆdeigma, eôkola elègqoume ìti h sun jhc akoloujða (e n ), ìpou e n = (0,..., 0, 1, 0,...) me th monˆda sth n-ost jèsh, eðnai bˆsh Schauder gia ton l p, 1 p < kai gia ton c 0 (ˆskhsh). Se aut thn Parˆgrafo ja suzht soume kˆpoia klasikˆ paradeðgmata bˆsewn se q rouc sunart sewn. H bˆsh tou Schauder gia ton C[0, 1] H bˆsh tou Schauder (f n ) ston C[0, 1] orðzetai wc ex c: oi pr tec pènte sunart seic thc akoloujðac eðnai oi (i) f 0 (t) = 1 sto [0, 1]. (ii) f 1 (t) = t sto [0, 1]. (iii) f 2 (t) = 2t sto [0, 1/2] kai f 2 (t) = 2 2t sto [1/2, 1]. (iv) f 3 (t) = 4t sto [0, 1/4], f 3 (t) = 2 4t sto [1/4, 1/2] kai f 3 (t) = 0 sto [1/2, 1]. (v) f 4 (t) = 0 sto [0, 1/2], f 4 (t) = 4t 2 sto [1/2, 3/4] kai f 4 (t) = 4 4t sto [3/4, 1]. Genikˆ, an k 1 kai i = 1,..., 2 k, orðzoume thn f 2 k +i jètontac f 2 k +i(t) = f 2 (2 k t i + 1) kai f 2 k +i(t) = 0 alli c (kˆnete èna sq ma). an i 1 2 k t i 2 k Basik parat rhsh. JewroÔme thn arðjmhsh t 0 = 0, t 1 = 1, t 2 = 1/2, t 3 = 1/4, t 5 = 3/4 klp twn duadik n rht n k/2 m tou [0, 1]. Apì ton trìpo orismoô twn f n èqoume f n (t n ) = 1 kai f m (t n ) = 0 an m > n. 'Epetai ìti oi f n eðnai grammikˆ anexˆrthtec (ˆskhsh). Basik sunèpeia tou trìpou orismoô twn f n eðnai h ex c: gia kˆje n N, o upìqwroc span{f 0, f 1,..., f 2 n} sumpðptei me ton upìqwro F n twn katˆ tm mata grammik n suneq n sunart sewn pou èqoun kìmbouc stouc duadikoôc rhtoôc k/2 n, k = 0, 1,..., 2 n. Autì prokôptei eôkola an parathr soume ìti kai oi dôo q roi èqoun diˆstash 2 n + 1. Mia bˆsh tou F n eðnai h {g 0, g 1,..., g 2 n}, ìpou h g i orðzetai monos manta apì tic g i (l/2 n ) = δ il, l = 0, 1,..., 2 n. H epìmenh parat rhsh eðnai ìti o upìqwroc pou parˆgoun oi f n eðnai puknìc ston C[0, 1]. Autì prokôptei apì thn prohgoômenh parat rhsh kai apì to gegonìc ìti n=1 F n = C[0, 1] (gia ton teleutaðo isqurismì, qrhsimopoi ste thn puknìthta twn duadik n rht n sto [0, 1]).

4.3 Paradeigmata basewn Schauder 85 SÔmfwna me thn Prìtash 4.2.1, prokeimènou na deðxoume ìti h (f n ) eðnai bˆsh gia ton C[0, 1], arkeð na deðxoume ìti upˆrqei stajerˆ K > 0 me thn idiìthta (1) n a i f i K m a i f i gia kˆje n < m kai kˆje a 1,..., a m R. Ja deðxoume ìti h (1) isqôei me K = 1 (h (f n ) eðnai monìtonh bˆsh tou C[0, 1]): jètoume P n = n a if i kai P m = m a if i. Parathr ste ìti, gia kˆje k n isqôei P m (t k ) = m a i f i (t k ) = i=0 n a i f i (t k ) = P n (t k ), i=0 afoô f n+1 (t k ) = = f m (t k ) = 0. 'Epetai ìti P n = max 0 k n P n(t k ) = max 0 k n P m(t k ) max 0 k m P m(t k ) = P m. Ta parapˆnw deðqnoun ìti h (f n ) n 0 eðnai bˆsh Schauder tou C[0, 1]. Kˆje f C[0, 1] grˆfetai monos manta sth morf f = n=0 a nf n. Qrhsimopoi ntac mˆlista thn basik parat rhsh, mporoôme na upologðsoume eôkola touc suntelestèc a n : èqoume n 1 a n = f(t n ) a k f k (t n ) gia kˆje n 0, ap' ìpou upologðzontai diadoqikˆ oi a 0, a 1,..., a n,.... k=0 To sôsthma Haar ston L p [0, 1], 1 p < To sôsthma Haar (h n ) ston L p [0, 1], 1 p < orðzetai wc ex c: oi pr tec tˆsseric sunart seic thc akoloujðac eðnai oi (i) h 0 (t) = 1 sto [0, 1]. (ii) h 1 (t) = 1 sto [0, 1/2] kai h 1 (t) = 1 sto (1/2, 1]. (iii) h 2 (t) = 1 sto [0, 1/4], h 2 (t) = 1 sto (1/4, 1/2] kai h 2 (t) = 0 sto (1/2, 1]. (iv) h 3 (t) = 0 sto [0, 1/2), h 3 (t) = 1 sto [1/2, 3/4] kai h 3 (t) = 1 sto (3/4, 1]. Genikˆ, an k 1 kai i = 0, 1,..., 2 k 1, orðzoume thn h 2 k +i jètontac h 2 k +i(t) = h 1 (2 k t i) an i 2 k t i + 1 2 k kai h 2 k +i(t) = 0 alli c (kˆnete èna sq ma). Parathr ste ìti to sôsthma Haar sundèetai me th bˆsh tou Schauder wc ex c: gia kˆje n 1, t f n (t) = 2 n 1 h n 1 (s) ds. 0

86 Baseic Schauder Basikèc parathr seic. (a) Gia kˆje n 1 èqoume 1 0 h n (t) dt = 0. (b) An n < m tìte sumbaðnei èna apì ta ex c dôo: eðte oi h n, h m èqoun xènouc foreðc o forèac thc h m perièqetai ston forèa thc h n kai h h n eðnai stajer (Ðsh me 1 1) ston forèa thc h m. Se kˆje perðptwsh, an n m èqoume 1 0 h n (t)h m (t) dt = 0. (g) Apì to (b) èpetai ìti oi h n, n 0 eðnai grammikˆ anexˆrthtec: an a 1 h 1 + + a n h n 0, tìte gia kˆje j = 1,..., n èqoume 0 = 1 0 (a 1 h 1 + + a n h n )(t)h j (t) dt = a j. Ja deðxoume ìti h (h n ) n 0 eðnai monìtonh bˆsh tou L p [0, 1]. ParathroÔme pr ta ìti o upìqwroc pou parˆgoun oi h n eðnai puknìc ston L p [0, 1]. 'Estw k N kai èstw H k o upìqwroc pou parˆgoun oi χ I, ìpou I diˆsthma thc morf c [ i 1 ] i, 2 k 2, i = 1,..., 2k. H diˆstash tou H k k eðnai 2 k kai oi h 0, h 1,..., h 2 k 1 an koun ston H k kai eðnai grammikˆ anexˆrthtec. 'Ara, H k = span{h n : 0 n 2 k 1} span{h n : n 0} p. Dedomènou ìti L p [0, 1] = k=0 H p k, sumperaðnoume ìti L p [0, 1] = span{h n : n 0} p. SÔmfwna me thn Prìtash 4.2.1, prokeimènou na deðxoume ìti h (h n ) eðnai monìtonh bˆsh gia ton L p [0, 1], arkeð na deðxoume ìti: gia kˆje g = n k=0 a kh k kai gia kˆje a n+1 R, isqôei g p p g + a n+1 h n+1 p p. ParathroÔme ìti h g eðnai stajer (Ðsh, ac poôme, me c) ston forèa I thc h n+1 en h g + a n+1 h n+1 paðrnei tic timèc c ± a n+1 se dôo diast mata m kouc I /2. Sto [0, 1] \ I oi g, g + a n+1 h n+1 sumpðptoun. 'Ara, g + a n+1 h n+1 p p g p p = I ( c + a n+1 h n+1 (t) p c p ) dt = I 2 ( c + a n+1 p + c a n+1 p 2 c p ). H teleutaða posìthta eðnai mh arnhtik : ìtan p 1, h x x p eðnai kurt, sunep c, qrhsimopoi ntac thn anisìthta x + y 2 p x p + y p 2 me x = c + a n+1 kai y = c a n+1 èqoume to zhtoômeno.

Ask seic 4.3 Paradeigmata basewn Schauder 87 1. 'Estw (X, ) ènac q roc Banach me bˆsh Schauder thn (x n). DeÐxte ìti h (x n) eðnai monìtonh bˆsh gia ton (X, ), ìpou x = sup n P n(x). 2. 'Estw X q roc Banach kai (x n) bˆsh Schauder tou X. DeÐxte ìti to sônolo {x n : n N} apoteleðtai apì memonwmèna shmeða. Akìma isqurìtera, deðxte ìti gia kˆje n N, x n / span{x m : m n}. 3. 'Estw (x n) kai (y n) bˆseic Schauder twn q rwn Banach X kai Y. DeÐxte ìti ta ex c eðnai isodônama: (a) H seirˆ n=1 anxn sugklðnei ston X an kai mìno an h seirˆ n=1 anyn sugklðnei ston Y. (b) Upˆrqei C > 0 me thn idiìthta: gia kˆje m N kai gia kˆje a 1,..., a m K, 1 C m m m a nx n a ny n C a nx n. n=1 n=1 4. (a) 'Estw X ènac q roc Banach me bˆsh Schauder kai èstw D èna puknì uposônolo tou X. DeÐxte ìti o X èqei bˆsh Schauder pou apoteleðtai apì stoiqeða tou D. (b) DeÐxte ìti o C[0, 1] èqei bˆsh Schauder pou apoteleðtai apì polu numa. 5. 'Estw (X, ) ènac q roc Banach me bˆsh Schauder thn (x n). An (x n) eðnai h akoloujða twn diorjog niwn sunarthsoeid n, deðxte ìti h (x n) eðnai basik akoloujða ston X. 6. Me ton sumbolismì thc prohgoômenhc 'Askhshc, deðxte ìti h (x n) eðnai bˆsh Schauder tou X an kai mìno an, gia kˆje x X isqôei lim sup n x S X n x (x) = 0, ìpou X n = span{x m : m n}. 7. OrÐzoume x n = e 1 + + e n ston c 0. Exetˆste an h (x n ) eðnai bˆsh Schauder gia ton c 0. ProsdiorÐste thn diorjog nia akoloujða (x n) kai exetˆste an eðnai bˆsh Schauder tou l 1. 8. 'Estw X ènac q roc Banach kai èstw (x n ) bˆsh Schauder gia ton X me x n = 1 gia kˆje n N. Upojètoume ìti upˆrqei x X ste x (x n ) = 1 gia kˆje n N. DeÐxte ìti h akoloujða (x n x n 1) (ìpou x 0 = 0) eðnai epðshc bˆsh Schauder tou X. 9. 'Estw X ènac q roc Banach. Mia akoloujða (x n) lègetai asjen c bˆsh gia ton X an gia kˆje x X upˆrqoun monadikoð a n K me thn idiìthta x ( ) n=1 anxn = x (x). DeÐxte ìti h (x n) eðnai asjen c bˆsh gia ton X an kai mìno an eðnai bˆsh Schauder gia ton X. n=1

Kefˆlaio 5 AsjeneÐc topologðec 5.1 Sunarthsoeidèc tou Minkowski 'Estw X ènac grammikìc q roc pˆnw apì to K. 'Ena mh kenì uposônolo A tou X lègetai isorrophmèno an gia kˆje x A kai gia kˆje λ K me λ 1 èqoume λx A. To A lègetai aporrofoôn an gia kˆje x X upˆrqei ε x > 0 ste tx A gia kˆje t [0, ε x ). Apì touc dôo orismoôc eðnai fanerì ìti kˆje isorrophmèno aporrofoôn uposônolo tou X perièqei to 0. An a A, to A lègetai aporrofoôn sto a an to sônolo A a eðnai aporrofoôn: dhlad, an an gia kˆje x X upˆrqei ε x > 0 ste a + tx A gia kˆje t [0, ε x ). 'Ena isorrophmèno kai aporrofoôn sônolo A X eðnai katˆ kˆpoion trìpo {perioq } tou 0: se kˆje {dieôjunsh} x 0 perièqei èna diˆsthma ( ε x x, ε x x) kai an x 0 A tìte olìklhro to diˆsthma [ x 0, x 0 ] perièqetai sto A. 'Estw t ra p : X R + mia hminìrma. JewroÔme to sônolo A = {x X : p(x) < 1}. L mma 5.1.1. To A eðnai kurtì, isorrophmèno kai aporrofoôn se kˆje shmeðo tou. Apìdeixh. (a) To A eðnai kurtì: an x, y A kai t (0, 1), tìte p((1 t)x + ty) p((1 t)x) + p(ty) = (1 t)p(x) + tp(y) < 1, ˆra (1 t)x + ty A. (b) To A eðnai isorrophmèno: an x A kai λ 1, tìte p(λx) = λ p(x) p(x) < 1, dhlad λx A. (g) To A eðnai aporrofoôn se kˆje shmeðo tou: èstw a A kai x X. An p(x) = 0, tìte p(a + tx) p(a) < 1 gia kˆje t 0. An p(x) > 0, tìte jètoume ε x = 1 p(a) 2p(x) > 0 kai, gia kˆje 0 t < ε x elègqoume ìti p(a + tx) p(a) + tp(x) < p(a) + 1 p(a) 2 = 1 + p(a) 2 < 1,

90 Asjeneic topologiec dhlad a + tx A. To epìmeno apotèlesma genikeôei to L mma 2.4.1: Prìtash 5.1.2. 'Estw X grammikìc q roc pˆnw apì to K kai èstw A èna mh kenì uposônolo tou X, to opoðo eðnai kurtì, isorrophmèno kai aporrofoôn se kˆje shmeðo tou. Tìte, upˆrqei monadik hminìrma p A : X R + me thn idiìthta A = {x X : p(x) < 1}. Apìdeixh. OrÐzoume p A (x) := inf{t 0 : x ta}. To A eðnai aporrofoôn sto 0, ˆra, gia kˆje x X to sônolo {t 0 : x ta} eðnai mh kenì. Autì deðqnei ìti h p A orðzetai kalˆ. DeÐqnoume pr ta tic idiìthtec thc hminìrmac: (a) p A (λx) = λ p(x) gia kˆje x X kai gia kˆje λ K. An λ = 0 to (a) isqôei profan c. Upojètoume loipìn ìti λ 0. Apì to gegonìc ìti to A eðnai isorrophmèno, mporoôme eôkola na doôme ìti βa = β A gia kˆje β K (ˆskhsh). Tìte, grˆfoume p A (λx) = inf{t 0 : λx ta} = inf {t 0 : x tλ } A { = inf t 0 : x t } λ A = λ inf {s 0 : x sa} = λ p A (x). (b) p A (x + y) p A (x) + p A (y) gia kˆje x, y X. 'Estw ε > 0. Upˆrqoun t, s 0 me t < p A (x) + ε, s < p A (y) + ε kai x ta, y sa. Apì thn kurtìthta tou A èpetai ìti ta + sa = (t + s)a, ˆra x + y (t + s)a. Sunep c, p A (x + y) p A (x) + p A (y) + 2ε. AfoÔ to ε > 0 tan tuqìn, èpetai to (b). DeÐqnoume t ra ìti A = {x X : p A (x) < 1}. An p A (x) = r < 1, upˆrqei s : r < s < 1 kai x sa. 'Omwc, sa A apì thn kurtìthta tou A kai thn 0 A. 'Ara, {x X : p A (x) < 1} A. AntÐstrofa, èstw x A. To A eðnai aporrofoôn sto x, mporoôme loipìn na broôme t > 0 ste x + tx A. 'Omwc, tìte, p A (x) 1 1+t < 1. Tèloc, an q eðnai mia ˆllh hminìrma me A = {x X : q(x) < 1}, eôkola blèpoume ìti q p A : gia parˆdeigma, ac upojèsoume ìti p A (x) > q(x) gia kˆpoio x X. Tìte, q ( x p A (x) ) < 1 = x p A (x) A, en p A ( ) x = 1 = x p A (x) p A (x) / A. Parat rhsh. H apeikìnish p A eðnai to sunarthsoeidèc Minkowski tou A. H apìdeixh thc Prìtashc 5.1.2 deðqnei ìti an to A upotejeð kurtì, isorrophmèno kai aporrofoôn sto 0, tìte h p A orðzetai kalˆ kai eðnai hminìrma. Tìte, {x X : p A (x) < 1} A qwrðc na isqôei aparaðthta isìthta.

5.2 Topika kurtoi qwroi 91 Ac upojèsoume t ra ìti to A eðnai kurtì kai aporrofoôn (tìte, 0 A). Me autèc tic upojèseic, den mporoôme na deðxoume ìti h p A ikanopoieð to (a), ikanopoieð ìmwc thn p A (λx) = λp A (x) an λ 0. To (b) exakoloujeð na isqôei, afoô h apìdeixh thc trigwnik c anisìthtac qrhsimopoieð mìno thn kurtìthta tou A. 'Eqoume loipìn to ex c sumpèrasma: Prìtash 5.1.3. 'Estw X grammikìc q roc pˆnw apì to K kai èstw A è- na mh kenì uposônolo tou X, to opoðo eðnai kurtì kai aporrofoôn. Tìte, to sunarthsoeidèc Minkowski p A : X R + eðnai upogrammikì sunarthsoeidèc, kai {x X : p A (x) < 1} A {x X : p(x) 1}. 5.2 Topikˆ kurtoð q roi Orismìc. Topologikìc grammikìc q roc eðnai ènac grammikìc q roc efodiasmènoc me mia topologða T ste ta monosônola na eðnai kleistˆ sônola kai oi prˆxeic tou grammikoô q rou (a) + : X X X me (x, y) x + y (b) : K X X me (λ, x) λx na eðnai suneqeðc wc proc tic antðstoiqec (se kˆje perðptwsh) topologðec. ParadeÐgmata. (a) Kˆje q roc me nìrma eðnai topologikìc grammikìc q roc, an pˆroume san T thn topologða pou epˆgetai apì th nìrma. (b) O L p [0, 1], 0 < p < 1, eðnai topologikìc grammikìc q roc, an pˆroume san T thn topologða pou epˆgetai apì thn metrik d(f, g) = 1 0 f g p (deðxte ìti oi prˆxeic + kai eðnai suneqeðc). 'Omwc, h T den proèrqetai apì nìrma, afoô den upˆrqoun mh tetrimmèna T suneq grammikˆ sunarthsoeid f : L p [0, 1] K. Parathr seic. Gia kˆje x X, h τ x : X X me τ x (y) = x + y eðnai omoiomorfismìc: eðnai suneq c, kai h antðstrof thc eðnai h τ x h opoða eðnai epðshc suneq c. 'Omoia, an λ 0, tìte h σ λ : X X me σ λ (x) = λx eðnai omoiomorfismìc (ed, σ 1 λ = σ 1/λ ). 'Epetai ìti: an B 0 eðnai mia bˆsh perioq n tou 0 gia thn T, tìte, gia kˆje x X, h B x = x + B 0 := {x + B : B B 0 } eðnai bˆsh perioq n tou x (ˆskhsh: deðxte autoôc touc isqurismoôc). An Y eðnai ènac kleistìc grammikìc upìqwroc tou X tìte o q roc phlðko gðnetai topologikìc grammikìc q roc me thn topologða pou epˆgetai apì thn topologða tou X: jewroôme thn Q : X X/Y me Q(x) = x+y kai sumfwnoôme ìti èna E X/Y eðnai anoiktì an to Q 1 (E) eðnai anoiktì ston X. Se aut thn Parˆgrafo ja mac apasqol sei mia eidik kathgorða topologik n grammik n q rwn, sthn opoða sumperilambˆnontai oi q roi me nìrma. 'Estw X ènac grammikìc q roc pˆnw apì to K, kai èstw P mia oikogèneia apì hminìrmec p : X R +, h opoða diaqwrðzei ta shmeða tou X: an x, y X kai x y, tìte upˆrqei p P ste p(x y) > 0. JewroÔme thn oikogèneia A 0 = {A = {x X : p(x) < ε} : p P, ε > 0}.

92 Asjeneic topologiec H A gðnetai upobˆsh gia mia topologða ston X, me ton ex c trìpo: jewroôme thn oikogèneia B = {(x 1 + A 1 ) (x n + A n ) : n N, A i A 0, x i X} {X}. H B eðnai bˆsh gia mia topologða T ston X: h T eðnai h oikogèneia ìlwn twn en sewn stoiqeðwn thc B. O (X, T ) eðnai topologikìc grammikìc q roc (ˆskhsh: oi + kai eðnai T suneqeðc). An x 0 X, mia bˆsh perioq n tou x 0 gia thn T eðnai h oikogèneia B x0 = {(x 0 + A 1 ) (x 0 + A n ) : n N, A i A 0 } (ˆskhsh). Me ˆlla lìgia, èna mh kenì sônolo U X eðnai T kˆje x 0 U upˆrqoun p 1,..., p n P kai ε 1,..., ε n > 0 ste n {x X : p i (x x 0 ) < ε i } U. anoiktì an gia H upìjesh ìti h P diaqwrðzei ta shmeða tou X, mac exasfalðzei ìti h topologða T eðnai Hausdorff: an x, y X kai x y, tìte upˆrqei p P me p(x y) > 0. 'Epetai ìti ta {z X : p(z x) < p(x y)/2} kai {z X : p(z y) < p(x y)/2} eðnai xènec T anoiktèc perioqèc twn x, y antðstoiqa. Orismìc. Topikˆ kurtìc q roc eðnai ènac topologikìc grammikìc q roc efodiasmènoc me mia topologða T h opoða orðzetai apì mia oikogèneia hminorm n P pou diaqwrðzei ta shmeða tou X. Ja mac apasqol soun dôo paradeðgmata topikˆ kurt n q rwn: (1) (X, w). 'Estw X ènac q roc me nìrma. Kˆje fragmèno grammikì sunarthsoeidèc x : X K orðzei me fusiologikì trìpo thn hminìrma p x : X R + me p x (x) = x (x). H oikogèneia P = {p x : x X } diaqwrðzei ta shmeða tou X: apì to je rhma Hahn Banach, an x, y X kai x y, tìte upˆrqei x X me x (x y) 0, dhlad p x (x y) = x (x y) > 0. Sunep c, h P orðzei topikˆ kurt topologða ston X, thn opoða sumbolðzoume me w. H w eðnai h asjen c topologða ston X. An x 0 X, mia bˆsh w perioq n tou x 0 apoteleðtai apì ta sônola thc morf c {x X : x i (x) x i (x 0 ) < ε, i = 1,..., n}, ìpou n N, x i X kai ε > 0 (ˆskhsh: giatð mporoôme na pˆroume to ε koinì gia ìla ta i?). (2) (X, w ). 'Estw X ènac q roc me nìrma kai èstw X o duðkìc tou. Kˆje x X orðzei me fusiologikì trìpo thn hminìrma p x : X R + me p x (x ) = x (x).

5.2 Topika kurtoi qwroi 93 H oikogèneia P = {p x : x X} diaqwrðzei ta shmeða tou X : an x, y X kai x y, tìte upˆrqei x X me x (x) y (x), dhlad p x (x y ) = x (x) y (x) > 0. Sunep c, h P orðzei topikˆ kurt topologða ston X, thn opoða sumbolðzoume me w. H w eðnai h asjen c- topologða ston X. An x 0 X, mia bˆsh w perioq n tou x 0 apoteleðtai apì ta sônola thc morf c {x X : x (x i ) x 0(x i ) < ε, i = 1,..., n}, ìpou n N, x i X kai ε > 0. To epìmeno je rhma mac dðnei ènan qarakthrismì twn topikˆ kurt n q rwn me bˆsh th morf twn perioq n tou 0: Je rhma 5.2.1. 'Estw X ènac topologikìc grammikìc q roc kai èstw U h oikogèneia twn anoikt n, kurt n kai isorrophmènwn uposunìlwn tou X. Tìte, o X eðnai topikˆ kurtìc an kai mìno an h U eðnai bˆsh perioq n tou 0. Apìdeixh. Upojètoume pr ta ìti o X eðnai topikˆ kurtìc. Apì ton orismì thc topologðac T tou X, h oikogèneia { n } B 0 = {x X : p i (x) < ε i } : n N, ε i > 0, p i P eðnai bˆsh perioq n tou 0. 'Omwc, kˆje sônolo sthn B 0 eðnai kurtì kai isorrophmèno. 'Ara, B 0 U, dhlad h U eðnai bˆsh perioq n tou 0. Upojètoume t ra ìti o X eðnai topologikìc grammikìc q roc me thn idiìthta h U na eðnai bˆsh perioq n tou 0. Gia kˆje A U jewroôme to sunarthsoeidèc Minkowski p A tou A. Parathr ste ìti kˆje A U eðnai kurtì, isorrophmèno kai aporrofoôn se kˆje shmeðo tou (to teleutaðo giatð to A eðnai anoiktì kai oi prˆxeic tou grammikoô q rou suneqeðc). 'Ara, kˆje p A, A U, eðnai hminìrma. JewroÔme thn oikogèneia hminorm n P = {p A : A U} kai thn topikˆ kurt topologða T pou aut orðzei ston X. Ja deðxoume ìti h T tautðzetai me thn arqik topologða T 0 tou X. H upobˆsh thc T apoteleðtai apì ta sônola thc morf c {x X : p A (x x 0 ) < ε} = x 0 + εa, A U, x 0 X, ε > 0, ta opoða eðnai T anoiktˆ. 'Ara, T T 0. AntÐstrofa, kˆje T 0 anoiktì sônolo eðnai ènwsh kˆpoiwn A i U. 'Omwc kˆje A U an kei sthn T, afoô A = {x X : p A (x) < 1}. 'Epetai ìti T 0 = T, dhlad h T 0 eðnai topikˆ kurt topologða. ParadeÐgmata. (a) Kˆje q roc me nìrma eðnai topikˆ kurtìc: h topologða tou parˆgetai apì mða hminìrma, th nìrma tou q rou.

94 Asjeneic topologiec (b) H topologða tou L p [0, 1], 0 < p < 1, den eðnai topikˆ kurt. EÐdame ìti to monadikì mh kenì, anoiktì kai kurtì uposônolo tou L p [0, 1] eðnai o Ðdioc o L p [0, 1]. Epomènwc, h U den eðnai bˆsh perioq n tou 0 se aut thn perðptwsh. To epìmeno apotèlesma dðnei ikan sunj kh gia th metrikopoihsimìthta enìc topikˆ kurtoô q rou (tìte, h sôgklish sto q ro mporeð na perigrafeð mèsw akolouji n). Prìtash 5.2.2. 'Estw X ènac topikˆ kurtìc q roc tou opoðou h topologða T orðzetai apì mia arijm simh oikogèneia hminorm n P = {p n : n N}. Tìte, o X eðnai metrikopoi simoc. Dhlad, mporoôme na orðsoume metrik d ston X me thn idiìthta: kˆje d anoiktì sônolo eðnai T anoiktì sônolo kai antðstrofa. Apìdeixh. OrÐzoume ston X mia metrik d wc ex c: an x, y X, jètoume d(x, y) := n=1 1 p n (x y) 2 n 1 + p n (x y). EÐnai eôkolo na deðxete ìti h d eðnai metrik ston X. Gia thn d(x, y) = 0 x = y ja qreiasteðte to gegonìc ìti h P diaqwrðzei ta shmeða tou X. 'Estw U èna d anoiktì sônolo kai èstw x 0 U. Upˆrqei δ > 0 ste: an x X kai d(x, x 0 ) < δ, tìte x U. Epilègoume N N ste n=n+1 1 2 < δ, n 2 kai jewroôme to sônolo To A eðnai T N A = {x X : p n (x x 0 ) < δ 2 }. n=1 anoikt perioq tou x 0 kai A U. Prˆgmati, an x A tìte d(x, x 0 ) = N n=1 ( N n=1 1 p n (x x 0 ) 2 n 1 + p n (x x 0 ) + 1 2 n ) n=n+1 max p n(x x 0 ) + n N 1 p n (x x 0 ) 2 n 1 + p n (x x 0 ) n=n+1 1 2 n < δ 2 + δ 2 = δ, ˆra x U. AfoÔ to x 0 tan tuqìn sto U, to U eðnai T anoiktì. AntÐstrofa, èstw U èna T anoiktì sônolo kai èstw x 0 U. Upˆrqoun N N kai ε > 0 ste N {x X : p n(x x 0 ) < ε} U. Jètoume δ = 1 ε 2 N 1+ε > 0. An d(x, x 0 ) < δ, tìte, gia kˆje n = 1,..., N èqoume 1 p n (x x 0 ) 2 n 1 + p n (x x 0 ) d(x, x 0) < 1 ε 2 N 1 + ε 1 ε 2 n 1 + ε, ap' ìpou èpetai ìti p n (x x 0 ) < ε gia kˆje n = 1,..., N. Dhlad, an d(x, x 0 ) < δ tìte x U. 'Ara, to U eðnai d anoiktì sônolo.

5.3 Diaqwristika jewrhmata se topika kurtouc qwrouc 95 5.3 Diaqwristikˆ jewr mata se topikˆ kurtoôc q rouc 'Estw X ènac topologikìc grammikìc q roc. MimoÔmenoi thn apìdeixh pou ègine sth perðptwsh twn q rwn me nìrma, mporeðte na deðxete ìti èna grammikì sunarthsoeidèc f : X K eðnai suneqèc an kai mìno an eðnai suneqèc sto 0, to opoðo me th seirˆ tou isqôei an kai mìno an o pur nac tou f eðnai kleistìc upìqwroc tou X (wc proc thn topologða tou X). Sthn perðptwsh pou o X eðnai topikˆ kurtìc, èqoume ton ex c qarakthrismì gia th sunèqeia enìc grammikoô sunarthsoeidoôc. Prìtash 5.3.1. 'Estw X ènac topikˆ kurtìc q roc pou h topologða tou orðzetai apì thn oikogèneia hminorm n P, kai èstw f : X K èna grammikì sunarthsoeidèc. Tìte, to f eðnai suneqèc an kai mìno an upˆrqoun p 1,..., p n P kai a > 0 ste, gia kˆje x X, ( ) f(x) a(p 1 (x) + + p n (x)). Apìdeixh. Upojètoume pr ta ìti h f eðnai suneq c. AfoÔ h f eðnai suneq c sto 0, upˆrqei basik perioq B tou 0 me thn idiìthta: an x B tìte f(x) < 1. Dhlad, upˆrqoun hminìrmec p 1,..., p n P kai ε > 0 ste (1) x 'Epetai ìti n {x X : p i (x) < ε} = f(x) < 1. f(x) 1 ε (p 1(x) + + p n (x)). [Prˆgmati, mporoôme na upojèsoume ìti f(x) = 1 kai apì thn (1) blèpoume ìti upˆrqei i n ste p i (x) ε, opìte f(x) = 1 1 ε (p 1(x) + + p n (x)).] To antðstrofo eðnai aplì: kˆje p P eðnai T suneq c. Upojètoume ìti isqôei h ( ) kai jewroôme tuqìn δ > 0. To B = n {x X : p i(x x 0 ) < δ an } eðnai T anoikt perioq tou x 0 kai, gia kˆje x B èqoume f(x) f(x 0 ) a(p 1 (x x 0 ) + + p n (x x 0 )) < δ. 'Ara, h f eðnai T suneq c. Sth sunèqeia perigrˆfoume tic genikeôseic twn diaqwristik n jewrhmˆtwn pou apodeðxame sthn Parˆgrafo 2.4. Basikì rìlo paðzei h ex c parallag twn Protˆsewn 5.2.2 kai 5.2.3. Prìtash 5.3.2. 'Estw X ènac topologikìc grammikìc q roc kai èstw A èna anoiktì kurtì uposônolo tou X pou perièqei to 0. Tìte, to sunarthsoeidèc Minkowski q A tou A eðnai mh arnhtikì suneqèc upogrammikì sunarthsoeidèc, kai A = {x X : q A (x) < 1}. Apìdeixh. To A eðnai kurtì kai aporrofoôn (giatð eðnai anoiktì kai perièqei to 0). Apì thn Prìtash 5.1.3, to q A eðnai upogrammikì sunarthsoeidèc, mh arnhtikì

96 Asjeneic topologiec ex orismoô. Gia thn isìthta A = {x X : q A (x) < 1}, mimhjeðte to epiqeðrhma thc Prìtashc 5.1.2: ja qreiasteðte to gegonìc ìti to A eðnai aporrofoôn se kˆje shmeðo tou, diìti eðnai anoiktì. Apì thn teleutaða isìthta prokôptei kai h sunèqeia tou q A : an x x 0 + εa, tìte q A (x) q A (x 0 ) q A (x x 0 ) < ε. Je rhma 5.3.3. 'Estw X ènac pragmatikìc topologikìc grammikìc q roc kai èstw A mh kenì, anoiktì kurtì uposônolo tou X pou den perièqei to 0. Tìte, upˆrqei suneqèc grammikì sunarthsoeidèc f : X R me thn idiìthta f(x) > 0 gia kˆje x A. Apìdeixh. 'Estw x 0 A. To A = x 0 A eðnai anoiktì, kurtì kai perièqei to 0. SÔmfwna me thn Prìtash 5.2.3, to mh arnhtikì upogrammikì sunarthsoeidèc q = q A : X R ikanopoieð thn q(x) < 1 an kai mìno an x A. Eidikìtera, q(x 0 ) 1 giatð x 0 / A. JewroÔme ton upìqwro W = span{x 0 } pou parˆgei to x 0, kai orðzoume f : W R me f(λx 0 ) = λq(x 0 ). H f frˆssetai apì to q: an λ 0, tìte f(λx 0 ) = q(λx 0 ), en an λ < 0, tìte f(λx 0 ) < 0 q(λx 0 ). Apì thn pr th morf tou jewr matoc Hahn Banach, h f epekteðnetai se grammikì sunarthsoeidèc f : X R, me f(x) q(x) gia kˆje x X. Gia kˆje x A èqoume x 0 x A, ˆra q(x 0 x) < 1, ap ìpou blèpoume ìti f(x 0 ) f(x) = f(x 0 x) q(x 0 x) < 1. PaÐrnontac up ìyin kai thn q(x 0 ) 1, sumperaðnoume ìti x A, f(x) > f(x0 ) 1 = q(x 0 ) 1 0. Tèloc, apì thn f(x) max{q(x), q( x)} prokôptei ìti h f eðnai suneq c sto 0, ˆra suneq c. Je rhma 5.3.4. 'Estw X pragmatikìc topologikìc grammikìc q roc kai èstw A, B xèna kurtˆ sônola, me to A anoiktì. Tìte, upˆrqoun suneqèc grammikì sunarthsoeidèc f : X R kai λ R ste: f(a) < λ an a A, kai f(b) λ an b B. An to B eðnai ki autì anoiktì, tìte ta A, B diaqwrðzontai gn sia. Apìdeixh. H apìdeixh eðnai akrib c ìmoia me aut n tou Jewr matoc 2.4.3. L mma 5.3.5. 'Estw X ènac topologikìc grammikìc q roc, K èna sumpagèc uposônolo tou X, kai A anoiktì uposônolo tou X me K A. Tìte, upˆrqei anoikt perioq U tou 0 me K + U + U A. Apìdeixh. DeÐqnoume pr ta ton akìloujo isqurismì: an W eðnai mia anoikt perioq tou 0 ston X, tìte upˆrqei summetrik anoikt perioq U tou 0 (dhlad, U = U) me U + U W. Prˆgmati, apì thn sunèqeia thc prìsjeshc mporoôme na broôme anoiktèc perioqèc U 1, U 2 tou 0 me U 1 + U 2 W. Jètoume U = U 1 U 2 ( U 1 ) ( U 2 ) kai èqoume ton isqurismì. Epagwgikˆ, blèpoume ìti an W eðnai mia anoikt perioq tou 0 ston X, tìte upˆrqei summetrik anoikt perioq U tou 0 me U + + U W (n forèc).

5.3 Diaqwristika jewrhmata se topika kurtouc qwrouc 97 DeÐqnoume t ra to L mma: 'Estw x K. Tìte, x A kai to A eðnai anoiktì, ˆra upˆrqei summetrik anoikt perioq U x tou 0 ste x + U x + U x + U x A. Apì th summetrða thc U x èpetai ìti (x + U x + U x ) ((X \ A) + U x ) =. JewroÔme thn anoikt kˆluyh {x + U x : x K} tou K. To K eðnai sumpagèc, ˆra upˆrqoun x 1,..., x n K ste K (x 1 + U x1 ) (x n + U xn ). An jèsoume U = U x1 U xn, blèpoume eôkola ìti K + U + U A. Je rhma 5.3.6. 'Estw X ènac topikˆ kurtìc q roc kai èstw A, B xèna kleistˆ kurtˆ uposônola tou X. An to B eðnai sumpagèc, tìte ta A, B diaqwrðzontai austhrˆ. Apìdeixh. AfoÔ ta A, B eðnai xèna, to sumpagèc B perièqetai sto anoiktì X\A. O q roc eðnai topikˆ kurtìc, ˆra upˆrqei bˆsh perioq n U pou apoteleðtai apì anoiktˆ, kurtˆ kai isorrophmèna (ˆra, summetrikˆ) sônola. Apì to L mma, upˆrqei tètoia perioq U me thn idiìthta B + U + U X \ A. 'Ara, (B + U) (A + U) =. Ta B + U, A + U eðnai anoiktˆ kai, apì to Je rhma 5.3.4, diaqwrðzontai gn sia. 'Epetai to je rhma (gia na deðxete ìti ta A, B diaqwrðzontai austhrˆ, ja qreiasteðte th sumpˆgeia tou B). To Je rhma 5.3.6 mac epitrèpei na deðxoume ìti o duðkìc enìc topikˆ kurtoô q rou eðnai arketˆ ploôsioc ste na diaqwrðzei shmeða, shmeðo apì kleistì upìqwro klp (kˆti pou eðqame dei sthn perðptwsh twn q rwn me nìrma). Pìrisma 5.3.7. 'Estw X ènac topikˆ kurtìc q roc, A kleistì kurtì uposônolo tou X kai x 0 / A. Upˆrqei suneqèc grammikì sunarthsoeidèc pou diaqwrðzei gn sia ta {x 0 } kai A. Ask seic 1. 'Estw X ènac topikˆ kurtìc q roc kai èstw (x n) akoloujða ston X me x n 0. DeÐxte ìti y n := x1 + + xn n 0. 2. 'Estw X ènac pragmatikìc topikˆ kurtìc q roc kai èstw A, B mh kenˆ, xèna kurtˆ uposônola tou X ste 0 / A B. DeÐxte ìti upˆrqei suneqèc grammikì sunarthsoeidèc f : X R ste sup{f(x) : x B} < inf{f(x) : x A}. 3. 'Estw L 0 o q roc twn Lebesgue metr simwn sunart sewn f : [0, 1] R (dôo sunart seic tautðzontai an eðnai sqedìn pantoô Ðsec). OrÐzoume mia topologða T ston L 0 paðrnontac san bˆsh perioq n tou 0 thn akoloujða twn sunìlwn B n = { f L 0 : λ({x : f(x) > 1/n}) < 1 n }

98 Asjeneic topologiec kai thn {f + B n : n N} san bˆsh perioq n thc f L 0. DeÐxte ìti: (a) O (L 0, T ) eðnai topologikìc grammikìc q roc. (b) O (L 0, T ) den eðnai topikˆ kurtìc (upìdeixh: gia kˆje n N, co(b n ) = L 0 ). (g) L 0 = {0}. 4. 'Estw L 0 o q roc twn Lebesgue metr simwn sunart sewn f : [0, 1] R (dôo sunart seic tautðzontai an eðnai sqedìn pantoô Ðsec). Ston L 0 jewroôme th metrik ρ(f, g) = 1 0 f(t) g(t) 1 + f(t) g(t) dλ(t). DeÐxte ìti ρ(f n, f) 0 an kai mìno an f n f katˆ mètro. SugkrÐnete thn topologða pou epˆgei h ρ ston L 0 me thn T thc prohgoômenhc ˆskhshc. 5. 'Estw 0 < p < 1. 'Estw l p o grammikìc q roc twn akolouji n x = (x n ) me n=1 x n p <. Ston l p jewroôme th metrik d(x, y) = n=1 x n y n p. DeÐxte ìti o (l p, d) eðnai pl rhc kai ìti o l p eðnai topologikìc grammikìc q roc, ìqi topikˆ kurtìc, me thn topologða pou epˆgei h d. Perigrˆyte ton q ro twn suneq n grammik n sunarthsoeid n ston l p. 5.4 H asjen c topologða Sthn Parˆgrafo 5.2 orðsame thn w topologða se ènan q ro X me nìrma, san thn topikˆ kurt topologða pou orðzetai ston X apì thn oikogèneia hminorm n P = {p x : x X }, ìpou p x (x) = x (x). Mia bˆsh perioq n tou 0 apoteleðtai apì ta sônola thc morf c B(x 1,..., x n; ε) = {x X : x i (x) < ε, i = 1,..., n}, ìpou n N, x i X, ε > 0. Parathr ste ìti, an o X eðnai apeirodiˆstatoc, tìte oi w-perioqèc tou 0 den eðnai fragmèna sônola: èqoume n Ker(x i ) B(x 1,..., x n; ε) gia kˆje ε > 0, dhlad kˆje w perioq tou 0 perièqei kˆpoion upìqwro tou X peperasmènhc sundiˆstashc. Prìtash 5.4.1. Kˆje w anoiktì sônolo eðnai -anoiktì. Apìdeixh. 'Estw U èna w anoiktì sônolo. Tìte, U = (x + B x ), x U ìpou B x einai w anoikt basik perioq tou 0. 'Omwc, kˆje B x eðnai sônolo thc morf c n {x X : x i (x) < ε} dhlad anoiktì sônolo (diìti, kˆje x i X eðnai suneqèc wc proc thn ). 'Epetai ìti to U eðnai anoiktì.

5.4 H asjenhc topologia 99 w Prìtash 5.4.2. 'Estw (x i ) dðktuo ston X kai x X. Tìte, x i x an kai mìno an x (x i ) x (x) gia kˆje x X. Apìdeixh. Parathr ste pr ta ìti an x X tìte to x eðnai w suneqèc: arkeð na elègxoume ìti to x eðnai w suneqèc sto 0, to opoðo eðnai profanèc afoô gia kˆje ε > 0 to sônolo {x X : x (x) < ε} eðnai w anoiktì. Aut h parat rhsh mac dðnei amèswc thn mða kateôjunsh: an x i x (x i ) x (x) gia kˆje x X. AntÐstrofa: upojètoume ìti x (x i ) x (x) gia kˆje x X. 'Estw U mia w anoikt perioq tou x. Upˆrqoun x 1,..., x n X kai ε > 0 ste {y X : x k(y) x k(x) < ε, k = 1,..., n} U. w x, tìte Gia kˆje k = 1,..., n upˆrqei i k ste: an i i k tìte x k (x i) x k (x) < ε. BrÐskoume i 0 i k gia kˆje k n. Tìte, gia kˆje i i 0 èqoume x k (x i) x k (x) < ε gia kˆje k = 1,..., n. Dhlad, x w i U. 'Epetai ìti x i x. (a) Asjen c suneq sunarthsoeid SumbolÐzoume me (X, w) ton grammikì q ro twn w suneq n grammik n sunarthsoeid n f : X K kai me X ton duðkì tou (X, ), ton gnwstì mac q ro twn fragmènwn grammik n sunarthsoeid n. Ja deðxoume ìti (X, w) = X. H apìdeixh basðzetai sto ex c L mma: L mma 5.4.3. 'Estw g, f 1,..., f n : X K grammikˆ sunarthsoeid me thn idiìthta n Ker(f k ) Ker(g). Tìte, upˆrqoun a 1,..., a n K ste g = a 1 f 1 + k=1 + a n f n. Apìdeixh. JewroÔme thn grammik apeikìnish T : X K n me T (x) = (f 1 (x),..., f n (x)) kai to grammikì sunarthsoeidèc ψ : T (X) K me ψ(t (x)) = g(x). O T (X) eðnai grammikìc upìqwroc tou K n kai to ψ eðnai kalˆ orismèno grammikì sunarthsoeidèc ston T (X): an T (x 1 ) = T (x 2 ) T (X), tìte f k (x 1 ) = f k (x 2 ) gia kˆje k = 1,..., n, ˆra x 1 x 2 n Ker(f k ). Apì thn upìjesh, x 1 x 2 Ker(g), k=1 dhlad g(x 1 ) = g(x 2 ). H grammikìthta elègqetai eôkola. H ψ epekteðnetai se èna grammikì sunarthsoeidèc ψ : K n K. Upˆrqoun a 1,..., a n K ste: gia kˆje t = (t 1,..., t n ) K n, Tìte, gia kˆje x X èqoume ψ(t 1,..., t n ) = a 1 t 1 + + a n t n. g(x) = ψ(t (x)) = ψ(f 1 (x),..., f n (x)) = a 1 f 1 (x) + + a n f n (x). Dhlad, g = a 1 f 1 + + a n f n.

100 Asjeneic topologiec Je rhma 5.4.4. 'Estw X ènac q roc me nìrma. Tìte, (X, w) = X. Apìdeixh. 'Eqoume parathr sei ìti X (X, w). AntÐstrofa, èstw f : X K èna w suneqèc grammikì sunarthsoeidèc. Upˆrqei basik perioq B(x 1,..., x n; ε) ste: an x B(x 1,..., x n; ε) tìte f(x) < 1. Dhlad, an x k (x) < ε gia kˆje k = 1,..., n, tìte f(x) < 1. Autì èqei san sunèpeia thn n Ker(x k ) Ker(f). Prˆgmati, an x n k=1 Ker(x k k=1 ), tìte, gia kˆje m N kai gia kˆje k = 1,..., n èqoume x k (mx) = 0 < ε. 'Ara, f(mx) < 1 = f(x) < 1 m gia kˆje m N. 'Epetai ìti f(x) = 0. Apì to L mma 5.4.3 upˆrqoun a 1,..., a n K ste f = a 1 x 1 + + a n x n. Autì deðqnei ìti f X. Sunep c, (X, w) X. (b) Asjen c kleist j kh je rhma Mazur 'Estw A X. AfoÔ h w topologða eðnai mikrìterh apì thn topologða, èqoume A = {B X : A B kai B kleistì} {B X : A B kai B w kleistì} = A w. An ìmwc to A eðnai kurtì, tìte isqôei isìthta: Je rhma 5.4.5 (Mazur). 'Estw X ènac q roc me nìrma kai èstw A kurtì uposônolo tou X. Tìte, A w = A. Apìdeixh. Upojètoume gia aplìthta ìti K = R. EÐdame ìti A A w. 'Estw ìti upˆrqei x 0 A w \ A. To {x 0 } eðnai sumpagèc kai to A eðnai kurtì kai kleistì ston (X, ). Sunep c, upˆrqoun x X kai λ R ste sup x (x) = sup x (x) < λ < x (x). x A x A 'Omwc, x 0 A w, ˆra upˆrqei dðktuo (x i ) sto A me x i w x, kai x (x 0 ) = lim i 'Etsi, katal goume se ˆtopo. x (x i ) sup x (x). x A 'Amesec (kai qr simec) sunèpeiec tou jewr matoc tou Mazur eðnai oi ex c. Pìrisma 5.4.6. 'Estw X ènac q roc me nìrma. 'Ena kurtì uposônolo tou X eðnai asjen c kleistì an kai mìno an eðnai kleistì. Pìrisma 5.4.7. 'Estw X ènac q roc me nìrma kai èstw Y ènac grammikìc upìqwroc tou X. Tìte, Y w = Y.

5.4 H asjenhc topologia 101 Pìrisma 5.4.8. 'Estw X ènac q roc me nìrma kai èstw (x n ) akoloujða ston w X me x i 0. Tìte, upˆrqei akoloujða (yk ) kurt n sunduasm n twn x n ste y k 0. ShmeÐwsh. Kˆje y k eðnai diˆnusma thc morf c y k = N k n=1 a knx n, ìpou a kn 0 kai N k n=1 a kn = 1. Apìdeixh. JewroÔme to sônolo A = co{x n : n N} ìlwn twn kurt n sunduasm n twn x n. To A eðnai kurtì, ˆra A w = A. Apì thn upìjesh, èqoume 0 {x n : n N} w A w. 'Epetai ìti 0 A, ˆra upˆrqei akoloujða (y k ) sto A me y k 0. (g) Asjen c suneqeðc telestèc 'Estw X, Y q roi Banach kai èstw T : X Y ènac grammikìc telest c. Lème ìti o T eðnai asjen c suneq c an eðnai suneq c sunˆrthsh wc proc tic asjeneðc topologðec twn X kai Y. Je rhma 5.4.9. 'Estw X, Y q roi Banach kai èstw T : X Y ènac grammikìc telest c. O T eðnai fragmènoc an kai mìno an eðnai asjen c suneq c. Apìdeixh. Upojètoume pr ta ìti o T eðnai fragmènoc. Gia na elègxoume ìti o T eðnai asjen c suneq c, arkeð na elègxoume thn w sunèqeia sto 0: èstw W mia asjen c perioq tou 0 ston Y. Upˆrqoun y 1,..., y n Y kai ε > 0 ste {y Y : y k(y) < ε, k = 1,..., n} W. OrÐzoume x k : X K me x k = y k T. Kˆje x k eðnai fragmèno grammikì sunarthsoeidèc kai x k y k T. OrÐzoume V := {x X : x k(x) < ε, k = 1,..., n}. H V eðnai w perioq tou 0 ston X kai an x V tìte yk (T x) = x k (x) < ε gia kˆje k = 1,..., n, dhlad T (x) W. 'Ara, 0 V T 1 (W ). AntÐstrofa, upojètoume ìti o T eðnai asjen c suneq c. An y Y = (Y, w), tìte (wc sônjesh asjen c suneq n sunart sewn) y T (X, w) = X. Dhlad, y Y = y T X. Qrhsimopoi ntac aut thn parat rhsh, ja deðxoume ìti o T èqei kleistì grˆfhma. 'Estw ìti x n x 0 kai T x n y 0. Gia kˆje y Y èqoume diìti T x n y, kai (y T )(x n ) = y (T x n ) y (y) (y T )(x n ) (y T )(x) = y (T x), diìti y T X kai x n x. 'Ara, gia kˆje y Y èqoume y (T x) = y (y). AfoÔ o Y diaqwrðzei ta shmeða tou Y, paðrnoume y = T x. Dhlad, to Γ(T ) eðnai kleistì. Apì to je rhma kleistoô graf matoc, o T eðnai fragmènoc.

102 Asjeneic topologiec 5.5 H asjen c- topologða 'Estw X ènac q roc me nìrma kai èstw X o duðkìc tou. OrÐzoume thn w topologða ston X san thn topikˆ kurt topologða pou epˆgetai apì thn oikogèneia hminorm n P = {p x : x X}, ìpou p x (x ) = x (x). Mia bˆsh perioq n tou 0 gia thn w topologða apoteleðtai apì ta sônola thc morf c B(x 1,..., x n ; ε) = {x X : x (x i ) < ε, i = 1,..., n}, ìpou n N, x i X, ε > 0. An o X eðnai apeirodiˆstatoc, tìte oi w -perioqèc tou 0 den eðnai fragmèna sônola: an τ eðnai h kanonik emfôteush tou X ston X, èqoume n Ker(τ(x i )) B(x 1,..., x n ; ε) gia kˆje ε > 0, dhlad kˆje w perioq tou 0 perièqei kˆpoion upìqwro tou X ˆpeirhc diˆstashc. Oi parakˆtw idiìthtec thc w topologðac eðnai anˆlogec me tic antðstoiqec thc w topologðac: Prìtash 5.5.1. 'Estw (x i ) dðktuo ston X kai x X. Tìte, x i kai mìno an x i (x) x (x) gia kˆje x X. Prìtash 5.5.2. Kˆje w anoiktì sônolo eðnai -anoiktì. Apìdeixh. 'Epetai apì to gegonìc ìti kˆje sônolo B(x 1,..., x n ; ε) = {x X : [τ(x i )](x ) < ε, i = 1,..., n} w x an eðnai anoiktì uposônolo tou X. Prìtash 5.5.3. 'Estw (X, w ) o grammikìc q roc twn w suneq n grammik n sunarthsoeid n f : X K. Tìte, (X, w ) = τ(x). Apìdeixh. Apì thn Prìtash 5.5.1 blèpoume ìti τ(x) (X, w ). AntÐstrofa, èstw f : X K èna w suneqèc grammikì sunarthsoeidèc. Upˆrqei basik perioq B(x 1,..., x n ; ε) ste: an x B(x 1,..., x n ; ε) tìte f(x ) < 1. Dhlad, an [τ(x k )](x ) < ε gia kˆje k = 1,..., n, tìte f(x ) < 1. 'Epetai ìti n Ker(τ(x k )) Ker(f). Apì to L mma 5.4.3 upˆrqoun a 1,..., a n K ste k=1 f = a 1 τ(x 1 ) + + a n τ(x n ) = τ(a 1 x 1 + + a n x n ). Autì deðqnei ìti f τ(x). Je rhma 5.5.4 (Alaoglu). 'Estw X ènac q roc me nìrma. H monadiaða mpˆla B X = {x : x 1} tou X eðnai w sumpagèc sônolo. Apìdeixh. JewroÔme to D = [ x, x ] me thn topologða ginìmeno: èna x B X dðktuo (t x i ) x B X sto D sugklðnei sto (t x ) x BX an kai mìno an lim t x i = t x i gia

5.5 H asjenhc- topologia 103 kˆje x B X. Apì to je rhma tou Tychonoff, o D eðnai sumpag c topologikìc q roc. OrÐzoume G : (B X, w ) D me G(x ) = (x (x)) x BX. Apì thn x (x) x x x èpetai ìti G(x ) D gia kˆje x B X. EpÐshc, an x 1, x 2 B X kai G(x 1) = G(x 2), tìte x 1(x) = x 2(x) gia kˆje x B X, opìte x 1 x 2. 'Ara, h G eðnai 1-1. H G eðnai suneq c: upojètoume ìti x i, x B X kai x w i x. Tìte, x i (x) x (x) gia kˆje x B X, opìte G(x i ) G(x ) sto D, apì ton orismì thc topologðac tou D. H eikìna G(B X ) thc G eðnai kleist sto D: èstw x i B me X G(x i ) (t x ) x BX D. Tìte, gia kˆje x B X èqoume t x = lim i x i (x). OrÐzoume f : X K me f(x) = x t x/ x an x 0 kai f(0) = 0. Qrhsimopoi ntac to gegonìc ìti f(x) = x t x/ x = x lim i x i ( ) x = lim x i (x) x i gia kˆje x X, elègqoume ìti h f eðnai grammikì sunarthsoeidèc. EpÐshc, f(x) = x t x/ x x, ˆra f B X kai x w i f. AfoÔ G(f) = (t x ) x BX, sumperaðnoume ìti to G(B X ) eðnai kleistì. Tèloc, elègqoume ìti h G 1 eðnai suneq c sto G(B X ). An G(x i ) G(x ) sto D, tìte x i (x) x (x) gia kˆje x B X, ˆra x w i x. 'Epetai ìti h B X = G 1 (G(B X )) eðnai w sumpag c. Pìrisma 5.5.5. 'Estw X ènac q roc Banach. 'Ena uposônolo A tou X eðnai w sumpagèc an kai mìno an eðnai w kleistì kai fragmèno. Apìdeixh. Upojètoume pr ta ìti to A eðnai w kleistì kai A ab X gia kˆpoion a > 0. H apeikìnish σ a : X X me σ a (x ) = ax eðnai w omoiomorfismìc. Apì to je rhma Alaoglu èpetai ìti to ab X eðnai w sumpagèc sônolo. To A eðnai w kleistì uposônolo tou ab X, ˆra eðnai w sumpagèc. AntÐstrofa: an to A eðnai w sumpagèc, tìte to A eðnai profan c w kleistì kai gia kˆje x X to sônolo { x (x) : x A} = { [τ(x)](x ) : x A} eðnai fragmèno. Apì thn arq omoiìmorfou frˆgmatoc èpetai ìti sup{ x : x A} <. Prìtash 5.5.6. Kˆje q roc Banach X eðnai isometrikˆ isìmorfoc me ènan kleistì upìqwro enìc q rou (C(M), ), ìpou M sumpag c topologikìc q roc Hausdorff. Apìdeixh. Jètoume M = B X. Apì to je rhma Alaoglu, o M eðnai sumpag c topologikìc q roc Hausdorff me thn w topologða.

104 Asjeneic topologiec OrÐzoume thn apeikìnish T : X (C(M), ) me (T x)(x ) = x (x), x M. H T eðnai kalˆ orismènh. 'Estw x X. An x w i x sto M, tìte x i (x) x (x), dhlad (T x)(x i ) (T x)(x ). 'Ara, T x C(M). H grammikìthta thc T elègqetai eôkola, kai T x = sup{ x (x) : x X } = x gia kˆje x X, d lad h T eðnai isometrða. 'Ara, h T eðnai isometrikìc isomorfismìc tou X me ènan kleistì upìqwro tou (C(M), ). Je rhma 5.5.7 (Goldstine). 'Estw X q roc me nìrma kai èstw τ : X X h kanonik emfôteush. Tìte, τ(b X ) w = B X. Apìdeixh. H τ eðnai grammik isometrða, ˆra to τ(b X ) eðnai kurtì uposônolo thc B X kai to τ(b X ) w eðnai w kleistì kurtì uposônolo thc B X (exhg ste giatð). Upojètoume ìti upˆrqei x 0 B x \ τ(b X ) w kai ja katal xoume se ˆtopo. Ston topikˆ kurtì q ro (X, w ) mporoôme na diaqwrðsoume to x 0 apì to τ(b X ) w : upˆrqei f (X, w ), dhlad upˆrqei x 0 X ste Apì thn ˆllh pleurˆ, x 0 (x 0) > sup x B X [τ(x)](x 0) = sup x B X x 0(x) = x 0. x 0 (x 0) x 0 x 0 x 0, to opoðo odhgeð se ˆtopo. 'Epetai ìti τ(b X ) w = B X. Pìrisma 5.5.8. 'Estw X q roc me nìrma kai èstw τ : X X h kanonik emfôteush. Tìte, τ(x) w = X. Apìdeixh. O τ(x) w eðnai upìqwroc tou X kai perièqei thn B X apì to prohgoômeno je rhma. Me th bo jeia twn asjen n topologi n mporoôme na d soume ènan qr simo qarakthrismì twn autopaj n q rwn. Ac upojèsoume pr ta ìti o q roc Banach X eðnai autopaj c. Tìte, h τ : (B X, w) (B X, w ) eðnai topologikìc omoiomorfismìc. Apì thn autopˆjeia tou X blèpoume ìti h τ eðnai 1-1 kai epð. EpÐshc, an x, x i B X, i I, èqoume x i w x x X x (x i ) x (x) x X [τ(x i )](x ) [τ(x)](x ) τ(x i ) w τ(x), dhlad, oi τ, τ 1 eðnai suneqeðc. Apì to je rhma Alaoglu, h B X eðnai w sumpag c. Sunep c, h B X eðnai sumpag c. IsqÔei ìmwc kai to antðstrofo:

5.5 H asjenhc- topologia 105 Je rhma 5.5.9. 'Estw X ènac q roc Banach. O X eðnai autopaj c an kai mìno an h B X eðnai w sumpag c. Apìdeixh. DeÐqnoume thn deôterh sunepagwg : èstw ìti h B X eðnai w sumpag c. Tìte, to τ(b X ) eðnai w sumpagèc sônolo (parathr ste ìti h τ eðnai (w, w ) suneq c). 'Ara, to τ(b X ) eðnai w kleistì sônolo. Apì to je rhma Goldstine èqoume τ(b X ) = τ(b X ) w = B X, ˆp ìpou èpetai ìti τ(x) = X. 'Ara, o X eðnai autopaj c. DÔo idiìthtec twn autopaj n q rwn (a) 'Estw X ènac q roc me nìrma. Mia akoloujða (x n ) ston X lègetai asjen c Cauchy an gia kˆje x X h akoloujða x (x n ) eðnai akoloujða Cauchy (isodônama, sugklðnei). O X lègetai asjen c akoloujiakˆ pl rhc an kˆje asjen c Cauchy akoloujða tou X eðnai asjen c sugklðnousa. Prìtash 5.5.10. Kˆje autopaj c q roc Banach X eðnai asjen c akoloujiakˆ pl rhc. Apìdeixh. 'Estw (x n ) mia w Cauchy akoloujða ston X. Gia kˆje x X, to sônolo { x (x n ) : n N} eðnai fragmèno, diìti to lim x (x n ) n upˆrqei. Apì thn arq omoiìmorfou frˆgmatoc, upˆrqei M > 0 ste x n M gia kˆje n N. To MB X eðnai w sumpagèc lìgw thc autopˆjeiac tou X. 'Ara, to sônolo {x n : n N} èqei asjenèc shmeðo suss reushc x MB X. 'Estw x X. AfoÔ to lim x (x n ) n upˆrqei, anagkastikˆ èqoume x (x) = lim x (x n ) n (exhg ste w giatð). 'Epetai ìti x n x. (b) 'Estw X ènac q roc me nìrma kai èstw Y ènac gn sioc kleistìc upìqwroc tou X. An x 0 X \ Y, h apìstash tou x 0 apì ton Y eðnai h d(x 0, Y ) = inf{ x 0 y : y Y }. Den eðnai genikˆ swstì ìti upˆrqei y 0 Y me d(x 0, Y ) = x 0 y 0. An ìmwc o X eðnai autopaj c, autì eðnai pˆnta swstì. Prìtash 5.5.11. 'Estw X autopaj c q roc, Y gn sioc kleistìc upìqwroc tou X kai x 0 X \ Y. Upˆrqei y 0 Y me d := d(x 0, Y ) = x 0 y 0. Apìdeixh. OrÐzoume M = {y Y : x 0 y 2d}. Tìte, d(x 0, Y ) = inf{ x 0 y : y M}. To sônolo M eðnai fragmèno (an y M tìte y x 0 + 2d) kai w kleistì: autì prokôptei eôkola apì to je rhma Mazur, afoô to M eðnai kurtì kai kleistì. O X èqei upotejeð autopaj c, ˆra to M eðnai w sumpagèc. Isqurismìc: H f : M R + me y x 0 y eðnai kˆtw hmisuneq c.

106 Asjeneic topologiec Prˆgmati, gia kˆje a R to sônolo {y M : x 0 y a} eðnai w kleistì: w an x 0 y i a kai y i y, tìte gia kˆje x B X èqoume x (x 0 y) = lim x (x 0 y i ) sup x 0 y i a, i i ˆra x 0 y a. ParathroÔme t ra ìti kˆje kˆtw hmisuneq c sunˆrthsh orismènh se sumpagèc sônolo paðrnei elˆqisth tim : jewr ste y i M me f(y i ) d. Upˆrqei w upodðktuo y j y0 M. Tìte, 'Ara, x 0 y 0 = f(y 0 ) = d(x 0, Y ). d x 0 y 0 = f(y 0 ) lim inf f(y j ) = d. j 5.6 Metrikopoihsimìthta kai diaqwrisimìthta H asjen c topologða se ènan apeirodiˆstato q ro Banach den eðnai potè metrikopoi simh: Ac upojèsoume ìti o (X, w) eðnai metrikopoi simoc. Kˆje metrikìc q roc èqei arijm simh bˆsh perioq n gia kˆje shmeðo tou. Sunep c, upˆrqei akoloujða peperasmènwn uposunìlwn D n tou X ste h akoloujða (U n ) me U n = {x X : x (x) < ε n, x D n } na eðnai w bˆsh perioq n tou 0. Tìte, gia to sônolo D = n=1 D n èqoume: ( ) X = span{x : x D}. Prˆgmati, an x X, to U x = {x : x (x) < 1} eðnai w perioq tou 0. Apì thn upìjes mac, upˆrqei n N ste U n U x. 'Omwc tìte, to x eðnai grammikìc sunduasmìc twn stoiqeðwn tou D n (deðte thn apìdeixh tou Jewr matoc 5.4.4). Apì thn ( ) sumperaðnoume ìti o X èqei arijm simh bˆsh san grammikìc q roc: èqoume ìmwc dei ìti autì den mporeð na isqôei gia ènan apeirodiˆstato q ro Banach. Anˆlogo sumpèrasma isqôei gia thn asjen - topologða. Upˆrqoun par ìla autˆ kˆpoia apotelèsmata metrikopoihsimìthtac gia fragmèna uposônola twn X, X. H qrhsimìthtˆ touc eðnai profan c: an h w (antðstoiqa, h w ) topologða sthn B X (antðstoiqa, sthn B X ) proèrqontai apì metrik, tìte h asjen c sôgklish sta fragmèna sônola perigrˆfetai apì akoloujðec. Je rhma 5.6.1. 'Estw X ènac q roc Banach. H w topologða sthn B X proèrqetai apì metrik an kai mìno an o X eðnai diaqwrðsimoc. Apìdeixh. Upojètoume pr ta ìti o X eðnai diaqwrðsimoc. 'Estw D = {x n : n N} èna arijm simo puknì uposônolo thc B X. OrÐzoume metrik d sthn B X me d(x, y ) = n=1 1 2 n x (x n ) y (x n ) 1 + x (x n ) y (x n ).

5.6 Metrikopoihsimothta kai diaqwrisimothta 107 JewroÔme thn tautotik apeikìnish I : (B X, w ) (B X, d). H I eðnai suneq c: èstw ìti x w i x kai èstw ε > 0. Upˆrqei N N ste n=n+1 1 2 < ε. Jètoume n 2 U = {y B X : y (x n ) x (x n ) < ε 2, n N}. Upˆrqei i 0 I ste x i U gia kˆje i i 0 (exhg ste giatð). Tìte, gia kˆje i i 0 èqoume d(x i, x ) N n=1 1 2 n ε 2 + n=n+1 1 2 n 1 < ε. 'Ara, x d i x. Apì to je rhma tou Alaoglu, h (B X, w ) eðnai sumpag c. AfoÔ h I eðnai suneq c, 1-1 kai epð, eðnai omoiomorfismìc. Dhlad, oi d kai w topologðec tautðzontai sthn B X. AntÐstrofa, an h w topologða sthn B X proèrqetai apì metrik, tìte u- pˆrqei arijm simh w bˆsh (U n ) perioq n tou 0, ìpou kˆje U n eðnai sônolo thc morf c U n = {x B X : x (x) < ε n, x D n } gia kˆpoio ε n > 0 kai kˆpoio peperasmèno D n X, kai n=1 U n = {0}. Jètoume D = n=1 D n. ParathroÔme ìti: an x B X kai x (x) = 0 gia kˆje x D, tìte x U n gia kˆje n, ˆra x = 0. Apì pìrisma tou jewr matoc Hahn Banach èpetai ìti span(d) = X. AfoÔ to D eðnai arijm simo, o X eðnai diaqwrðsimoc. Je rhma 5.6.2. 'Estw X ènac q roc Banach. H w topologða sthn B X proèrqetai apì metrik an kai mìno an o X eðnai diaqwrðsimoc. Apìdeixh. Upojètoume pr ta ìti o X eðnai diaqwrðsimoc. 'Estw τ : X X h kanonik emfôteush. Apì to prohgoômeno je rhma, h w topologða sthn B X proèrqetai apì metrik. H τ : (B X, w) (τ(b X ), w ) eðnai omoiomorfismìc (exhg ste giatð), ˆra h (B X, w) eðnai metrikopoi simh, afoô h τ(b X ) me thn sqetik w topologða eðnai metrikìc q roc. AntÐstrofa, an h w topologða sthn B X proèrqetai apì metrik, tìte upˆrqei arijm simh w bˆsh perioq n (U n ) tou 0, ìpou kˆje U n eðnai sônolo thc morf c U n = {x B X : x (x) < ε n, x D n} gia kˆpoio ε n > 0 kai kˆpoio peperasmèno Dn X, kai n=1 U n = {0}. Jètoume D = n=1 D n kai X1 = span(d ). O X1 eðnai diaqwrðsimoc upìqwroc tou X. Ja deðxoume ìti X1 = X : èstw ìti upˆrqei y X \ X1. Tìte, d = d(y, X1 ) > 0 kai, apì pìrisma tou jewr matoc Hahn Banach, upˆrqei x X pou ikanopoieð ta ex c: (a) x = 1/d, (b) x (y ) = 1 kai x (x ) = 0 gia kˆje x X1.

108 Asjeneic topologiec JewroÔme to w anoiktì sônolo (sthn B X ) V = { x B X : y (x) < d }. 2 AfoÔ h (U n ) eðnai w bˆsh perioq n tou 0, upˆrqei n N ste U n V. 'Eqoume dx B X kai apì to je rhma Goldstine h τ(b X ) eðnai w pukn sthn B X. 'Ara, upˆrqei x B X to opoðo ikanopoieð ta ex c: (i) dx (x ) τ(x)(x ) < ε n gia kˆje x D n. (ii) dx (y ) τ(x)(y ) < d 2. Apì thn pr th sunj kh èpetai ìti x (x) < ε n gia kˆje x D n, dhlad x U n. Apì thn deôterh sunj kh èqoume d y (x) = dx (y ) y (x) < d 2, dhlad y (x) > d. 'Ara, 2 x / V. SumperaÐnoume loipìn ìti x U n \ V, to opoðo eðnai ˆtopo afoô U n V. Parathr ste ìti, an h w topologða sthn B X proèrqetai apì metrik, tìte o X eðnai diaqwrðsimoc, ˆra o X eðnai diaqwrðsimoc. To antðstrofo den eðnai pˆnta swstì: Je rhma 5.6.3 (Schur). Ston l 1, kˆje w sugklðnousa akoloujða eðnai sugklðnousa. Apìdeixh. Apì to Je rhma 5.6.1 kai apì to je rhma Alaoglu, h (B l, w ) eðnai sumpag c metrikìc q roc. Mia metrik pou orðzei thn w topologða eðnai h d(a, b) = n=1 a n b n 2 n. w 'Estw x k = (x kn ) l 1 me x k 0. Ja deðxoume ìti xk 1 0. Gia kˆje ε > 0 kai gia kˆje m N orðzoume { } F m = a = (a n ) B l : a n x kn ε 3, k m. Kˆje F m eðnai w kleistì sônolo kai apì thn x k w 0 èpetai ìti Bl = m=1 F m (exhg ste giatð). Apì to je rhma tou Baire, upˆrqei m N ste to F m na èqei mh kenì w eswterikì. Dhlad, upˆrqoun a = (a n ) F m kai δ > 0 ste d(a, b) < δ = b F m. Epilègoume N N ste n=n+1 1 2 n < δ 2 kai, gia kˆje k N orðzoume bk B l me b k n = a n an 1 n N kai b k n = sign(x kn ) an n > N. Parathr ste ìti d(a, b k ) = n=n+1 n=1 a n b k n 2 n n=n+1 2 2 n < δ,

dhlad b k F m gia kˆje k N. 'Estw k m. AfoÔ b k F m, èqoume 'Omwc x k isqôei 5.6 Metrikopoihsimothta kai diaqwrisimothta 109 b k N nx kn = a n x kn + n=1 n=1 n=n+1 x kn ε 3. w 0, ˆra mporoôme na broôme m1 m ste gia kˆje k m 1 na N x kn < ε 3. n=1 Sunduˆzontac tic dôo anisìthtec blèpoume ìti, gia kˆje k m 1, x k 1 = N N N x kn = x kn + x kn + a n x kn a n x kn n=1 n=n+1 n=1 n=1 N N N x kn + x kn + a n x kn + a n x kn n=1 n=1 ε 3 + ε 3 + ε 3 = ε, n=n+1 ìpou, gia na frˆxoume ton teleutaðo ìro, qrhsimopoi same to gegonìc ìti a n 1 gia kˆje n N. To Je rhma 5.6.3 deðqnei ìti h (B l1, w) den eðnai metrikopoi simh. An tan, tìte h w topologða kai h topologða ja tautðzontan ston l 1 (exhg ste giatð). Autì ìmwc den mporeð na isqôei se ènan apeirodiˆstato q ro. To antðstoiqo tou Jewr matoc 5.6.3 den isqôei ston l p, 1 < p <. n=1 n=1 Ask seic 1. 'Estw X ènac apeirodiˆstatoc q roc Banach. w (a) DeÐxte ìti S X = BX. (b) DeÐxte ìti h sunˆrthsh : (X, w) R me x x den eðnai suneq c se kanèna shmeðo tou X. 2. 'Estw {x n} akoloujða ston l p, 1 p <. Grˆfoume x n = (x nk ) k N. (a) An 1 < p <, deðxte ìti x w n 0 an kai mìno an (i) upˆrqei M > 0 ste x n p M gia kˆje n, kai (ii) lim x nk = 0 gia kˆje k. n w (b) An p = 1, deðxte ìti x n 0 an kai mìno an (i) upˆrqei M > 0 ste x n 1 M gia kˆje n, kai (ii) lim x nk = 0 gia kˆje k. n 3. 'Estw (f n ) akoloujða ston C[0, 1] me f n 1 gia kˆje n. DeÐxte ìti f w n 0 an kai mìno an f n (t) 0 gia kˆje t [0, 1]. 4. Ston l 1 jewroôme th sun jh basik akoloujða {e n }. DeÐxte ìti e n w 0 allˆ den upˆrqei akoloujða (y k ) kurt n sunduasm n twn e n me y k 1 0.

110 Asjeneic topologiec 5. 'Estw (e n ) h sun jhc bˆsh tou l 2. JewroÔme to sônolo A = {e m + me n : 1 m < n, m, n N}. DeÐxte ìti to 0 an kei sthn w kleist j kh tou A allˆ den upˆrqei akoloujða (a k ) sto A me a w k 0. 6. 'Estw X ènac q roc Banach. DeÐxte ìti o X eðnai w akoloujiakˆ pl rhc: an x n X kai gia kˆje x X h akoloujða (x n(x)) eðnai akoloujða Cauchy, tìte upˆrqei x 0 X ste x w n x 0. 7. SÔmfwna me to je rhma Mazur, an x w n 0 ston q ro Banach X, tìte upˆrqei akoloujða (y k ) kurt n sunduasm n twn x n me y k 0. Den eðnai ìmwc swstì ìti mporoôme na pˆroume y k = 1 (x k 1 + + x k ) ston parapˆnw isqurismì. (a) 'Ena parˆdeigma eðnai to ex c: Ston L 2 ( π, π) jewroôme thn akoloujða x n (t) = 1 e ikt. n DeÐxte ìti x w n 0. BreÐte sugkekrimènh akoloujða (y k ) kurt n sunduasm n twn x n me y k 2 0. DeÐxte ìmwc ìti x 1 + + x n n 2 n 2 k=1 0 ìtan n. (b) DeÐxte ìti an f n L 2 ( π, π) kai f n (f n ) ste 0 ìtan n. 2 f k 1 + + f kn n w 0, tìte upˆrqei upakoloujða (f kn ) thc 8. 'Estw X ènac autopaj c q roc Banach, èstw (x n ) mia fragmènh akoloujða ston X kai x 0 X. OrÐzoume K n = co{x m : m n}. DeÐxte ìti x w n x 0 an kai mìno an n=1 K n = {x 0 }. 9. 'Estw (x n ) fragmènh akoloujða se ènan q ro Banach X. Upojètoume ìti upˆrqei akoloujða (x k) ston X me X = span{x k : k N} kai lim n x k(x n ) = 0 gia kˆje k. DeÐxte ìti x w n 0. 10. 'Estw (x n) akoloujða Cauchy se ènan q ro X me nìrma. An x w n 0 tìte x n 0. 11. 'Estw x n c 0 me x w n 0. DeÐxte ìti upˆrqei upakoloujða (x kn ) thc (x n) ste x k 1 + + x kn n 0 ìtan n. 12. 'Estw X ènac q roc Banach kai èstw A X. DeÐxte ìti to A diaqwrðzei ta shmeða tou X an kai mìno an span(a) w = X.

Kefˆlaio 6 Je rhma Krein Milman 6.1 AkraÐa shmeða 'Estw X ènac pragmatikìc (gia aplìthta) grammikìc q roc. Upojètoume ìti K eðnai èna mh kenì kurtì uposônolo tou X. (b) 'Ena shmeðo x K lègetai akraðo shmeðo tou K an gia kˆje y, z K kai gia kˆje λ (0, 1) isqôei (1 λ)y + λz A = x = y = z. Dhlad, to x den eðnai eswterikì shmeðo kˆpoiou eujôgrammou tm matoc [y, z] me ˆkra y z K. Grˆfoume ex(k) gia to sônolo twn akraðwn shmeðwn tou K. (a) 'Ena mh kenì uposônolo A tou K lègetai akraðo an: gia kˆje y, z K kai gia kˆje λ (0, 1) isqôei (1 λ)y + λz A = y A kai z A. Dhlad, an to A eðnai akraðo kai an, gia kˆpoia y, z K, kˆpoio eswterikì shmeðo tou eujôgrammou tm matoc [y, z] an kei sto A, tìte ta y, z (ˆskhsh: kai ìla ta shmeða tou [y, z]) an koun sto A. Parathr ste ìti èna shmeðo x K eðnai akraðo an kai mìno an to {x} eðnai akraðo uposônolo tou K. ParadeÐgmata 1. JewroÔme ton l n 2 = (R n, 2 ) kai ta sônola B n = {x : x 2 1}, D n = {x : x 2 < 1}, S n = {x : x 2 = 1}. Parathr ste ìti: ex(b n ) = S n kai ex(d n ) =. 2. An P eðnai èna kurtì polôgwno sto epðpedo, tìte kˆje akm tou P eðnai akraðo uposônolo tou P. To ex(p ) eðnai to sônolo twn koruf n tou P. 3. Se kˆje q ro X me nìrma isqôei ex(b X ) S X = {x : x = 1}. Upˆrqoun ìmwc q roi X gia touc opoðouc ex(b X ) =. Gia parˆdeigma, an X = L 1 [0, 1],

112 Jewrhma Krein Milman tìte kˆje f S X grˆfetai sth morf f = g+h 2 ìpou g h kai g, h S X : gia to skopì autì, arkeð na jewr soume x (0, 1) ste x f dλ = 1/2 kai na 0 orðsoume h = 2fχ [0,x], g = 2fχ (x,1]. Tìte, 1 0 g dλ = 1 h dλ = 1, dhlad 0 h, g S X, kai f = g+h. AfoÔ 2 h, g f, èpetai ìti f / ex(b X). Sunep c, ex(b X ) =. H epìmenh Prìtash deðqnei ìti, parˆ to Parˆdeigma 3, ta mh kenˆ kurtˆ sumpag uposônola opoioud pote topikˆ kurtoô q rou èqoun akraða shmeða. Prìtash 6.1.1. 'Estw X ènac topikˆ kurtìc q roc kai èstw K èna mh kenì, sumpagèc kai kurtì uposônolo tou X. Tìte, ex(k). Apìdeixh. JewroÔme thn oikogèneia A ìlwn twn kleist n, mh ken n, akraðwn uposunìlwn tou K. H A eðnai mh ken, afoô K A. OrÐzoume merik diˆtaxh sthn A wc ex c: an A 1, A 2 A, tìte A 1 A 2 an kai mìno an A 1 A 2. To merikˆ diatetagmèno sônolo (A, ) ikanopoieð tic upojèseic tou L mmatoc tou Zorn: an (A i ) i I eðnai mia alusðda sthn A, tìte to sônolo A 0 := A i eðnai i I ˆnw frˆgma thc sthn A. Prˆgmati: (i) To A 0 = A i eðnai mh kenì. An upojèsoume to antðjeto, tìte ta (anoiktˆ i I sto K) sônola K \ A i, i I, sqhmatðzoun anoikt kˆluyh tou K. Qrhsimopoi ntac th sumpˆgeia tou K pernˆme se peperasmènh upokˆluyh. IsodÔnama, upˆrqoun i 1,..., i n I ste A i1 A in =. Apì to gegonìc ìti h (A i ) i I eðnai alusðda, blèpoume t ra ìti kˆpoio apì ta A ij, 1 j n, eðnai kenì. Autì eðnai ˆtopo. (ii) Profan c, A j A 0 = A i gia kˆje j I (o orismìc thc ). i I (iii) To sônolo A 0 eðnai akraðo uposônolo tou K, dhlad A 0 A. Prˆgmati, èstw a A 0 kai b, c K, 0 < λ < 1 me a = (1 λ)b + λc. 'Estw i I. AfoÔ a A i kai kˆje A i eðnai akraðo uposônolo tou K, èpetai ìti b, c A i. AfoÔ to i I tan tuqìn, blèpoume telikˆ ìti b, c A 0. Apì to L mma tou Zorn upˆrqei megistikì A A. Ja deðxoume ìti to A eðnai monosônolo, dhlad upˆrqei x K ste A = {x}. Tìte, x ex(k). Ac upojèsoume ìti upˆrqoun x, y A me x y. Upˆrqei suneqèc grammikì sunarthsoeidèc f : X R ste f(x) f(y). Dhlad, h f den eðnai stajer sto A. JewroÔme to sônolo A 1 = {z A : f(z) = min{f(w) : w A}}. To sônolo A eðnai sumpagèc, ˆra to A 1 eðnai mh kenì kai kleistì uposônolo tou K, apì th sunèqeia thc f. Epiplèon, to A 1 eðnai akraðo uposônolo tou K: 'Estw z 1, z 2 K, z A 1 kai z = (1 λ)z 1 + λz 2 gia kˆpoio λ (0, 1). Tìte, z 1, z 2 A, diìti to A eðnai akraðo kai z A 1 A. 'Omwc, min w A f(w) = f(z) = (1 λ)f(z 1 ) + λf(z 2 ) kai z 1, z 2 A, ˆra f(z 1 ) = f(z 2 ) = min w A f(w).

6.2 Jewrhma Krein Milman 113 Dhlad, z 1, z 2 A 1. AfoÔ to A 1 eðnai akraðo uposônolo tou K kai A 1 A, èqoume A A 1. 'Omwc to A eðnai megistikì, ˆra A = A 1. Autì eðnai ˆtopo, afoô tìte h f ja tan stajer sto A = A 1. DeÐxame ìti kˆje megistikì stoiqeðo A A eðnai monosônolo. H A èqei toulˆqiston èna megistikì stoiqeðo, ˆra upˆrqei toulˆqiston èna x K ste to {x} na eðnai akraðo uposônolo tou K. Kˆje tètoio x an kei sto ex(k). 'Ara, ex(k). ShmeÐwsh. Parathr ste ìti h sumpˆgeia tou K qrhsimopoi jhke ousiastikˆ gia na deðxoume ìti h A ikanopoieð tic upojèseic tou L mmatoc tou Zorn. To parˆdeigma 3 deðqnei ìti den mporoôme na antikatast soume thn upìjesh ìti to K eðnai sumpagèc me thn asjenèsterh upìjesh ìti to K eðnai kleistì. 6.2 Je rhma Krein Milman H Prìtash 6.1.1 thc prohgoômenhc Paragrˆfou mac lèei ìti kˆje mh kenì, kurtì kai sumpagèc uposônolo K enìc topikˆ kurtoô q rou èqei akraða shmeða. To je rhma Krein Milman isqurðzetai ìti to K èqei {pollˆ} akraða shmeða: Je rhma 6.2.1 (Krein Milman). 'Estw X ènac topikˆ kurtìc q roc kai è- stw K èna mh kenì, sumpagèc kai kurtì uposônolo tou X. Tìte, K = co(ex(k)). Apìdeixh. Ac upojèsoume ìti upˆrqei x K \ co(ex(k)). Tìte, ta sônola {x} kai co(ex(k)) diaqwrðzontai gn sia. Upˆrqoun suneqèc grammikì sunarthsoeidèc f : X R kai λ R ste ( ) OrÐzoume sup{f(y) : y ex(k)} = sup{f(z) : z co(ex(k))} < λ < f(x). K 1 = {z K : f(z) = max w K f(w)}. ParathroÔme ta ex c: (i) To K 1 eðnai mh kenì, sumpagèc, kurtì (apì th sunèqeia kai th grammikìthta thc f). (ii) K 1 ex(k) = (apì thn ( )). (iii) Kˆje akraðo shmeðo tou K 1 eðnai akraðo shmeðo tou K (qrhsimopoi ste epiqeðrhma anˆlogo me ekeðno thc apìdeixhc thc Prìtashc 6.1.1). Apì to (i) kai thn Prìtash 6.1.1 èqoume ex(k 1 ). 'Omwc, ex(k 1 ) ex(k) K 1 lìgw tou (iii). Apì to (ii) odhgoômaste se ˆtopo. Sthn epìmenh Parˆgrafo ja doôme kˆpoiec efarmogèc tou jewr matoc Krein Milman. DÐnoume ed èna pr to parˆdeigma: sômfwna me to je rhma Alaoglu, an X eðnai ènac q roc me nìrma tìte h B X eðnai èna mh kenì, sumpagèc kai kurtì uposônolo tou topikˆ kurtoô q rou (X, w ). 'Etsi, èqoume to ex c. Prìtash 6.2.2. 'Estw X ènac q roc me nìrma. Tìte, B X = co(ex(b X )) w. Eidikìtera, ex(b X ).

114 Jewrhma Krein Milman Dhlad, h monadiaða mpˆla tou duðkoô q rou X opoioud pote q rou me nìrma X prèpei na èqei (arketˆ) akraða shmeða. ParadeÐgmata (a) H monadiaða mpˆla tou L 1 [0, 1] den èqei akraða shmeða (deðte to Parˆdeigma 3 sthn prohgoômenh parˆgrafo). SÔmfwna me thn Prìtash 6.2.2, o L 1 [0, 1] den mporeð na eðnai isometrikˆ isìmorfoc me ton duðkì q ro kˆpoiou q rou me nìrma. (b) To Ðdio isqôei gia ton c 0 : èstw x B c0, dhlad x = sup x n 1. AfoÔ n x n 0, upˆrqei m N ste x m < 1. OrÐzoume 2 y, z c 0 me y n = z n = x n an n m kai y m = x m + 1, 2 z m = x m 1. Tìte, 2 y, z B c 0, y, z x kai x = y+z. 2 'Ara, x / ex(b c0 ). Dhlad, h monadiaða mpˆla tou c 0 den èqei akraða shmeða. Ta epìmena dôo jewr mata qrhsimopoioôntai suqnˆ ìtan efarmìzoume to je- rhma Krein Milman. Je rhma 6.2.3. 'Estw X ènac topikˆ kurtìc q roc, K èna mh kenì, sumpagèc kurtì uposônolo tou X kai F K me thn idiìthta K = co(f ). Tìte, ex(k) F. Apìdeixh. MporoÔme na upojèsoume ìti to F eðnai kleistì. Ac upojèsoume ìti x 0 eðnai èna akraðo shmeðo tou K pou den an kei sto F. MporoÔme na broôme anoikt, kurt kai summetrik perioq U tou 0 ste (x 0 + U) (F + U) = (exhg ste giatð). Tìte, x 0 / F + U. To F eðnai sumpagèc, ˆra upˆrqoun y 1,..., y n F ste F n (y j + U). Gia j = 1,..., n orðzoume K j = co(f (y j + U)). To sônolo co(f (y j + U)) perièqetai sto kurtì sônolo y j + U, ˆra K j y j + U = y j + U, j = 1,..., n. Isqurismìc. K = co(k 1 K n ). Apìdeixh. EÐnai fanerì ìti K j co(f ) = K gia kˆje j = 1,..., n kai to K eðnai kurtì, epomènwc co(k 1 K n ) K. Gia ton antðstrofo egkleismì, parathr ste ìti F n (F (y j + U)), ˆra F K 1 K n, ap ìpou èpetai ìti j=1 K = co(f ) co(k 1 K n ). 'Omwc, to sônolo co(k 1 K n ) eðnai sumpagèc: eðnai h eikìna tou sumpagoôc C = K 1 K n {t = (t j ) j n : 0 t j 1, n j=1 t j = 1} mèsw thc suneqoôc apeikìnishc Φ(x 1,..., x n, t) = n j=1 t jx j. 'Ara, K co(k 1 K n ) = co(k 1 K n ). Apì ton isqurismì èpetai ìti to x 0 K grˆfetai sth morf x 0 = a 1 x 1 + + a n x n, ìpou x j K j, a j 0 kai n j=1 a j = 1. 'Omwc, to x 0 eðnai akraðo shmeðo tou K, sunep c x 0 = x j gia kˆpoio j n (exhg ste giatð). Tìte, x 0 K j y j + U F + U, j=1

to opoðo eðnai ˆtopo. 6.3 Jewrhma anaparastashc tou Riesz 115 ShmeÐwsh. Den eðnai genikˆ swstì ìti to sônolo ex(k) twn akraðwn shmeðwn enìc sumpagoôc kurtoô sunìlou eðnai kleistì. 'Ena aplì parˆdeigma ston R 3 dðnei o diplìc k noc K = co({a, B} S), ìpou A = (0, 0, 1), B = (0, 0, 1) kai S = {(x, y, 0) : (x 1/2) 2 + (y 1/2) 2 1/4}. To sônolo twn akraðwn shmeðwn tou K eðnai to {A, B} (S \ {0}) (ˆskhsh) to opoðo den eðnai kleistì. Je rhma 6.2.4. 'Estw X, Y dôo topikˆ kurtoð q roi, K mh kenì, sumpagèc kai kurtì uposônolo tou X kai T : X Y suneq c grammik apeikìnish. Tìte, to T (K) eðnai sumpagèc kai kurtì uposônolo tou Y kai gia kˆje y ex(t (K)) upˆrqei x ex(k) ste T (x) = y. Apìdeixh. O pr toc isqurismìc eðnai profan c. 'Estw y ex(t (K)). To sônolo K y = {x K : T (x) = y} eðnai kleistì (ˆra, sumpagèc) kai kurtì. Upˆrqei loipìn x 0 to opoðo eðnai akraðo shmeðo tou K y. Tìte, x 0 ex(k): an x 0 = (1 λ)x 1 + λx 2, ìpou 0 < λ < 1 kai x 1, x 2 K, tìte y = T (x 0 ) = (1 λ)t (x 1 ) + λt (x 2 ), kai autì shmaðnei ìti T (x 1 ) = T (x 2 ) = y, diìti y ex(t (K)). 'Ara, x 1, x 2 K y kai, afoô x 0 ex(k y ), èpetai ìti x 0 = x 1 = x 2. ShmeÐwsh. Den isqôei kˆpoiou eðdouc antðstrofo. Jewr ste, gia parˆdeigma, thn probol T (x, y) = x apì to monadiaðo dðsko sto [ 1, 1]. 6.3 Je rhma anaparˆstashc tou Riesz 'Estw K ènac sumpag c metrikìc q roc kai èstw C(K) o q roc twn suneq n pragmatik n sunart sewn sto K, me thn topologða thc omoiìmorfhc sôgklishc. 'Ena grammikì sunarthsoeidèc I : C(K) R lègetai jetikì an I(f) 0 gia kˆje f C(K) me f(x) 0 gia kˆje x K. Je rhma 6.3.1 (Riesz). 'Estw I : C(K) R fragmèno jetikì grammikì sunarthsoeidèc. Upˆrqei monadikì kanonikì Borel mètro µ sto K ste ( ) I(f) = gia kˆje f C(K). K f dµ ShmeÐwsh. 'Ena mètro µ sthn B(K) lègetai kanonikì an µ(k) < kai gia kˆje A B(K) isqôoun ta ex c: (i) µ(a) = inf{µ(u) : U anoiktì kai A U}, (ii) µ(a) = sup{µ(c) : C sumpagèc kai C A}. An µ eðnai èna proshmasmèno Borel mètro sto K, tìte to µ lègetai kanonikì an ta µ + kai µ eðnai kanonikˆ. O q roc M(K) twn proshmasmènwn kanonik n Borel mètrwn eðnai grammikìc q roc. H µ = µ (K) = µ + (K) + µ (K) eðnai nìrma ston M(K) kai o (M(K), ) eðnai q roc Banach.

116 Jewrhma Krein Milman Apìdeixh tou Jewr matoc 6.3.1. DeÐqnoume pr ta th monadikìthta tou µ. 'Estw ν èna ˆllo kanonikì Borel mètro sto K ste ( ) I(f) = K f dν = K f dµ gia kˆje f C(K). 'Estw C sumpagèc uposônolo tou K kai U anoiktì uposônolo tou K me K U. Apì to L mma tou Urysohn upˆrqei f C(K) me 0 f 1, f 1 sto C kai supp(f) U. Tìte, µ(c) = K χ C dµ K f dµ = K f dν K χ U dν = ν(u). PaÐrnontac infimum wc proc ìla ta U C kai qrhsimopoi ntac thn kanonikìthta tou ν, sumperaðnoume ìti µ(c) ν(c) gia kˆje C. PaÐrnontac supremum wc proc ìla ta C U kai qrhsimopoi ntac thn kanonikìthta tou µ, sumperaðnoume ìti µ(u) ν(u) gia kˆje U. Lìgw summetrðac, èqoume µ(c) = ν(c) kai µ(u) = ν(u) gia kˆje sumpagèc C kai kˆje anoiktì U. Apì thn kanonikìthta twn µ kai ν èpetai ìti µ ν. DeÐqnoume t ra thn Ôparxh: gia kˆje U K anoiktì, orðzoume µ(u) := sup{i(f) : f U}, ìpou f U shmaðnei ìti 0 f 1 kai supp(f) U. AfoÔ to I eðnai jetikì sunarthsoeidèc, apì ton orismì eðnai profanèc ìti µ(u) 0. EpÐshc, an U 1, U 2 eðnai anoiktˆ uposônola tou K kai U 1 U 2 tìte µ(u 1 ) µ(u 2 ). Apì thn teleutaða parat rhsh èpetai ìti an, gia kˆje A K, orðsoume (1) µ (A) = inf{µ(u) : U anoiktì kai A U}, èqoume µ (U) = µ(u) gia kˆje anoiktì U K. EpÐshc, an f U èqoume I(f) I f I, ˆra µ (U) I. 'Epetai ìti µ (A) I gia kˆje A K. 1. DeÐqnoume ìti to µ eðnai exwterikì mètro. To µ eðnai monìtono: an A, B B(K) kai A B, tìte µ (A) µ (B). Autì eðnai fanerì apì thn (1). EÐnai epðshc fanerì ìti µ ( ) = 0. An (A i ) eðnai mia akoloujða uposunìlwn tou K, tìte ( ) (2) µ A i µ (A i ). Apìdeixh. DeÐqnoume pr ta ìti µ (U 1 U 2 ) µ (U 1 )+µ (U 2 ) an ta U 1, U 2 eðnai anoiktˆ. 'Estw g U 1 U 2. Apì to L mma 1.4.3 upˆrqoun h i U i, i = 1, 2 ste h 1 +h 2 1 sto supp(g). Tìte, g = gh 1 +gh 2, ˆra I(g) = I(gh 1 )+I(gh 2 ). 'Omwc, gh i U i, ˆra I(g) µ (U 1 ) + µ (U 2 ). 'Epetai ìti µ (U 1 U 2 ) = sup{i(g) : g U 1 U 2 } µ (U 1 ) + µ (U 2 ). Epagwgikˆ, an U 1,..., U n eðnai anoiktˆ uposônola tou K èqoume µ (U 1 U n ) n µ (U i ).

6.3 Jewrhma anaparastashc tou Riesz 117 'Estw t ra (A i ) mia akoloujða uposunìlwn tou K. JewroÔme tuqìn ε > 0. Upˆrqoun anoiktˆ sônola U i A i me µ (U i ) < µ (A i ) + ε/2 i. Tìte, 'Estw f f n U i. Tìte, H f tan tuqoôsa, ˆra ( ) ( ) µ A i µ U i. U i. Lìgw thc sumpˆgeiac tou supp(f), upˆrqei n N ste ( n ) I(f) µ U i n µ (U i ) < µ (A i ) + ε. ( ) µ A i µ (A i ) + ε, kai èpetai to zhtoômeno. Apì to je rhma tou Karajeodwr, orðzetai h σ ˆlgebra twn µ metr simwn uposunìlwn tou K kai o periorismìc tou µ se aut n eðnai mètro. 2. DeÐqnoume ìti kˆje anoiktì U K eðnai µ metr simo. 'Estw A K. BrÐskoume anoiktì V A me µ (V ) = µ(v ) < µ (A) + ε kai f U V me I(f) > µ(u V ) ε. To V \ supp(f) eðnai anoiktì, ˆra upˆrqei g V \ supp(f) ste I(g) > µ(v \ supp(f)) ε. Tìte, f + g V, ˆra µ (A) + ε > µ(v ) I(f + g) = I(f) + I(g) > µ(u V ) + µ(v \ supp(f)) 2ε µ (A U) + µ (A \ U) 2ε. Autì apodeiknôei ìti µ (A) µ (A U) + µ (A \ U), dhlad to U eðnai µ metr simo. 'Ara, o periorismìc tou µ sthn B(K) eðnai mètro. 3. An C K sumpagèc, tìte µ(c) = inf{i(f) : C f}, ìpou C f shmaðnei ìti 0 f 1 kai f 1 sto C. Apìdeixh. 'Estw C f. JewroÔme to anoiktì sônolo U = {x K : f(x) > 1 ε} C, ìpou 0 < ε < 1. An g U, tìte g 1 1 1 εf sto K, ˆra I(g) I(f). 1 ε 'Ara, µ(c) µ(u) 1 1 εi(f). 'Epetai ìti µ(c) I(f), dhlad µ(c) inf{i(f) : C f}. AntÐstrofa, gia tuqìn ε > 0, upˆrqei anoiktì U C me µ(u) < µ(c) + ε. MporoÔme na broôme g C(K) me C g U. Tìte, AfoÔ to ε > 0 tan tuqìn, èpetai ìti I(g) µ(u) < µ(c) + ε. inf{i(f) : C f} µ(c).

118 Jewrhma Krein Milman 4. To µ eðnai kanonikì. Apìdeixh. Apì ton orismì èqoume automˆtwc thn pr th idiìthta tou kanonikoô mètrou: gia kˆje A B(K) isqôei µ(a) = µ (A) = inf{µ(u) : U anoiktì kai A U}. Gia thn deôterh idiìthta, jewroôme tuqìn A B(K) kai ε > 0, kai brðskoume anoiktì U A ste µ(u) < µ(a) + ε. Upˆrqei g U me I(g) > µ(u) ε. Jètoume C = supp(g) U. An C f, tìte g f ˆra I(g) I(f). Sunep c, µ(c) = inf{i(f) : C f} I(g). Dhlad, upˆrqei sumpagèc C U me µ(c) > µ(u) ε. AfoÔ µ(u \ A) < ε, upˆrqei anoiktì V U \ A me µ(v ) < 2ε. Tìte, to F = C \ V eðnai sumpagèc uposônolo tou A kai A \ F (U \ C) V, ˆra µ(a) µ(f ) µ(u \ C) + µ(v ) < 3ε. 'Epetai ìti µ(a) = sup{µ(f ) : F sumpagèc kai F A}. 5. IsqÔei I(f) = f dµ gia kˆje f C(K). K Apìdeixh. MporoÔme (jewr ntac tic f + kai f ) na upojèsoume ìti f 0, kai (pollaplasiˆzontac me katˆllhlh stajerˆ) mporoôme na upojèsoume ìti 0 f 1. StajeropoioÔme N N kai, gia 0 k N, orðzoume B k = {x K : f(x) k/n}. 'Eqoume B 0 = K kai kˆje B k eðnai sumpagèc sônolo. Gia s = 0, 1,..., N 1 orðzoume f s = min { { max f, s }, s + 1 } s N N N. 1 Kˆje f s eðnai suneq c sto K kai N χ B s+1 f s 1 N χ B s. Parathr ste ìti f = f 0 + f 1 + + f N 1. 'Epetai ìti 1 N (µ(b 1) + + µ(b N )) K f dµ 1 N (µ(b 0) + + µ(b N 1 )). Gia kˆje 0 s N 1 èqoume χ Bs+1 Nf s, ˆra µ(b s+1 ) I(Nf s ) = NI(f s ). EpÐshc, Nf s χ Bs, ˆra Nf s U gia kˆje anoiktì U B s, to opoðo mac dðnei NI(f s ) µ(u) gia kˆje anoiktì U B s. 'Epetai ìti µ(b s ) = µ (B s ) NI(f s ). SunoyÐzontac, ap' ìpou blèpoume ìti Dhlad, µ(b s+1 ) N I(f s ) µ(b s) N, 1 N (µ(b 1) + + µ(b N )) I(f) 1 N (µ(b 0) + + µ(b N 1 )). K f dµ I(f) µ(b 0) µ(b N ) N kai afoô to N tan tuqìn, I(f) = K f dµ. µ(k) N, MporoÔme t ra na deðxoume ìti kˆje fragmèno grammikì sunarthsoeidèc I : C(K) R anaparðstatai apì monadikì proshmasmèno kanonikì Borel mètro sto K.

6.3 Jewrhma anaparastashc tou Riesz 119 Je rhma 6.3.2. 'Estw I : C(K) R fragmèno grammikì sunarthsoeidèc. Upˆrqei monadikì proshmasmèno kanonikì Borel mètro µ sto K ste ( ) I(f) = gia kˆje f C(K). Apìdeixh. An f C(K) me f 0, orðzoume K f dµ I + (f) = sup{i(g) : g C(K), 0 g f sto K}. An t ra f C(K), grˆfoume thn f sth morf f = f + f kai orðzoume Tèloc, orðzoume I + (f) = I + (f + ) I + (f ). I (f) = I + (f) I(f). DeÐxte ìti ta I +, I eðnai fragmèna jetikˆ grammikˆ sunarthsoeid ston C(K). Apì to Je rhma 6.3.1, upˆrqoun monadikˆ kanonikˆ Borel mètra µ +, µ sto K ste I + (f) = f dµ + kai I (f) = f dµ. K To I anaparðstatai apì to proshmasmèno mètro µ = µ + µ. Oi leptomèreiec af nontai wc ˆskhsh. Prìtash 6.3.3. 'Estw µ M(K). OrÐzoume I µ : C(K) R me I µ (f) = K f dµ. Tìte, to I µ eðnai fragmèno grammikì sunarthsoeidèc kai I = µ. Apìdeixh. H grammikìthta tou I µ eðnai faner. EpÐshc, I µ (f) K K f d µ µ f gia kˆje f C(K), ˆra I µ µ. Gia thn antðstrofh anisìthta qrhsimopoioôme to gegonìc ìti upˆrqei µ metr simh sunˆrthsh h : K R me h(x) = 1 gia kˆje x K, h opoða ikanopoieð thn µ(a) = A hd µ gia kˆje A B(K). Gia tuqìn ε > 0 brðskoume (ˆskhsh) g C(K) me g 1 kai Tìte, µ = µ (K) = K K h g d µ < ε. h 2 d µ = K h dµ < I µ (g) + ε. AfoÔ g 1 kai to ε > 0 tan tuqìn, èpetai ìti µ sup{ I µ (g) : g 1} = I µ. Sunduˆzontac ta parapˆnw paðrnoume to je rhma anaparˆstashc tou Riesz:

120 Jewrhma Krein Milman Je rhma 6.3.4. 'Estw K ènac sumpag c metrikìc q roc. Tìte, o M(K) eðnai isometrikˆ isìmorfoc me ton [C(K)]. Apìdeixh. Apì ta prohgoômena jewr mata, h Φ : M(K) [C(K)] me µ I µ eðnai isometrikìc isomorfismìc. 6.4 Efarmogèc 6.4aþ Je rhma Stone Weierstrass 'Estw K ènac sumpag c metrikìc q roc kai èstw C(K) o q roc twn suneq n pragmatik n sunart sewn sto K, me thn topologða thc omoiìmorfhc sôgklishc. 'Ena uposônolo A tou C(K) lègetai upoˆlgebra tou C(K) an gia kˆje f, g A kai gia kˆje t R èqoume tf, f + g, f g A. Je rhma 6.4.1 (Stone Weierstrass). 'Estw A mia upoˆlgebra tou C(K) me tic ex c idiìthtec: (i) H stajer sunˆrthsh 1 A. (ii) H A diaqwrðzei ta shmeða tou K: an x y sto K, tìte upˆrqei f A ste f(x) f(y). Tìte, A = C(K). Apìdeixh. ArkeÐ na deðxoume ìti: an µ (C(K), ) kai f dµ = 0 gia K kˆje f A, tìte µ 0. JewroÔme loipìn ton upìqwro N(A) = {µ : f A, f dµ = 0} kai th K monadiaða tou mpˆla B. To B eðnai w kleistì uposônolo thc monadiaðac mpˆlac tou (C(K), ), kai sômfwna me to je rhma Alaoglu eðnai w sumpagèc. Apì to je rhma Krein Milman èpetai ìti ex(b). Ac upojèsoume ìti N(A) {0}. 'Estw µ ex(b). 'Eqoume µ 0 (exhg ste giatð) kai an C = K \ {V : V anoiktì kai µ (V ) = 0} eðnai o forèac tou µ, tìte (i) µ (C) = 1 = µ (K), ˆra C, (ii) K f dµ = C f dµ gia kˆje f C(K). 'Estw x 0 C. Ja deðxoume ìti C = {x 0 }: JewroÔme tuqìn x K me x x 0. Upˆrqei f 1 A pou diaqwrðzei ta x 0, x, dhlad f 1 (x 0 ) f 1 (x) =: b. H stajer sunˆrthsh b A, ˆra h sunˆrthsh f 2 = f 1 b A kai f 2 (x 0 ) 0 = f 2 (x). EpÐshc, h sunˆrthsh f 3 = f2 2 A kai ikanopoieð tic f 3 0, f 3 (x) = 0, f 3 (x 0 ) > 0. OrÐzoume f := f 3 1 + f 3. Tìte, f A, 0 f < 1 sto K kai f(x 0 ) > 0, f(x) = 0. Gia kˆje g A èqoume gf A kai g(1 f) A, ˆra gf dµ = 0 kai g(1 f) dµ = 0.

6.4 Efarmogec 121 Dhlad, ta mètra µ 1, µ 2 me dµ 1 = f dµ kai dµ 2 = (1 f) dµ an koun ston N(A). EpÐshc, µ 1 (K) = K f d µ kai µ 2 (K) = (1 f) d µ, diìti 0 f < 1 sto K K. Isqurismìc. µ 1 (K) = α (0, 1). Apìdeixh. AfoÔ f(x 0 ) > 0, upˆrqoun perioq U tou x 0 kai ε > 0 ste f(y) > ε gia kˆje y U. 'Eqoume U C diìti x 0 U C, ˆra α = K f d µ U f d µ ε µ (U) > 0. Me ton Ðdio trìpo, qrhsimopoi ntac thn f(x 0 ) < 1, blèpoume ìti EpÐshc, Grˆfoume µ 2 (K) = K α = K f d µ < 1. (1 f) d µ = µ (K) α = 1 α. ( ) ( ) µ1 µ2 µ = α + (1 α) = αν 1 + (1 α)ν 2. µ 1 (K) µ 2 (K) AfoÔ µ 1, µ 2 N(A), èqoume ν 1, ν 2 B. 'Omwc, µ ex(b). Sunep c, µ = ν 1 = ν 2. Dhlad, K h dµ = K hf α dµ gia kˆje h C(K). Gia na sumbaðnei kˆti tètoio, prèpei upoqrewtikˆ na èqoume f = α µ sqedìn pantoô. 'Omwc h f eðnai suneq c, ˆra f α sto C (exhg ste giatð). AfoÔ f(x) = 0, sumperaðnoume ìti x / C. To x x 0 tan tuqìn, ˆra C = {x 0 }. Autì me th seirˆ tou shmaðnei ìti µ (K) = 1 kai µ({x 0 }) = ±1. 'Omwc, 1 A kai µ N(A), ˆra 0 = K 1 dµ = 1 µ({x 0 }) = ±1, to opoðo eðnai ˆtopo. H upìjes mac tan ìti N(A) {0}, ˆra N(A) = {0} kai autì deðqnei ìti A = C(K). ShmeÐwsh. To Je rhma 6.4.1 efarmìzetai stic ex c peript seic: (a) K = [0, 1], A = o q roc twn poluwnômwn sto [0, 1]: kˆje suneq c sunˆrthsh f : [0, 1] R proseggðzetai omoiìmorfa apì polu numa. (b) K = [ π, π], A = o q roc twn trigwnometrik n poluwnômwn sto [ π, π]: h A eðnai upoˆlgebra tou C[ π, π] (parathr ste ìti to ginìmeno dôo trigwnometrik n poluwnômwn eðnai trigwnometrikì polu numo kai ìti to {cos x, sin x} diaqwrðzei ta shmeða tou [ π, π]). 'Ara, kˆje suneq c sunˆrthsh f : [ π, π] R proseggðzetai omoiìmorfa apì trigwnometrikˆ polu numa.

122 Jewrhma Krein Milman 6.4bþ Oloklhrwtikèc anaparastˆseic 'Estw X ènac topikˆ kurtìc q roc, K èna mh kenì sumpagèc uposônolo tou X kai µ èna kanonikì Borel mètro pijanìthtac sto K. Lème ìti to x 0 X anaparðstatai apì to µ (eðnai to kèntro bˆrouc tou µ) an gia kˆje suneqèc grammikì sunarthsoeidèc f : X R isqôei K f(x) dµ(x) = f(x 0 ). To pr to mac je rhma exasfalðzei (upì proôpojèseic) thn Ôparxh kèntrou bˆrouc gia to µ. Je rhma 6.4.2. 'Estw F mh kenì kleistì uposônolo tou topikˆ kurtoô q rou X ste to K = co(f ) na eðnai sumpagèc. Tìte, kˆje kanonikì Borel mètro pijanìthtac sto F èqei monadikì kèntro bˆrouc pou an kei sto K. Apìdeixh. To I(f, µ) := f(x) dµ(x) orðzetai kalˆ gia kˆje suneqèc grammikì F sunarthsoeidèc f : X R. Gia kˆje tètoia f orðzoume to kleistì uperepðpedo An deðxoume ìti E f = {y X : f(y) = I(f, µ)}. ( ) K f E f, tìte kˆje shmeðo autoô tou sunìlou eðnai kèntro bˆrouc tou µ (exhg ste giatð). EÐnai fanerì ìti an up rqan dôo tètoia shmeða x 1 x 2, tìte gia kˆje f X ja eðqame f(x 1 ) = f(x 2 ) = F f(x) dµ(x), to opoðo eðnai ˆtopo afoô o X diaqwrðzei ta shmeða tou X se kˆje topikˆ kurtì q ro. ApodeiknÔoume loipìn thn ( ): afoô to K eðnai sumpagèc, arkeð na deðxoume ìti an f 1,..., f n X tìte K E f1 E fn (exhg ste giatð). JewroÔme thn grammik apeikìnish T : X R n me T (y) = (f 1 (y),..., f n (y)). ArkeÐ na deðxoume ìti ( w := F ) f 1 dµ,..., f n dµ T (K). F 'Estw ìti w / T (K). AfoÔ to T (K) eðnai kurtì sumpagèc uposônolo tou R n, upˆrqei suneqèc grammikì sunarthsoeidèc h : R n R ste ( ) max{h(t (y)) : y K} < h(w). Upˆrqoun b 1,..., b n R ste h(x) = b 1 x 1 + +b n x n gia kˆje x = (x 1,..., x n ) R n. Tìte, h ( ) grˆfetai max y K n b j f j (y) < j=1 n b j f j (x) dµ(x). j=1 F

6.4 Efarmogec 123 An jèsoume m := n j=1 b jf j, apì thn F K kai thn prohgoômenh anisìthta blèpoume ìti max m(y) < m(x) dµ(x). y f F Autì eðnai ˆtopo, afoô to µ eðnai mètro pijanìthtac. To epìmeno je rhma deðqnei ìti kˆje shmeðo thc co(f ) anaparðstatai apì toulˆqiston èna mètro pijanìthtac sto F. Je rhma 6.4.3. 'Estw F mh kenì sumpagèc uposônolo tou topikˆ kurtoô q rou X kai èstw x X. Tìte, x co(f ) an kai mìno an upˆrqei kanonikì Borel mètro pijanìthtac µ sto F pou èqei kèntro bˆrouc to x. Apìdeixh. ( =) 'Estw µ èna mètro pijanìthtac sto F kai èstw x to kèntro bˆrouc tou. An x / co(f ), upˆrqoun f X kai λ R ste max f(y) < λ < f(x). x F AfoÔ to x eðnai to kèntro bˆrouc tou µ, λ < f(x) = F f(y) dµ(y) max f(y) < λ, y F to opoðo eðnai ˆtopo. (= ) 'Estw x co(f ). Upˆrqei dðktuo apì kurtoôc sunduasmoôc n i x i = a ij y ij, j=1 y ij F, i I ste x i x. Gia kˆje i I jewroôme to mètro µ i = n i j=1 a ijδ yij pou dðnei mˆza a ij se kˆje y ij. Kˆje µ i eðnai kanonikì Borel mètro pijanìthtac sto F. AfoÔ h monadiaða mpˆla B tou (C(F ), ) eðnai w sumpag c kai µ i B, w upˆrqei upodðktuo µ l, l L, me µ l µ B. To µ eðnai mètro pijanìthtac sto F kai èqei kèntro bˆrouc to x: gia kˆje f X èqoume n l f(x) = lim f(x l ) = lim a lj f(y lj ) l l = lim l F j=1 f(y) dµ l (y) = F f(y) dµ(y). An to F eðnai dh sumpagèc kai kurtì, tìte kˆje x co(f ) = F anaparðstatai apì to mètro Dirac δ x. To epìmeno je rhma deðqnei ìti to δ x eðnai to monadikì mètro pou èqei kèntro bˆrouc to x an kai mìno an to x eðnai akraðo shmeðo tou F. Je rhma 6.4.4 (Bauer). 'Estw F èna sumpagèc kai kurtì uposônolo tou topikˆ kurtoô q rou X kai èstw x F. Tìte, x ex(f ) an kai mìno an to δ x eðnai to monadikì kanonikì Borel mètro pijanìthtac sto F pou èqei kèntro bˆrouc to x.

124 Jewrhma Krein Milman Apìdeixh. ( =) 'Estw ìti x / ex(f ). Upˆrqoun y z F kai λ (0, 1) ste x = (1 λ)y + λz. Tìte, to x eðnai kèntro bˆrouc tou (1 λ)δ y + λδ z δ x. (= ) 'Estw x ex(f ) kai èstw µ kanonikì Borel mètro pijanìthtac pou èqei kèntro bˆrouc to x. Ja deðxoume ìti µ(f \ {x}) = 0 deðqnontac ìti µ(c) = 0 gia kˆje sumpagèc C F \ {x}. Ac upojèsoume ìti upˆrqei tètoio C me µ(c) > 0. Apì th sumpˆgeia tou C sumperaðnoume ìti upˆrqei y C me thn idiìthta: µ(c U) > 0 gia kˆje perioq U tou y (exhg ste giatð). AfoÔ x y, mporoôme na broôme kurt perioq U tou y ste x / U F (exhg ste gðatð). To U F ikanopoieð thn 0 < µ(u F ) < 1. H dexiˆ anisìthta isqôei giatð, an eðqame µ(u F ) = 1 tìte to kèntro bˆrouc tou µ ja an ke sto sumpagèc kurtì U F apì to Je rhma 6.4.2, to opoðo den mporeð na sumbeð afoô x / U F. OrÐzoume ta mètra pijanìthtac µ 1, µ 2 sto F me µ 1 (B) = µ(b U F ) µ(u F ), µ 2 (B) = µ(b \ U F ), B Borel F. 1 µ(u F ) 'Estw x 1, x 2 F ta kèntra bˆrouc twn µ 1, µ 2. 'Eqoume x 1 U F, ˆra x 1 x. 'Omwc, µ = µ(u F ) µ 1 + (1 µ(u F )) µ 2, ˆra (exhg ste giatð) to opoðo eðnai ˆtopo afoô x ex(f ). x = µ(u F ) x 1 + (1 µ(u F )) x 2, Sunduˆzontac to Je rhma 6.4.3 me to je rhma Krein Milman paðrnoume to ex c qr simo je rhma oloklhrwtik c anaparˆstashc: Je rhma 6.4.5. 'Estw X ènac topikˆ kurtìc q roc kai èstw K èna mh kenì, sumpagèc kai kurtì uposônolo tou X. Gia kˆje x K upˆrqei kanonikì Borel mètro pijanìthtac µ x sto ex(k) ste: gia kˆje suneqèc grammikì sunarthsoeidèc f : X R, f(x) = ex(k) f(y) dµ x (y). Apìdeixh. Apì to je rhma Krein Milman èqoume K = co(ex(k)), opìte efarmìzetai to Je rhma 6.4.3. Ask seic 1. JewroÔme ton q ro X = L [0, 1]. DeÐxte ìti f ex(b X) an kai mìno an f(t) = 1 λ sqedìn pantoô kai sumperˆnate ìti B X co(ex(b X)). 2. 'Estw X ènac q roc Banach me thn idiìthta to ex(b X ) na eðnai peperasmèno sônolo. An dim(x) =, deðxte ìti o X den eðnai isometrikˆ isìmorfoc me duðkì q ro.

6.4 Efarmogec 125 3. JewroÔme ton X = (C[0, 1], ). BreÐte to ex(b X ). DeÐxte ìti o X den eðnai isometrikˆ isìmorfoc me duðkì q ro. 4. 'Estw X ènac pragmatikìc topikˆ kurtìc q roc kai èstw K sumpagèc kai kurtì uposônolo tou X. An E K, deðxte ìti ta ex c eðnai isodônama: (a) K = co(e). (b) Gia kˆje suneqèc grammikì sunarthsoeidèc f : X R, sup x K f(x) = sup f(x). x E 5. 'Estw X diaqwrðsimoc q roc Banach kai èstw (x n ) fragmènh akoloujða ston X. DeÐxte ìti ta ex c eðnai isodônama: (a) x w n x. (b) x (x n ) x (x) gia kˆje x ex(b X ) w. 6. DeÐxte ìti B l = co(ex(b l )). 7. 'Estw X ènac q roc Banach kai Y ènac kleistìc upìqwroc tou X. 'Eqoume dei ìti o Y eðnai isometrikˆ isìmorfoc me ton X /N(Y ) (deðte tic ask seic thc paragrˆfou 2.3). DeÐxte ìti gia kˆje [x] ex(b X /N(Y )) upˆrqei x 1 [x] me x 1 ex(b X ). 8. 'Estw X ènac pragmatikìc topikˆ kurtìc q roc kai èstw K sumpagèc kai kurtì uposônolo tou X. 'Estw f : K R kurt kai ˆnw hmisuneq c: gia kˆje s R to sônolo {x K : f(x) < s} eðnai anoiktì sto K. DeÐxte ìti upˆrqei x 0 ex(k) ste f(x) f(x 0) gia kˆje x K. 9. 'Estw X, Y sumpageðc metrikoð q roi. DeÐxte ìti gia kˆje f C(X Y ) kai gia kˆje ε > 0 upˆrqoun n N, g 1,..., g n C(X) kai h 1,..., h n C(Y ) ste n f(x, y) g j(x)h j(y) < ε j=1 gia kˆje (x, y) X Y. 10. 'Estw K ènac sumpag c metrikìc q roc kai èstw {V n : n N} mia bˆsh gia thn topologða tou K. OrÐzoume f n : K R me f n (x) = d(x, K \V n ). DeÐxte ìti h ˆlgebra pou parˆgetai apì thn (f n) kai apì th stajer sunˆrthsh f 0 1 eðnai pukn ston C(K), kai sumperˆnate ìti o C(K) eðnai diaqwrðsimoc.

Kefˆlaio 7 Jewr mata stajeroô shmeðou 7.1 Sustolèc se pl reic metrikoôc q rouc 'Estw S èna mh kenì sônolo kai èstw f : S S. To x S lègetai stajerì shmeðo thc f an f(x) = x. Se autì to Kefˆlaio ja doôme merikˆ jewr mata pou exasfalðzoun thn Ôparxh stajeroô shmeðou upì proôpojèseic gia thn f kai to S. To aploôstero Ðswc tètoio je rhma eðnai to je rhma stajeroô shmeðou tou Banach, sto plaðsio twn pl rwn metrik n q rwn. Je rhma 7.1.1 (Banach). 'Estw (X, d) ènac pl rhc metrikìc q roc kai èstw f : X X sustol : dhlad, upˆrqei 0 α < 1 ste d(f(x), f(y)) α d(x, y) gia kˆje x, y X. Tìte, h f èqei monadikì stajerì shmeðo. Apìdeixh. Epilègoume tuqìn x 0 X kai orðzoume akoloujða (x n ) ston X jètontac x n = f(x n 1 ) gia n = 1, 2,.... Epagwgikˆ deðqnoume ìti d(f(x n+1 ), f(x n )) α n d(x 1, x 0 ), kai qrhsimopoi ntac thn trigwnik anisìthta gia thn d blèpoume ìti an n < m tìte d(x m, x n ) m 1 s=n m 1 d(x s+1, x s ) d(x 1, x 0 ) α n 1 α d(x 1, x 0 ). s=n m n 1 α s α n d(x 1, x 0 ) 'Epetai ìti h (x n ) eðnai akoloujða Cauchy ston X. O X eðnai pl rhc, ˆra upˆrqei x X ste x n x. H f eðnai suneq c, opìte x = lim n x n = lim n f(x n 1 ) = s=0 α s

128 Jewrhmata stajerou shmeiou f(lim x n 1 ) = n f(x). 'Ara, to x eðnai stajerì shmeðo thc f. Tèloc, h f den mporeð na èqei dôo diaforetikˆ stajerˆ shmeða: an up rqan x y ste f(x) = x kai f(y) = y, tìte ja eðqame 0 < d(x, y) = d(f(x), f(y)) αd(x, y), dhlad α 1. Mia apl epèktash tou Jewr matoc 7.1.1 eðnai h ex c. Je rhma 7.1.2. 'Estw (X, d) ènac pl rhc metrikìc q roc kai èstw f : X X mia sunˆrthsh me thn idiìthta h f n = f f (n forèc) na eðnai sustol. Tìte, h f èqei monadikì stajerì shmeðo. Apìdeixh. Apì to prohgoômeno je rhma, h f n èqei monadikì stajerì shmeðo x X. Parathr ste ìti f n (f(x)) = f(f n (x)) = f(x). 'Ara, to f(x) eðnai epðshc stajerì shmeðo thc f n. Anagkastikˆ, f(x) = x. Tèloc, afoô kˆje stajerì shmeðo thc f eðnai kai stajerì shmeðo thc f n, h f den mporeð na èqei ˆllo stajerì shmeðo. 7.2 Jewr mata stajeroô shmeðou se q rouc me nìrma Ta apotelèsmata aut c thc Paragrˆfou basðzontai sto je rhma tou Brouwer (gia mia apìdeixh, deðte to: N. Dunford and J. Schwartz, Linear Operators). Je rhma 7.2.1. 'Estw B n = {x R n : x 2 1} h EukleÐdeia monadiaða mpˆla ston R n kai èstw φ : B n B n suneq c sunˆrthsh. Upˆrqei x B n ste φ(x) = x. To je rhma tou Brouwer genikeôetai wc ex c. Je rhma 7.2.2. 'Estw X ènac q roc peperasmènhc diˆstashc me nìrma, èstw K èna mh kenì sumpagèc kai kurtì uposônolo tou X kai èstw f : K K suneq c sunˆrthsh. Tìte, upˆrqei x 0 K ste f(x 0 ) = x 0. Apìdeixh. Upojètoume pr ta ìti X = (R m, ). H eðnai isodônamh me thn 2 kai to K eðnai fragmèno, ˆra upˆrqei r > 0 ste K rb m. OrÐzoume φ : rb m K me φ(x) to monadikì y K gia to opoðo x y 2 = d(x, K). H φ eðnai suneq c sunˆrthsh: 'Estw x n, x rb m me x n x 2 0. An φ(x kn ) y K, tìte x y 2 = lim n x kn φ(x kn ) 2 = lim n d(x kn, K) = d(x, K) diìti h d(, K) eðnai suneq c, ˆra y = φ(x). 'Epetai ìti φ(x n ) φ(x). EpÐshc, φ(x) = x an x K.

7.2 Jewrhmata stajerou shmeiou se qwrouc me norma 129 OrÐzoume h : B m 1 r K B m me h(x) = 1 r f(φ(rx)). H h eðnai suneq c, opìte to Je rhma 7.2.1 mac dðnei x B m ste 1 r f(φ(rx)) = x 1 r K. Jètoume x 0 = rx K. AfoÔ φ(x 0 ) = x 0, paðrnoume f(x 0 ) = f(φ(x 0 )) = f(φ(rx)) = rx = x 0. Dhlad, to x 0 eðnai stajerì shmeðo thc f. An o X eðnai tuq n m-diˆstatoc q roc me nìrma, mporoôme na orðsoume ston R m wste na upˆrqei isometrikìc isomorfismìc σ : X (R m, ). OrÐzoume g : σ(k) σ(k) me g(σ(x)) = σ(f(x)). H g èqei stajerì shmeðo, dhlad upˆrqei x K ste g(σ(x)) = σ(x). Tìte, σ(f(x)) = σ(x), ˆra f(x) = x. To je rhma stajeroô shmeðou tou Schauder genikeôei to Je rhma 7.2.2 sto plaðsio twn apeirodiˆstatwn q rwn me nìrma. Je rhma 7.2.3. 'Estw F èna kleistì, kurtì kai fragmèno uposônolo enìc q rou X me nìrma. 'Estw f : F F sunˆrthsh me thn ex c idiìthta: gia kˆje A F to f(a) eðnai sumpagèc. Tìte, upˆrqei x F ste f(x) = x. ShmeÐwsh. An to F upotejeð sumpagèc, tìte kˆje suneq c sunˆrthsh f : F X ikanopoieð thn upìjesh tou Jewr matoc: gia kˆje A F to f(a) eðnai sumpagèc. Gia thn apìdeixh tou Jewr matoc 7.2.3 ja qreiastoôme to ex c L mma. L mma 7.2.4. 'Estw K mh kenì sumpagèc uposônolo enìc q rou X me nìrma. 'Estw ε > 0 kai A peperasmèno uposônolo tou K ste K {D(a, ε) : a A}. Gia kˆje a A orðzoume m a : K R + me m a (x) = ε x a an x D(a, ε) kai m a (x) = 0 alli c. Tìte, h sunˆrthsh φ A : K X me φ A (x) = a A m a(x)a a A m a(x) orðzetai kalˆ, eðnai suneq c kai φ A (x) x < ε gia kˆje x K. Apìdeixh tou L mmatoc. Gia kˆje x K èqoume m a (x) > 0: an tan a A m a (x) = 0 gia kˆje a A, tìte ja eðqame x / D(a, ε), ˆtopo. Autì deðqnei ìti h φ A orðzetai kalˆ. Kˆje m a : K [0, ε] eðnai suneq c, ˆra h φ A eðnai suneq c. 'Estw x K. Tìte, φ A (x) x = a A a A m a(x)(a x) a A m. a(x) ParathroÔme ìti an m a (x) > 0 tìte a x < ε. 'Ara, m a (x)(a x) m a (x) ε gia kˆje a A. Apì thn trigwnik anisìthta èpetai ìti φ A (x) x < ε.

130 Jewrhmata stajerou shmeiou Apìdeixh tou Jewr matoc 7.2.3. Jètoume K = f(f ). Apì thn upìjesh gia thn f, to K eðnai sumpagèc. Gia kˆje n N jewroôme peperasmèno uposônolo A n tou K me thn idiìthta K a A n D(a, 1/n) kai orðzoume th sunˆrthsh φ n := φ An ìpwc sto L mma 7.2.4. Apì ton orismì thc φ n, gia kˆje x K èqoume φ n (x) co(k) F, afoô to F eðnai kurtì. Sunep c, h f n := φ n f apeikonðzei to F sto F. EpÐshc, apì to L mma 7.2.4, gia kˆje x F èqoume f n (x) f(x) = φ n (f(x)) f(x) < 1 n. JewroÔme ton peperasmènhc diˆstashc q ro X n = span(a n ) kai jètoume F n := F X n. To F n eðnai sumpagèc kurtì uposônolo tou X n kai gia kˆje x F n èqoume f n (x) = φ n (f(x)) co(a n ) K X n F n. AfoÔ h f n : F n F n eðnai suneq c, to Je rhma 7.2.3 deðqnei ìti upˆrqei x n F n ste f(x n ) = x n. H akoloujða (f(x n )) perièqetai sto sumpagèc sônolo K. 'Ara, upˆrqei upakoloujða f(x kn ) x 0 K. Tìte, x kn x 0 = f kn (x kn ) x 0 = φ kn (f(x kn )) x 0 φ kn (f(x kn )) f(x kn ) + f(x kn ) x 0 1 + f(x kn ) x 0 0. k n Apì thn x kn x 0 sumperaðnoume ìti f(x 0 ) = lim n f(x kn ) = x 0. ShmeÐwsh. To Je rhma tou Schauder genikeôetai kai sto plaðsio twn topikˆ kurt n q rwn. To epìmeno je rhma exasfalðzei koinì stajerì shmeðo gia mia oikogèneia affinik n apeikonðsewn pou orðzontai se èna sumpagèc kurtì sônolo kai antimetatðjentai. Mia apeikìnish T : X X lègetai affinik an upˆrqei x 0 X ste h S : X X me S(x) = T (x) x 0 na eðnai grammikìc telest c. Je rhma 7.2.5 (Markov Kakutani). 'Estw X ènac topikˆ kurtìc q roc, èstw K èna mh kenì, kurtì kai sumpagèc uposônolo tou X kai èstw {T i : i I} mia oikogèneia suneq n affinik n apeikonðsewn T i : K K oi opoðec antimetatðjentai: T i T j = T j T i gia kˆje i, j I. Tìte, upˆrqei x 0 K ste T i (x 0 ) = x 0 gia kˆje i I.

7.2 Jewrhmata stajerou shmeiou se qwrouc me norma 131 Apìdeixh. Gia kˆje i I kai n N orðzoume T (n) i T (n) : K K me i = I + T i + Ti 2 + + T n 1 i, n ìpou Ti k = T i T i (k forèc). Apì thn upìjesh ìti oi T i antimetatðjentai, èpetai ìti ( ) T (n) i T (m) j = T (m) j T (n) i gia kˆje i, j I kai n, m N. JewroÔme thn oikogèneia K := {T (n) i (K) : i I, n N}. Kˆje sônolo sthn K eðnai sumpagèc kai kurtì. Qrhsimopoi ntac thn ( ) blèpoume ìti, gia kˆje i 1,..., i k I kai n 1,..., n k N, T (n1) i 1 T (n k) (K) i k k j=1 T (n j) i j (K). Dhlad, h K èqei thn idiìthta peperasmènhc tom c. Apì th sumpˆgeia tou K èpetai ìti upˆrqei x 0 K ste x 0 T (n) i (K) gia kˆje i I kai kˆje n N. Autì shmaðnei ìti, gia kˆje i I kai n N upˆrqei x = x(i, n) K ste dhlad x 0 = x + T i(x) + + T n 1 i (x), n T i (x 0 ) x 0 = T (x) + T 2 i (x) + + T n i (x) n x + T i(x) + + T n 1 i (x) n = 1 n (T i n (x) x) 1 (K K). n AfoÔ to K eðnai sumpagèc, to K K eðnai epðshc sumpagèc kai 0 K K. 'Estw U anoikt perioq tou 0. Upˆrqei n N ste K K nu (ˆskhsh). 'Ara, T i (x 0 ) x 0 1 n (K K) U. 'Epetai ìti T i(x 0 ) = x 0 gia kˆje i I (to i I tan tuqìn).