# Rearrangement of series. The theorem of Levy-Steiniz.

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1 Rearrangement of series. The theorem of Levy-Steiniz. Instituto Universitario de Matemática Pura y Aplicada Universidad Politécnica de Valencia, SPAIN Poznań (Poland). October 20

2 Rearrangement of series. Surprising phenomenon in mathematical analysis: Infinite sums of numbers do NOT satisfy the commutative property.

3 Series a k. Sequence of real number a, a 2,..., a k,..., called terms of the series. We want to associate to them a sum. Idea of Cauchy. Partial Sums s := a, s 2 := a + a 2,..., s k := a + a a k. The series converges if the limit ĺım s k = s exists and this limit is called the sum of the series s = k= a k. x k = + x + x converges if and only if x <. Its sum is k=0 x k = x.

4 Aspects to study about series. () Convergence. k= k = π 2 /6 (Euler). 2 (2) Asymptotic behaviour. The harmonic series k diverges (Euler). Its partial sums s k behave asymptotically like log k. This means ĺım k s k log k =. Euler proved also that the series p, the sum extended to the prime numbers p diverges. (3) An aspect that is exclusive to series is rearrangement.

5 Rearrangement of series A rearrangement of the series a k is the series a π(k), where is a bijection. π : N N A series a k is unconditionally convergent if the series a π(k) converges for each bijection π. If the series a k converges, the set of sums is S( a k ) := {x R x = a π(k) for some π} k= It is the set of sums of all the rearrangements of the series.

6 The Theorem of Riemann Theorem (Riemann, 857) Let a k be a series of real numbers. ak is unconditionally convergent if and only if a k is convergent, that is the series is absolutely convergent. If a k converges, but not unconditionally, then S( a k ) = R.

7 Riemann ( ). Riemann integral, Riemann s surfaces, the Cauchy Riemann equation, the theorem of Riemann Lebesgue, the theorem of Riemann s function in complex analysis, Riemann s zeta function,...

8 The center of Mathematics between 800 y 933 was Göttingen (Germany). Gauss, Dirichlet, Riemann, Hilbert y Klein were there.

9 The Theorem of Riemann Idea of the proof: If a k converges absolutely, Cauchy s criterion ensures that every rearrangement converges to the same sum. Suppose that a k converges but not absolutely. Let p k y q k be the positive and negative terms of the series respectively. Possible cases: pk converges, q k converges, then a k converges. pk =, q k converges, then a k =. pk converges, q k =, then a k =. Thus p k = y q k =. Fix α R, select the first n() with p p n() > α; then the first n(2) with p p n() + q q n(2) < α. Since ĺım a k = 0, the rest of the proof is ε-δ.

10 The alternate harmonic series. The alternate harmonic series = k= ( )k+ k = log 2. Leibniz s criterion shows that this series is convergent. It is not absolutely convergent. The result was known to Mercator (S. XVII). There are many different proofs, for example by a theorem of Abel on power series using the development of log( + x).

11 The alternate harmonic series. An elementary proof: Put I n := π/4 0 tg n xdx. We have: () (I n ) n is decreasing. (2) I n = n I n 2. Integrating by parts. (3) 2(n+) I n 2(n ). (4) Using induction in (2) and I = 2 log 2, we get 4(n + ) I 2n+ = n ( ) k+ 2k 2 k= Multiplying by 2 we obtain the result. log 2 4n.

12 The alternate harmonic series. The rearrangement of Laurent = = ( 2 ) 4 + ( 3 6 ) 8 + ( 5 0 ) 2... = = = = 2 ( ) = log 2. 2

13 The alternate harmonic series. A rearrangement of the alternate harmonic series is called simple if the positive and negative terms separately are in the same order as in the original series. For example, Laurent s rearrangement is simple. In a simple rearrangement we denote by r n the number of positive terms between the first n of the rearrangement. Theorem of Pringsheim, 883 A simple rearrangement a π(k) of the alternate harmonic series converges if and only if ĺım n r nn =: α <. In this case k= a π(k) = log log(α( α) ). For the Laurent s rearrangement we have α = /3 y log log( ) = log 2. 2

14 Series of vectors. What happens if we consider series of vectors? Example in R 2 : (( ) k+ k, 0). The set of sums is R {0}. It is not all the space R 2, but it is an affine subspace R 2. This phenomenon was observed by Levy for n = 2 in 905 and by Steinitz for n 3 in 93.

15 The Theorem Levy Steinitz. Notation. E is a real locally convex Hausdorff space. Examples: R n, l p, p, L p, p (Banach spaces), H(Ω), C (Ω) (Fréchet spaces: metrizable and complete), D, D, H(K), A(Ω),...(more complicated spaces). uk is a convergent series and S( u k ) is its set of sums (of all its convergent rearrangements). Set of summing functionals Γ( u k ) := {x E x, u k < } E. The annihilator of G E is G := {x E x, g = 0 g G}.

16 The Theorem Levy Steinitz. The Theorem Levy Steinitz. 905, 93. If u k is a convergent series of vectors in R n, then S( u k ) = u k + Γ( u k ) is an affine subspace of R n. P. Rosenthal, in an article in the American Mathematical Monthly in 987 explaining this theorem, remarked that it is a beautiful result, which deserves to be better known, but that the difficulty of its proof is out of proportion of the statement.

17 The Theorem Levy Steinitz. The inclusion in the statement is easy and holds in general: Let x = u π(k) S( u k ). We want to see that x u k Γ( u k ). To do this, fix x Γ( u k ). By Riemann s theorem, we get x, x u k = x, u π(k) x, u k = 0, since the series x, u k is absolutely convergent.

18 The Theorem Levy Steinitz. Idea of the proof of the other inclusion: Let E be a complete metrizable space (A) S( u k ) S e ( u k ). Expanded set of sums S e ( u k ) := {x E π (j m ) m : x = ĺım m j m u π(k) }. (B) S e ( u k ) = u k + m= Z m. Z m = Z m ( u k ) := { k J u k J {m, m +, m + 2,...} finite}.

19 The Theorem Levy Steinitz. Idea of the proof of the other inclusion: Continued: (C) u k + m=z m u k + m=co(z m ). co(c) is the convex hull of C. (D) u k + m=co(z m ) = u k + Γ( u k ), by the Hahn-Banach theorem. The problem is to find conditions to ensure that the inclusions (A) y (C) are equalities.

20 The Theorem Levy Steinitz. The equality in (A) follows in the finite dimensional case from the following lemma. Lemma of polygonal confinement of Steinitz For every real Banach space E of finite dimension m there is a constant 0 < C(E) m such that for every finite set of vectors x, x 2,..., x n satisfying n x k = 0 there s a bijection σ on {, 2,..., n} such that for all r =, 2,...n. r j= x σ(j) C(E) máx j=,...n x j The exact value of the constant C(E) is unknown even for Hilbert spaces of finite dimension m > 2. For m = 2, C(l 2 2 ) = 5 2.

21 The Theorem Levy Steinitz. The equality in (C) follows in the finite dimensional case from the following lemma. Round-off coefficients Lemma. Let E be a real Banach space of finite dimension m. Let x, x 2,..., x n be a finite set of vectors such that x j for all j =,..., n. For each x co( k I x k I {,..., n}) there is J {, 2,..., n} such that x k J x k m 2.

22 The work of Nash-Williams and White Nash-Williams and White ( ) obtained the following results applying graph theory: Let π be a bijection on N. They defined the width w(π) of π in a combinatorial way with values in N {0, }. w(π) = if and only if there is a series a k of real numbers such that a π(k) converges to a different sum. w(π) = 0 if and only if a π(k) converges to the sum a k when aπ(k) converges. w(π) N if and only if for a convergent series a k the series aπ(k) has the same sum or diverges. In this case, if we fix π, they determine the set of accumulation points of the sequences of partial sums of series of the form a π(k) with a k = 0. They also extend their results for series in R n, n 2.

23 Series in infinite dimensional Banach spaces. The study of series in infinite dimensional spaces was initiated by Orlicz in Banach and his group used to meet in the Scottish Café in Lvov (now Ukraine). The problems they formulated were recorded in the Scottish Book, that was saved and published by Ulam. Problem 06: Does a result analogous to the Levy Steinitz theorem hold for Banach spaces of infinite dimension? The prize was a bottle of wine; smaller by the way than the prize for the approximation problem of Mazur, that was solved by Enflo. In that case the prize was a goose. The negative answer was obtained by Marcinkiewicz with an example in L 2 [0, ].

24 The Scottish Cafe, Lvov.

25 The Scottish Cafe, Lvov. 200.

26 The example of Marcinkiewicz. Consider the following functions in L 2 [0, ]. Here χ A is the characteristic function of A. x i,k := χ [ k 2 i, k+ 2 i ], y i,k := x i,k, 0 i <, 0 k < 2 i. Clearly x i,k 2 = 2 i for each i, k. One has (x 0,0 + y 0,0 ) + (x,0 + y,0 ) + (x, + y, ) + (x 2,0 + y 2,0 ) +... = 0 x 0,0 + (x,0 + x, + y 0,0 ) + (x 2,0 + x 2, + y,0 ) + (x 2,2 + x 2,3 + y, ) +... = No rearrangement converges to the constant function /2, since all the partial sums are functions with entire values.

27 The example of Marcinkiewicz. x 00 x 0 x 0 0 /2 0 /2

28 The example of Marcinkiewicz. x 20 x x 22 x

29 Series in infinite dimensional Banach spaces. Dvoretsky-Rogers Theorem A Banach space E is finite dimensional if and only if every unconditionally convergent series in E is absolutely convergent. This is a very important result in the theory of nuclear locally convex spaces of Grothendieck and in the theory of absolutely summing operators of Pietsch. Example. In l 2, we set u k := (0,..,0, /k, 0,...). The series u k is not absolutely convergent since u k = =. But, unconditionally in l 2. k u k = (, /2, /3,..., /k,...)

30 Series in infinite dimensional Banach spaces. Theorem of Mc Arthur Every Banach space E of infinite dimension contains a series whose set of sums reduces to a point, but is not unconditionally convergent. Idea in l 2. Denote by e i the canonical basis. e e + (/2)e 2 (/2)e 2 + (/2)e 2 (/2)e 2 + (/4)e 3... = 0. We have 2 n terms of the form 2 n+ e n with alternate signs. If a rearrangement converges, its sum must be 0, as can be seen looking at each coordinate. However, it is not unconditionally convergent, for if it were, then (2, 2, 2,...) l 2. For an arbitrary Banach space, one uses basic sequences.

31 Series in infinite dimensional Banach spaces. Theorem of Kadets and Enflo Every Banach space E of infinite dimension contains a series whose set of sums consists exactly of two different points. Theorem of J.O. Wojtaszczyk Every Banach space E of infinite dimension contains a series whose set of sums is an arbitrary finite set which is affinely independent. The theorem of Levy Steinitz fails in a drastic way for infinite dimensional Banach spaces.

32 Series in infinite dimensional Banach spaces. Theorem of Ostrovski There is a series in L 2 ([0, ] [0, ]) whose set of sums is not closed. Problem. Is there a series u k in a Banach space whose set of sums S( u k ) is a non-closed affine subspace? Theorem. Every separable Banach space contains a series whose set of sums is all the space.

33 Series in infinite dimensional spaces. Problem. Is it possible to extend the theorem of Levy Steinitz for some infinite dimensional spaces? YES. A locally convex Hausdorff space E is called nuclear if every unconditionally convergent series is absolutely convergent. For Fréchet of (DF)-spaces this coincides with the original definition of Grothendieck. In general this is not the case. Examples. H(Ω), C (Ω), S, S, H(K), D, D, A(Ω).

34 The theorem of Banaszczyk. Theorem of Banaszczyk. 990, 993. Let E be a Fréchet space. The following conditions are equivalent: () E is nuclear. (2) For each convergent series u k in E we have S( u k ) = u k + Γ( u k ). This is a very deep result. Both directions are difficult. extensions of the lemmas of confinement and of rounding-off coefficients with Hilbert-Schmidt operators, a characterization of nuclear Fréchet spaces with volume numbers, topological groups, etc are needed.

35 Series in non-metrizable spaces. Bonet and Defant studied in 2000 the set of sums of series in non-metrizable spaces and, in particular, in (DF)-spaces, like the space S of Schwartz or the space H(K) of germs of holomorphic functions on the compact set K in the complex plane. The notation E = ind n E n means that E is the increasing union of the Banach spaces E n E n+ with continuous inclusions, and E is endowed with the finest locally convex topology such that all the inclusions E n E are continuous.

36 Series in non-metrizable spaces. Theorem of Bonet and Defant Let u k be a convergent series in the nuclear (DF)-space E = ind n E n (then it converges in a Banach step E n(0) ). The following holds: (a) S( u k ) = u k + Γ loc ( u k ), where Γ loc( u k ) := {x E n x, x = 0 x E n with n n(0) u k, x < } is a subspace of E. (b) If E is not isomorphic to the direct sum ϕ of copies of R, then there is a convergent series in E whose set of sums is a non-closed subspace of E.

37 Series in non-metrizable spaces. Theorem of Bonet and Defant Let E = ind n E n be a complete (DF)-space such that every convergent sequence in E converges in one of the Banach spaces E n. If we have S( u k ) = u k + Γ loc( u k ) for every convergent series u k in E, then the space E is nuclear.

38 Series in non-metrizable spaces. The proof of the theorem requires new improvements in the lemmas of confinement and round-off. The techniques of proof for the positive part can be utilized for more general spaces, including the space of distributions D or the space of real analytic functions A(Ω), thus obtaining that the set of sums of a convergent series is an affine subspaces that need not be closed. The result about nuclear (DF)-spaces not isomorphic to ϕ requires deep results due to Bonet, Meise, Taylor (99) and Dubinski, Vogt (985) about the existence of quotients of nuclear Fréchet spaces without the bounded approximation property and their duals.

39 Other open problems. Does every non-nuclear Fréchet space contain a convergent series uk such that its set of sums consists exactly of two points? Improve the converse for non-metrizable spaces. Find concrete spaces E and conditions on a convergent series u k in E to ensure that the set of sums S( u k ) has exactly the form of the Theorem of Levy and Steinitz. Chasco and Chobayan have results of this type for spaces L p of p-integrable functions.

40 Some references. W. Banaszczyk, The Steinitz theorem on rearrangement of series for nuclear spaces, J. reine angew. Math. 403 (990), W. Banaszczyk, Rearrangement of series in nonnuclear spaces, Studia Math. 07 (993), J. Bonet, A. Defant, The Levy-Steinitz rearrangement theorem for duals of metrizable spaces, Israel J. Math. 7 (2000), M.I. Kadets, V.M. Kadets, Series in Banach Spaces Operator Theory: Advances and Applications 94, Birkhäuser Verlag, Basel 997. P. Rosenthal, The remarkable theorem of Levy and Steinitz, Amer. Math. Monthly 94 (987),

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