# On Nicely Smooth Banach Spaces

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 E extracta mathematicae Vol. 16, Núm. 1, (2001) On Nicely Smooth Banach Spaces Pradipta Bandyopadhyay, Sudeshna Basu Stat-Math Unit, Indian Statistical Institute, 203 B.T. Road, Calcutta , India Department of Mathematics, Howard University, Washington DC 20059, USA (Research paper presented by David Yost) AMS Subject Class. (2000): 46B20, 46B22 Received June 3, Introduction We work with real Banach spaces. We will denote by B(X), S(X) and B[x, r] respectively the closed unit ball, the unit sphere and the closed ball of radius r > 0 around x X. We will identify any element x X with its canonical image in X. All subspaces we usually consider are norm closed. Definition 1.1. (a) We say A B(X ) is a norming set for X if x = sup{x (x) : x A}, for all x X. A closed subspace F X is a norming subspace if B(F ) is a norming set for X. (b) A Banach space X is (i) nicely smooth if X contains no proper norming subspace; (ii) has the Ball Generated Property (BGP) if every closed bounded convex set in X is ball-generated, i.e., intersection of finite union of balls; (iii) has Property (II) if every closed bounded convex set in X is the intersection of closed convex hulls of finite union of balls, or equivalently, w*-points of continuity (w*-pcs) of B(X ) are norm dense in S(X ) [5]; (iv) has the Mazur Intersection Property (MIP) (or, Property (I)) if every closed bounded convex set in X is intersection of balls, or equivalently, w*-denting points of B(X ) are norm dense in S(X ) [8]. 27

2 28 p. bandyopadhyay, s. basu Clearly, the MIP implies both Property (II) and the BGP. In this work, we obtain a sufficient condition for the BGP, and show that Property (II) implies the BGP, which, in turn, implies nice smoothness. Notice that if a separable space is nicely smooth then it has separable dual. (The converse is false, since a dual space is nicely smooth if and only if it is reflexive.) It follows that if nice smoothness was inherited by subspaces, a nicely smooth space would necessarily be Asplund. However, recent work of Jiménez Sevilla and Moreno [15] shows that this is not true. More precisely, they showed that any Banach space can be isomorphically embedded in a Banach space with the MIP, a property stronger than nice smoothness. In this work, we obtain some necessary and/or sufficient conditions for a space to be nicely smooth, and show that they are all equivalent for separable or Asplund spaces. These sharpen known results. We observe that every equivalent renorming of a space is nicely smooth if and only if it is reflexive. We also show that if X is nicely smooth, X E X and E has the finite-infinite intersection property (.f.ip) (Definition 2.15), then E = X. In particular, X is nicely smooth with.f.ip if and only if X is reflexive. Coming to stability results, we show that the class of nicely smooth spaces is stable under c 0 and l p sums (1 < p < ) and also under finite l 1 sums. We show that while nice smoothness is not a three space property, existence of a nicely smooth renorming is a three space property. We show that the Bochner L p spaces (1 < p < ) are nicely smooth if and only if X is both nicely smooth and Asplund. And for a compact Hausdorff space K, C(K, X) is nicely smooth if and only if K is finite and X is nicely smooth. We also study nice smoothness of certain operator and tensor product spaces. 2. Main Results We record the following corollary of the proof of the Hahn-Banach Theorem (see e.g., [18, Section 48]) for future use. Lemma 2.1. Let X be a normed linear space and let Y be a subspace. Suppose x 0 / Y and y S(Y ). Then sup{y (y) x 0 y : y Y } inf{y (y) + x 0 y : y Y } and α lies between these two numbers if and only if there exists a Hahn-Banach (i.e., norm preserving) extension x of y with x (x 0 ) = α.

3 on nicely smooth banach spaces 29 For a Banach space X, let us define O(X) = {x X : x + x x for all x X}. And we have the following characterization: Proposition 2.2. For a Banach space X, the following are equivalent: (a) X is nicely smooth. (b) For all x X, x X B[x, x x ] = {x }. (c) O(X) = {0}. (d) For all nonzero x X, there exists x S(X ) such that every Hahn-Banach extension of x to X takes nonzero value at x. (e) Every norming set A B(X ) separates points of X. Proof. Equivalence of (a) and (b) is [12, Lemma 2.4]. (a) (c) [10, Lemma I.1] shows that F is a norming subspace of X if and only if F O(X). Hence the result. (c) (d) Each of the statements below is clearly equivalent to the next: (i) x O(X), (ii) x x x for all x X, (iii) for every x S(X ), sup{x (x) x x : x X} 0 inf{x (x) + x x : x X}, (iv) every x S(X ) has a Hahn-Banach extension x with x (x ) = 0 (Lemma 2.1). Hence the result. (a) (e) Since any norming set spans a norming subspace, this is clear. We note a characterization of the BGP.

4 30 p. bandyopadhyay, s. basu Theorem 2.3. A Banach space X has the BGP if and only if for every x X and ε > 0, there exists w*-slices S 1, S 2,..., S n of B(X ) such that for any (x 1, x 2,..., x n) n S i, there are scalars a 1, a 2,..., a n such that x n a ix i ε. Proof. Observe that X has the BGP if and only if every x X is ballcontinuous on B(X) [11, Theorem 8.3]. Now the result follows from the characterization of such functionals obtained in [5, Theorem 2]. This leads to the following, more tractable sufficient condition for the BGP. Definition 2.4. A point x 0 in a convex set K X is called a w*-small combination of slices (w*-scs) point of K, if for every ε > 0, there exist w*-slices S 1, S 2,..., S n of K, and a convex combination S = n λ is i such that x 0 S and diam(s) < ε. Proposition 2.5. If X is the closed linear span of the w*-scs points of B(X ), then X has the BGP. Proof. Let x X and ε > 0. Since the set of w*-scs points of B(X ) is symmetric and spans X, there exist w*-scs points x 1, x 2,..., x n of B(X ), and positive scalars a 1, a 2,..., a n such that x n a ix i ε/2. By definition of w*-scs points, for each i = 1, 2,..., n, there exist w*- slices S i1, S i2,..., S ini of B(X ), and a convex combination S i = n i k=1 λ iks ik such that x i S i and diam(s i ) < ε/(2 n a i). Now, for any (x ik ) n ni k=1 S ik, x n n i a i λ ik x ik k=1 x ε/2 + n a i x n i + a i x i n a i diam(s i ) ε. Hence by Theorem 2.3, X has the BGP. n λ ik x ik Corollary 2.6. Property (II) implies the BGP, which, in turn, implies nicely smooth. Proof. Since X has Property (II), w*-pcs of B(X ) are norm dense in S(X ), and a w*-pc is necessarily a w*-scs point (this follows from Bourgain s Lemma, see e.g., [17, Lemma 1.5]). Thus, Property (II) implies the BGP. k=1

5 on nicely smooth banach spaces 31 That the BGP implies nicely smooth is proved in [11, Theorem 8.3]. But here is an elementary proof. Let F be a norming subspace of X. Then B(X) is σ(x, F )-closed, so that every ball-generated set is also σ(x, F )-closed. But if every closed bounded convex set is σ(x, F )-closed, then F = X. We now obtain a localization of [5, Theorem 3.1] and [9, Lemma 6]. Proposition 2.7. Let X be a Banach space. Let x 0 S(X ) and x 0 X. The following are equivalent: (a) every Hahn-Banach extension of x 0 to X takes the same value at x 0 ; (b) x 0, considered as a function on (B(X ), w ), is continuous at x 0 ; (c) sup{x 0 (x) x 0 x : x X} = inf{x 0 (x) + x 0 x : x X}; (d) for any α R, if x 0 (x 0 ) α, then there exists a ball B in X with centre in X such that x 0 B and B and x 0 lies in the same open half space determined by x 0 and α. Proof. (a) (b) This is a natural localization of the proof of [13, Lemma III.2.14]. We omit the details. (a) (c) Follows from Lemma 2.1. (c) (d) Suppose x 0 (x 0 ) > α. By Lemma 2.1, inf{x 0 (x)+ x 0 x : x X} > α. By (c), it follows that sup{x 0 (x) x 0 x : x X} > α. So, there exists x X such that x 0 (x) x 0 x > α. Put B = B [x, x 0 x ]. Clearly, x 0 B, and inf x 0 (B ) = x 0 (x) x 0 x > α. Similarly for x 0 (x 0 ) < α. (d) (c) Suppose x 0 (x 0 ) > α. By (d), there exists a ball B = B [x, r] in X such that x 0 B and inf x 0 (B ) > α. This implies x 0 x r and inf x 0 (B ) = x 0 (x) r > α. It follows that x 0 (x) x 0 x > α, whence sup{x 0 (x) x 0 x : x X} > α. Since α was arbitrary, it follows that sup{x 0 (x) x 0 x : x X} x 0 (x 0 ). Similarly, inf{x 0 (x) + x 0 x : x X} x 0 (x 0 ). The result now follows from Lemma 2.1. Corollary 2.8. [5, Theorem 3.1] For a Banach space X and f 0 S X, the following are equivalent: (i) f 0 is a w*-w PC of B X ; (ii) f 0 has a unique Hahn-Banach extension in X ;

6 32 p. bandyopadhyay, s. basu (iii) for any x 0 X and α R, if f 0 (x 0 ) α, then there exists a ball B in X with centre in X such that x 0 B and B and x 0 lies in the same open half space determined by f 0 and α. We now identify some necessary and some sufficient conditions for a space to be nicely smooth. Definition 2.9. For x S(X), let D(x) = {f S(X ) : f(x) = 1}. The set valued map D is called the duality map and any selection of D is called a support mapping. Theorem For a Banach space X, consider the following statements: (a) X is the closed linear span of the w*-weak PCs of B(X ). (b) Any two distinct points in X are separated by disjoint closed balls having centre in X. (b 1 ) For every x X, the points of w*-continuity of x in S(X ) separates points of X. (b 2 ) For every nonzero x X, there is a point of w*-continuity x S(X ) of x such that x (x ) 0. (c) X is nicely smooth. (d) For every norm dense set A S(X) and every support mapping φ, the set φ(a) separates points of X. Then (a) (b) (c) (d) and (a) (b 1 ) (b 2 ) (c) (d). Moreover, if X is an Asplund space (or, separable), all the above conditions are equivalent, and equivalent to each of the following: (e) X is the closed linear span of the w*-strongly exposed points of B(X ). (f) X is the closed linear span of the w*-denting points of B(X ). (g) X is the closed linear span of the w*-scs points of B(X ). (h) X has the BGP. Proof. (a) (b) Let x 0 y 0. By (a), there exists a w*-w PC x 0 B X, such that (x 0 y 0 )(x 0 ) > 0. Let α R be such that x 0 (x 0) > α > y 0 (x 0).

7 on nicely smooth banach spaces 33 Now applying Corollary 2.8, it follows that there exists a ball B1 with centre in X with x 0 B 1 and inf x 0 (B 1 ) > α. And there exists a ball B 2 with centre in X such that y0 B 2 and sup x 0 (B 2 ) < α. Clearly, B 1 B 2 =. (b) (c) Clearly, (b) implies condition (b) of Proposition 2.2. (a) (b 1 ) (b 2 ) follows from definitions. (b 2 ) (c) By (b 2 ), for every nonzero x X, there is a point of w*-continuity x S(X ) of x such that x (x ) 0. By Proposition 2.7, every Hahn-Banach extension of x to X takes the same value at x. The result now follows from Proposition 2.2(d). (c) (d) We simply observe that φ(a) is a norming set for X. Clearly, even without X being Asplund, (e) (f) (g) (h) (c), and (f) (a). Now if X is Asplund (if X is separable, (d) implies X is separable), then for A = {x S(X) : the norm is Fréchet differentiable at x}, and any support mapping φ, φ(a) = {w*-strongly exposed points of B(X )}. Hence, (d) (e). Remark (a) If in the first part, we simply assume that the w*-weak PCs of B(X ) form a norming set, then (a) (c) are equivalent. And under the stronger assumption that the set {x S(X) : D(x) intersects the w*-weak PCs of B(X )} is dense in S(X), (a) (d) are equivalent. This happens if X is Asplund. Can any of the implications be reversed in general? (b) In [9, Lemma 5], Godefroy proves (a) (c) by actually showing (b 1 ) (c). (b 2 ) is a weaker sufficient condition. (c) In [4, Theorem 7], it is observed that (f) (h). (g) is a weaker sufficient condition. Proposition If every separable subspace of X is nicely smooth, then X has the BGP, and hence, is nicely smooth. Proof. Since for separable spaces nice smoothness is equivalent to the BGP, the result follows from the characterization of ball-continuous functionals obtained in [11, Theorem 2.4 and 2.5]. Theorem A Banach space X is reflexive if and only if every equivalent renorming is nicely smooth.

8 34 p. bandyopadhyay, s. basu Proof. The converse being trivial, suppose X is not reflexive. Let x X \ X and let F = {x X : x (x ) = 0}. Define a new norm on X by x 1 = sup{x (x) : x B(F )} for x X. It follows from the proof of [11, Theorem 8.2] that 1 is an equivalent norm on X with F as a proper norming subspace. Remark (a) In [14] the authors showed that X is reflexive if and only if for any equivalent norm, X is Hahn-Banach smooth and has ANP-III. This was strengthened in [3, Corollary 2.5] to just Hahn-Banach smooth. The above is an even stronger result with even easier proof. (b) In [11, Theorem 8.1 and 8.2], it is shown that a subset W X is ballgenerated for every equivalent renorming if and only if it is weakly compact. The global analogue of this local result would read: every equivalent renorming of X has the BGP if and only if X is reflexive. Thus in the global version, our result is somewhat stronger, except for separable spaces, though, as the proof shows, it is implicit in [11, Theorem 8.2]. Definition A Banach space X is said to have the finite-infinite intersection property (.f.ip) if every family of closed balls in X with empty intersection contains a finite subfamily with empty intersection. It is well known that all dual spaces and their 1-complemented subspaces have.f.ip Proof. Sufficiency is obvious. For necessity, recall from [11, Theorem 2.8] that X has.f.ip if and only if X = X +O(X). Since X is nicely smooth, O(X) = {0} and consequently, X is reflexive. Theorem X is nicely smooth with.f.ip if and only if X is reflexive. Remark Since Hahn-Banach smooth spaces (respectively, spaces with Property (II)) are nicely smooth, [3, Theorem 2.11] (respectively, [3, Theorem 3.4]) follows as an immediate corollary with a simpler proof. Indeed, we have stronger results.

9 on nicely smooth banach spaces 35 Theorem If X is nicely smooth, X E X and E has the.f.ip, then E = X. Proof. Using the Principle of Local Reflexivity and w*-compactness of dual balls, it is easy to see that E has.f.ip if and only if any family of closed balls with centres in E that intersects in E also intersects in E. Now if X E X, let x 0 X \ E. Consider the family B = {B[x, x 0 x ] : x X}. Clearly, they intersect at x 0 X E. And since X is nicely smooth, by Proposition 2.2 (b), the intersection in X is singleton {x 0 }. But then, this family cannot intersect in E. Remark Can one strengthen this to show that if X is nicely smooth, E has.f.ip and X E, then X E? Recall that c 0 is nicely smooth, and it is known [16] that if c 0 E and E is 1-complemented in a dual space, then l is isomorphic to a subspace of E. Our question is isometric. In the special case of X = c 0, we can prove: Proposition If c 0 E and E has.f.ip, then l is a quotient of E. Proof. Since c 0 E, l E. Let F = E l. Since l is injective, let T : E l be the norm preserving extension of the inclusion map from F to l. Note that T is identity on c 0. We will prove that T is onto. Let x 0 l. Let B = {B[x, x 0 x ] : x c 0 }. As before, this family intersects in E, and hence in E. And since T = 1, if e E belongs to this intersection, so does T e. But the intersection in l is singleton {x 0 }, since c 0 is nicely smooth. Thus, T e = x 0. And in general, we have the result under stronger assumptions on both X and E : Proposition If X has Property (II), E is a dual space and X E, then X E.

10 36 p. bandyopadhyay, s. basu Proof. Let E = Y. Let i : X Y be the inclusion map. Then i : Y X is onto. We will show that i Y : Y X is onto. It will follow that X is a quotient of Y and hence, X is a subspace of Y = E. To show that i Y : Y X is onto, it suffices to check that i (B(Y )) = B(X ). Since X has Property (II), it suffices to check that any w*-pc x B(X ) belongs to i (B(Y )). Let Λ Y be a norm preserving extension of x to Y. Let {y α } B(Y ) such that y α Λ in the w*-topology of B(Y ). Then y α X x in the w*-topology of B(X ). Since x is a w*-pc, y α X x in norm, i.e., x i (B(Y )). 3. Stability Results Theorem 3.1. Let {X α } α Γ be a family of Banach spaces. Then X = l p X α (1 < p < ) is nicely smooth if and only if for each α Γ, X α is nicely smooth. Proof. We show that O(X) = {0} if and only if for every α Γ, O(X α ) = {0}. Now, X = l p X α implies X = l p X α, and x O(X) if and only if α Γ x + x p x p for all x X x α + x α p α Γ x α p for all x X. It is immediate that if for every α Γ, x α O(X α ), then x O(X). And hence, O(X) = {0} implies for every α Γ, O(X α ) = {0}. Conversely, suppose for every α Γ, O(X α ) = {0}. Let x X \ {0}. Let α 0 Γ be such that x α 0 0. Then x α 0 / O(X α0 ). Hence, there exists x α0 X α0 such that x α 0 + x α0 < x α0. Choose ε > 0 such that x α 0 + x α0 p + ε < x α0 p. Then there exists a finite Γ 0 {α Γ : x α 0} such that α 0 Γ 0 and α/ Γ 0 x α p < ε. If α Γ 0, then x α / O(X α ). Hence, there exists x α X α such that x α + x α < x α. Define y X by y α = { xα if α Γ 0 0 otherwise.

11 on nicely smooth banach spaces 37 Then we have, x + y p p = α Γ x α + y a p = xα + x a p + x α 0 + x α0 p + x α p α Γ0 α/ Γ 0 α α 0 < x a p + x α 0 + x α0 p + ε α Γ0 α α 0 < α Γ 0 x α p = y p p which shows that x / O(X). Remark 3.2. The above argument also works for finite l 1 (or l ) sums and shows that if X is the l 1 (or l ) sum of X 1, X 2,..., X n, then X is nicely smooth if and only if for every coordinate space X i is so. However, if Γ is infinite, X = l 1 X α is never nicely smooth as c 0 X α is a proper norming subspace of X = l X α. A similar argument also shows that nice smoothness is not stable under infinite l sums. We now show that nice smoothness is stable under c 0 sums. Theorem 3.3. Let {X α } α Γ be a family of Banach spaces. Then X = c 0 X α is nicely smooth if and only if for each α Γ, X α is nicely smooth. Proof. As before, we will show that O(X) = {0} if and only if for every α Γ, O(X α ) = {0}. Necessity is similar to that in Theorem 3.1. Conversely, suppose for every α Γ, O(X α ) = {0}. And let x X \ {0}. Let α 0 Γ be such that x α 0 0. Then x α 0 / O(X α0 ). Hence, there exists x α0 X α0 such that x α 0 + x α0 < x α0. The triangle inequality shows that for any λ 1, x α 0 + λx α0 < λx α0. Thus, replacing x α0 by λx α0 for some λ 1, if necessary, we may assume x α0 > x. Define y X by { xα0 if α = α y α = 0 0 otherwise.

12 38 p. bandyopadhyay, s. basu Then, x + y = max{sup{ x α α α0 }, x α 0 + x α0 } < x α0 = y whence x / O(X). Corollary 3.4. Nice smoothness is not a three space property. Proof. Let X = c, the space of all convergent sequences with the sup norm. Recall that c = l 1 and that l 1 acts on c as a, x = a 0 lim x n + a n+1 x n, a = {a n } n=0 l 1, x = {x n } n=0 c. n=0 It follows that {a l 1 : a 0 = 0} is a proper norming subspace for c. Put Y = c 0. Then, by Theorem 3.3, Y is nicely smooth and dim(x/y ) = 1, so that X/Y is also nicely smooth. But, by above, X is not nicely smooth. Remark 3.5. Since nice smoothness is not an isomorphic property, perhaps a more pertinent question here would be whether having a nicely smooth renorming is a three space property. The answer in this case is yes. Theorem 3.6. Let X be a Banach space. Let Y be a subspace of X. If both Y and X/Y have a nicely smooth renorming, then so does X. Proof. Observe that a Banach space X has a nicely smooth renorming if and only if for any proper subspace M X, the norm interior of the w*-closure of its unit ball is empty. Now suppose Y and X/Y both have a nicely smooth renorming. Suppose there exists a subspace M X such that B(M) w has nonempty interior, i.e., there exists m M and ε > 0 such that B(m, ε) B(M) w. We will show that M is not proper. Consider the inclusion map i : Y X. Then i : X Y is the natural quotient map and is w*-continuous. It follows that i (B(m, ε)) B(i (M)) w. Therefore, i (M) Y is a subspace the w*-closure of whose unit ball has nonempty interior. Because of our assumption on Y, we have i (M) = Y, and hence, X = M + Y. Put N = M Y. The natural isomorphism between X /M and Y /N shows that the relative norm interior of B(N) w is also nonempty. Since

13 on nicely smooth banach spaces 39 Y = (X/Y ), our assumption on X/Y implies that M Y = Y, i.e., M contains Y. Thus, X = M + Y = M. Remark 3.7. Since finite l 1 sums of infinite dimensional Banach spaces fail Property (II) [3, Proposition 3.7], the space c 0 l 1 c 0 produces an example of a nicely smooth space, which being Asplund, also has the BGP, but lacks Property (II). Recall that a closed subspace M X is said to be an M-summand if there is a projection P on X with range M such that x = max{ P x, x P x } for all x X. Proposition 3.8. The BGP is inherited by M-summands. Proof. We follow the arguments of [1, Proposition 2]. Let Y be an M-summand in X with P the corresponding projection. Let K be a closed bounded convex set in Y. Since X has the BGP, K = i I n i k=1 B[x ik, r ik ], where for each i and k, K B[x ik, r ik ]. Given i and k, let x K B[x ik, r ik ] Y, then x x ik r ik, so that x ik P x ik = (x x ik ) P (x x ik ) x x ik r ik. Claim : K = n i B Y [P x ik, r ik ]. ( ) i I k=1 Since P = 1, we have n i K = P (K) P ( B[x ik, r ik ]) i I k=1 i I n i k=1 B Y [P x ik, r ik ]. Conversely, if x is in the RHS of ( ), for all i I, there exists k such that x x ik = max{ x P x ik, x ik P x ik } r ik, as x ik P x ik r ik. Thus, x n i B[x ik, r ik ] = K. i I k=1 Theorem 3.9. Let X be a Banach space, µ denote the Lebesgue measure on [0, 1] and 1 < p <. The following are equivalent:

14 40 p. bandyopadhyay, s. basu (a) L p (µ, X) has BGP. (b) L p (µ, X) is nicely smooth. (c) X is nicely smooth and Asplund. Proof. Clearly (a) (b). (b) (c) Since L q (µ, X ) is always a norming subspace of L p (µ, X), 1 p + 1 q = 1, and they coincide if and only if X has the RNP with respect to µ [7, Chapter IV], (b) implies X has the RNP, or, X is Asplund. Also for any norming subspace F X, L q (µ, F ) is a norming subspace of L p (µ, X). Hence, (b) also implies X is nicely smooth. (c) (a) If X is nicely smooth and Asplund, by Theorem 2.10, X is the closed linear span of the w*-denting points of B(X ). And it suffices to show that L p (µ, X) = L q (µ, X ) is the closed linear span of the w*-denting points of B(L q (µ, X )). Let F = n α ix i χ A i with x i S(X ) for all i = 1, 2,..., n be a simple function in S(L q (µ, X )). Let ε > 0. Now, for each i = 1, 2,..., n, there exists λ ik R, and x ik, w*-denting points of B(X ), k = 1, 2,..., N, such that x i N k=1 λ ikx ik < ε. For k = 1, 2,..., N. Define F k = n α i λ ik x ik χ A i. Since each x ik is a w*-denting point of B(X ), for each k, F k / F k is a w*- denting point of B(L q (µ, X )) [1, Lemma 10]. And, N F q n N n q F k = α i x i χ Ai α i λ ik x ik χ A i k=1 q k=1 q n q N = α i q x i λ ik x ik µ(a i ) < k=1 n ε q α i q µ(a i ) ε q F q q ε. The analogues of the following results for Property (II) were obtained in [3]. Proposition Let K be a compact Hausdorff space, then C(K, X) is nicely smooth if and only if K is finite and X is nicely smooth.

15 on nicely smooth banach spaces 41 Proof. For a compact Hausdorff space K and a Banach space X, the set A = {δ(k) x : k K, x S(X )} B(C(K, X) ) is a norming set for C(K, X). So, if C(K, X) is nicely smooth, C(K, X) = span(a). It follows that K admits no nonatomic measure, whence K is scattered. Now, let K 1 denote the set of isolated points of K. Then K 1 is dense in K, so, the set A 1 = {δ(k) x : k K 1, x S(X )} is also norming. Thus, C(K, X) = span(a 1 ). But if k K \K 1, then for any x S(X ), δ(k) x / span(a 1 ). Hence, K = K 1, whence K must be finite. And if k 0 K 1, x χ {k0 }x is an isometric embedding of X into C(K, X) as an M-summand. Thus, X is nicely smooth. The converse is immediate from Theorem 3.3. Remark (a) It is immediate that for C(K) spaces Property (II), the BGP and nice smoothness (indeed, any of the conditions of Theorem 2.10) are equivalent, and are equivalent to reflexivity. (b) It follows from the above and [3, Theorem 3.9] that C(K, X) has Property (II) if and only if K is finite and X has Property (II). Only the special case of C(K) is noted in [3]. Proposition Let X be a Banach space such that there exists a bounded net {K α } of compact operators such that K α Id in the weak operator topology. If L(X) is nicely smooth, then X is finite dimensional. Proof. For x X, x X, let x x denote the functional defined on L(X) by (x x )(T ) = x (T (x)). Then x x = x x. And, since T = sup{x (T (x) : x = 1, x = 1} = sup{(x x )(T ) : x = 1, x = 1}, it follows that A = {x x : x = 1, x = 1} is a norming set, and hence, L(X) = span(a). Claim : K α Id weakly. Since {K α } is bounded, it suffices to check that K α Id on A, i.e., to check x (K α (x)) x (x) for all x = 1, x = 1. But, K α (x) x weakly, hence the claim. Thus, Id is a compact operator, so that X is finite dimensional. Proposition For a compact Hausdorff space K, L(X, C(K)) is nicely smooth if and only if K(X, C(K)) is nicely smooth if and only if X is reflexive and K is finite.

16 42 p. bandyopadhyay, s. basu Proof. Suppose L(X, C(K)) is nicely smooth. By definition of the norm, A = {δ(k) x : x B(X), k K} is a norming set for L(X, C(K)), and hence, L(X, C(K)) = span(a). It follows that L(X, C(K)) = K(X, C(K)) and that K(X, C(K)) is nicely smooth. Now, from the easily established identification, K(X, C(K)) = C(K, X ) and Proposition 3.10, it follows that K(X, C(K)) is nicely smooth if and only if K is finite and X is nicely smooth, which, in turn, is equivalent to K is finite and X is reflexive. Also, if K is finite, C(K) is finite dimensional, so that L(X, C(K)) = K(X, C(K)). This completes the proof. Coming to general tensor product spaces, the proof of [12, Theorem 5.2] combined with Theorem 2.10 actually shows that: Theorem If X, Y are nicely smooth Asplund spaces, then X ε Y is nicely smooth. We prove the converse for general Banach spaces. Theorem Let X, Y be Banach spaces such that X ε Y is nicely smooth. Then both X and Y are nicely smooth. Proof. Let M and N be norming subspaces of X and Y respectively. Then B(M) and B(N) are norming sets for X and Y respectively. Hence, co w (B(M)) = B(X ) and co w (B(N)) = B(Y ). Thus B(X ) B(Y ) = co w (B(M)) co w (B(N)). By definition of the injective norm, B(X ) B(Y ) is a norming set for X ε Y. Thus it follows that co(b(m)) co(b(n)) is a norming set for X ε Y. Since co(b(m) B(N)) co(b(m)) co(b(n)), it follows that co(b(m) B(N)) and hence B(M) B(N) is a norming set for X ε Y. And since this space is nicely smooth, (X ε Y ) = span(b(m) B(N)). Suppose x X, then for any y S(Y ) and ε > 0, there exist f i B(M), e i B(N) and λ i R such that x y n λ i f i e i < ε.

17 on nicely smooth banach spaces 43 Applying to elementary tensors, this implies n (x y λ i f i e i )(x y) < ε x y = = x (x)y (y) x (x)y n λ i f i (x)e i (y) < ε x y n λ i f i (x)e i < ε x for all x X. for all x X, y Y for all x X, y Y Let x E = kerf i. Then, x (x)y < ε x, i.e., x (x) < ε x. That is, x E < ε. This implies d(x, span{f i }) < ε. It follows that x M and hence X is nicely smooth. Similarly for Y. It seems difficult to obtain analogues of Theorems 3.14 and 3.15 for the projective tensor product. However, we have the following Proposition Suppose X, Y are Banach spaces such that X has the approximation property and L(X, Y ) = K(X, Y ), i.e., any bounded linear operator from X to Y is compact. Then the following are equivalent: (a) K(X, Y ) is nicely smooth. (b) X, Y are reflexive (and hence nicely smooth). (c) X π Y is reflexive (and hence nicely smooth). Proof. (a) (b) Since X has the approximation property, K(X, Y ) = X ε Y and it follows from Theorem 3.15 that X and Y are nicely smooth, and therefore, X and Y are reflexive. (b) (c) This is a well-known result of Holub (see [7]). (c) (a) X and Y being closed subspaces of the reflexive space X π Y are themselves reflexive and from K(X, Y ) = (X π Y ) = X π Y it follows that K(X, Y ) is reflexive, and hence, nicely smooth.

18 44 p. bandyopadhyay, s. basu Acknowledgements The authors would like to thank the referee for his suggestions that improved the presentation of the paper and also for observing Theorem 3.6. A major portion of this work is contained in the second author s Ph. D. Thesis [2] written under the supervision of Professor A. K. Roy. It is a pleasure to thank him. The first author would also like to thank Professors Gilles Godefroy and TSSRK Rao for many useful discussions and Department of Mathematics of the Pondicherry University for bringing them together. References [1] Bandyopadhyay, P., Roy, A.K., Some stability results for Banach spaces with the Mazur intersection property, Indagatione Math. New Series 1 (2) (1990), [2] Basu, S., The Asymptotic Norming Properties and Related Themes in Banach Spaces, Ph. D. Thesis, Indian Statistical Institute, Calcutta, [3] Basu, S., Rao, TSSRK, Some stability results for asymptotic norming properties of Banach spaces, Colloq. Math. 75 (1998), [4] Chen, D., Hu, Z., Lin, B.L., Balls intersection properties of Banach spaces, Bull. Austral. Math. Soc. 45 (1992), [5] Chen, D., Lin, B.L., Ball separation properties in Banach spaces, Rocky Mountain J. Math. 28 (1998), [6] Chen, D., Lin, B.L., Ball topology on Banach spaces, Houston J. Math. 22 (1996), [7] Diestel, J., Uhl, J.J., Jr., Vector Measures, Math. Surveys No. 15, Amer. Math. Soc., Providence, R.I., [8] Giles, J.R., Gregory, D.A., Sims, B., Characterisation of normed linear spaces with the Mazur s intersection property, Bull. Austral. Math. Soc. 18 (1978), [9] Godefroy, G., Nicely Smooth Banach Spaces, Longhorn Notes, The University of Texas at Austin, Functional Analysis Seminar ( ), [10] Godefroy, G., Existence and uniqueness of isometric preduals: a survey, Contemp. Math. 85 (1989), [11] Godefroy, G., Kalton, N.J., The ball topology and its application, Contemp. Math. 85 (1989), [12] Godefroy, G., Saphar, P.D., Duality in spaces of operators and smooth norms on Banach spaces, Illinois J. Math. 32 (1988), [13] Harmand, P., Werner, D., Werner, W., M-Ideals in Banach Spaces and Banach Algebras, Lecture Notes Math., 1547, Springer-Verlag, [14] Hu, Z., Lin, B.L., Smoothness and asymptotic norming properties in Banach spaces, Bull. Austral. Math. Soc. 45 (1992),

19 on nicely smooth banach spaces 45 [15] Jiménez Sevilla, M., Moreno, J.P., Renorming Banach spaces with the Mazur intersection property, J. Funct. Anal. 144 (1997), [16] Rosenthal, H., On relatively disjoint families of measures, with some applications to Banach space theory, Studia Math. 37 (1970), [17] Rosenthal, H., On the structure of non-dentable closed bounded convex sets, Advances in Math. 70 (1988), [18] Simmons, G.F., Introduction to Topology and Modern Analysis, McGraw- Hill, New York, 1963.

### ON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE. 1. Introduction

ON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE J.M. CALABUIG, J. RODRÍGUEZ, AND E.A. SÁNCHEZ-PÉREZ Abstract. Let m be a vector measure taking values in a Banach space X. We prove that

### Metric Spaces. Chapter 1

Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergence

### SOLUTIONS TO ASSIGNMENT 1 MATH 576

SOLUTIONS TO ASSIGNMENT 1 MATH 576 SOLUTIONS BY OLIVIER MARTIN 13 #5. Let T be the topology generated by A on X. We want to show T = J B J where B is the set of all topologies J on X with A J. This amounts

### Duality of linear conic problems

Duality of linear conic problems Alexander Shapiro and Arkadi Nemirovski Abstract It is well known that the optimal values of a linear programming problem and its dual are equal to each other if at least

### FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES

FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES CHRISTOPHER HEIL 1. Cosets and the Quotient Space Any vector space is an abelian group under the operation of vector addition. So, if you are have studied

### INTEGRAL OPERATORS ON THE PRODUCT OF C(K) SPACES

INTEGRAL OPERATORS ON THE PRODUCT OF C(K) SPACES FERNANDO BOMBAL AND IGNACIO VILLANUEVA Abstract. We study and characterize the integral multilinear operators on a product of C(K) spaces in terms of the

### BANACH AND HILBERT SPACE REVIEW

BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but

### F. ABTAHI and M. ZARRIN. (Communicated by J. Goldstein)

Journal of Algerian Mathematical Society Vol. 1, pp. 1 6 1 CONCERNING THE l p -CONJECTURE FOR DISCRETE SEMIGROUPS F. ABTAHI and M. ZARRIN (Communicated by J. Goldstein) Abstract. For 2 < p

### Separation Properties for Locally Convex Cones

Journal of Convex Analysis Volume 9 (2002), No. 1, 301 307 Separation Properties for Locally Convex Cones Walter Roth Department of Mathematics, Universiti Brunei Darussalam, Gadong BE1410, Brunei Darussalam

### 1. Prove that the empty set is a subset of every set.

1. Prove that the empty set is a subset of every set. Basic Topology Written by Men-Gen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since

### 16.3 Fredholm Operators

Lectures 16 and 17 16.3 Fredholm Operators A nice way to think about compact operators is to show that set of compact operators is the closure of the set of finite rank operator in operator norm. In this

### The Dirichlet Unit Theorem

Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if

### Geometrical Characterization of RN-operators between Locally Convex Vector Spaces

Geometrical Characterization of RN-operators between Locally Convex Vector Spaces OLEG REINOV St. Petersburg State University Dept. of Mathematics and Mechanics Universitetskii pr. 28, 198504 St, Petersburg

### Chapter 1. Metric Spaces. Metric Spaces. Examples. Normed linear spaces

Chapter 1. Metric Spaces Metric Spaces MA222 David Preiss d.preiss@warwick.ac.uk Warwick University, Spring 2008/2009 Definitions. A metric on a set M is a function d : M M R such that for all x, y, z

### Further Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1

Further Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1 J. Zhang Institute of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing

### IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction

IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible

### MA651 Topology. Lecture 6. Separation Axioms.

MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples

### Metric Spaces. Chapter 7. 7.1. Metrics

Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some

### Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011

Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely

### Mathematical Methods of Engineering Analysis

Mathematical Methods of Engineering Analysis Erhan Çinlar Robert J. Vanderbei February 2, 2000 Contents Sets and Functions 1 1 Sets................................... 1 Subsets.............................

### Notes on metric spaces

Notes on metric spaces 1 Introduction The purpose of these notes is to quickly review some of the basic concepts from Real Analysis, Metric Spaces and some related results that will be used in this course.

### Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

### CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.

CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,

### SOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 3. Spaces with special properties

SOLUTIONS TO EXERCISES FOR MATHEMATICS 205A Part 3 Fall 2008 III. Spaces with special properties III.1 : Compact spaces I Problems from Munkres, 26, pp. 170 172 3. Show that a finite union of compact subspaces

### EDWARD W. ODELL BIOGRAPHICAL DATA. EDUCATION Ph.D. Massachusetts Institute of Technology 1975 B.A. SUNY at Binghamton 1969

EDWARD W. ODELL BIOGRAPHICAL DATA EDUCATION Ph.D. Massachusetts Institute of Technology 1975 B.A. SUNY at Binghamton 1969 PROFESSIONAL EXPERIENCE 2005- John T. Stuart III Centennial Professor of Mathematics

### POWER SETS AND RELATIONS

POWER SETS AND RELATIONS L. MARIZZA A. BAILEY 1. The Power Set Now that we have defined sets as best we can, we can consider a sets of sets. If we were to assume nothing, except the existence of the empty

### Math212a1010 Lebesgue measure.

Math212a1010 Lebesgue measure. October 19, 2010 Today s lecture will be devoted to Lebesgue measure, a creation of Henri Lebesgue, in his thesis, one of the most famous theses in the history of mathematics.

### THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS

THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear

### Ideal Class Group and Units

Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal ideals

### Metric Spaces Joseph Muscat 2003 (Last revised May 2009)

1 Distance J Muscat 1 Metric Spaces Joseph Muscat 2003 (Last revised May 2009) (A revised and expanded version of these notes are now published by Springer.) 1 Distance A metric space can be thought of

### {f 1 (U), U F} is an open cover of A. Since A is compact there is a finite subcover of A, {f 1 (U 1 ),...,f 1 (U n )}, {U 1,...

44 CHAPTER 4. CONTINUOUS FUNCTIONS In Calculus we often use arithmetic operations to generate new continuous functions from old ones. In a general metric space we don t have arithmetic, but much of it

### The Positive Supercyclicity Theorem

E extracta mathematicae Vol. 19, Núm. 1, 145 149 (2004) V Curso Espacios de Banach y Operadores. Laredo, Agosto de 2003. The Positive Supercyclicity Theorem F. León Saavedra Departamento de Matemáticas,

### Finite dimensional topological vector spaces

Chapter 3 Finite dimensional topological vector spaces 3.1 Finite dimensional Hausdorff t.v.s. Let X be a vector space over the field K of real or complex numbers. We know from linear algebra that the

University of Oslo MAT2 Project The Banach-Tarski Paradox Author: Fredrik Meyer Supervisor: Nadia S. Larsen Abstract In its weak form, the Banach-Tarski paradox states that for any ball in R, it is possible

### Chapter 5. Banach Spaces

9 Chapter 5 Banach Spaces Many linear equations may be formulated in terms of a suitable linear operator acting on a Banach space. In this chapter, we study Banach spaces and linear operators acting on

### Surface bundles over S 1, the Thurston norm, and the Whitehead link

Surface bundles over S 1, the Thurston norm, and the Whitehead link Michael Landry August 16, 2014 The Thurston norm is a powerful tool for studying the ways a 3-manifold can fiber over the circle. In

### CHAPTER 5. Product Measures

54 CHAPTER 5 Product Measures Given two measure spaces, we may construct a natural measure on their Cartesian product; the prototype is the construction of Lebesgue measure on R 2 as the product of Lebesgue

### MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set.

MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. Vector space A vector space is a set V equipped with two operations, addition V V (x,y) x + y V and scalar

### CURVES WHOSE SECANT DEGREE IS ONE IN POSITIVE CHARACTERISTIC. 1. Introduction

Acta Math. Univ. Comenianae Vol. LXXXI, 1 (2012), pp. 71 77 71 CURVES WHOSE SECANT DEGREE IS ONE IN POSITIVE CHARACTERISTIC E. BALLICO Abstract. Here we study (in positive characteristic) integral curves

### Point Set Topology. A. Topological Spaces and Continuous Maps

Point Set Topology A. Topological Spaces and Continuous Maps Definition 1.1 A topology on a set X is a collection T of subsets of X satisfying the following axioms: T 1.,X T. T2. {O α α I} T = α IO α T.

### Ri and. i=1. S i N. and. R R i

The subset R of R n is a closed rectangle if there are n non-empty closed intervals {[a 1, b 1 ], [a 2, b 2 ],..., [a n, b n ]} so that R = [a 1, b 1 ] [a 2, b 2 ] [a n, b n ]. The subset R of R n is an

### Properties of BMO functions whose reciprocals are also BMO

Properties of BMO functions whose reciprocals are also BMO R. L. Johnson and C. J. Neugebauer The main result says that a non-negative BMO-function w, whose reciprocal is also in BMO, belongs to p> A p,and

### PROPERTIES OF SOME NEW SEMINORMED SEQUENCE SPACES DEFINED BY A MODULUS FUNCTION

STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume L, Number 3, September 2005 PROPERTIES OF SOME NEW SEMINORMED SEQUENCE SPACES DEFINED BY A MODULUS FUNCTION YAVUZ ALTIN AYŞEGÜL GÖKHAN HIFSI ALTINOK Abstract.

### Introduction to Topology

Introduction to Topology Tomoo Matsumura November 30, 2010 Contents 1 Topological spaces 3 1.1 Basis of a Topology......................................... 3 1.2 Comparing Topologies.......................................

### TOPOLOGY: THE JOURNEY INTO SEPARATION AXIOMS

TOPOLOGY: THE JOURNEY INTO SEPARATION AXIOMS VIPUL NAIK Abstract. In this journey, we are going to explore the so called separation axioms in greater detail. We shall try to understand how these axioms

### MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix.

MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix. Nullspace Let A = (a ij ) be an m n matrix. Definition. The nullspace of the matrix A, denoted N(A), is the set of all n-dimensional column

### Three observations regarding Schatten p classes

Three observations regarding Schatten p classes Gideon Schechtman Abstract The paper contains three results, the common feature of which is that they deal with the Schatten p class. The first is a presentation

### The Ideal Class Group

Chapter 5 The Ideal Class Group We will use Minkowski theory, which belongs to the general area of geometry of numbers, to gain insight into the ideal class group of a number field. We have already mentioned

### 1 if 1 x 0 1 if 0 x 1

Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

### A simple criterion on degree sequences of graphs

Discrete Applied Mathematics 156 (2008) 3513 3517 Contents lists available at ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam Note A simple criterion on degree

### CHAPTER 1 BASIC TOPOLOGY

CHAPTER 1 BASIC TOPOLOGY Topology, sometimes referred to as the mathematics of continuity, or rubber sheet geometry, or the theory of abstract topological spaces, is all of these, but, above all, it is

### No: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics

No: 10 04 Bilkent University Monotonic Extension Farhad Husseinov Discussion Papers Department of Economics The Discussion Papers of the Department of Economics are intended to make the initial results

### 2.3 Convex Constrained Optimization Problems

42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions

### x if x 0, x if x < 0.

Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the

### An example of a computable

An example of a computable absolutely normal number Verónica Becher Santiago Figueira Abstract The first example of an absolutely normal number was given by Sierpinski in 96, twenty years before the concept

### ON A GLOBALIZATION PROPERTY

APPLICATIONES MATHEMATICAE 22,1 (1993), pp. 69 73 S. ROLEWICZ (Warszawa) ON A GLOBALIZATION PROPERTY Abstract. Let (X, τ) be a topological space. Let Φ be a class of realvalued functions defined on X.

### THE BANACH CONTRACTION PRINCIPLE. Contents

THE BANACH CONTRACTION PRINCIPLE ALEX PONIECKI Abstract. This paper will study contractions of metric spaces. To do this, we will mainly use tools from topology. We will give some examples of contractions,

### Math 312 Homework 1 Solutions

Math 31 Homework 1 Solutions Last modified: July 15, 01 This homework is due on Thursday, July 1th, 01 at 1:10pm Please turn it in during class, or in my mailbox in the main math office (next to 4W1) Please

### ON FIBER DIAMETERS OF CONTINUOUS MAPS

ON FIBER DIAMETERS OF CONTINUOUS MAPS PETER S. LANDWEBER, EMANUEL A. LAZAR, AND NEEL PATEL Abstract. We present a surprisingly short proof that for any continuous map f : R n R m, if n > m, then there

### SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH

31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,

### Row Ideals and Fibers of Morphisms

Michigan Math. J. 57 (2008) Row Ideals and Fibers of Morphisms David Eisenbud & Bernd Ulrich Affectionately dedicated to Mel Hochster, who has been an inspiration to us for many years, on the occasion

### 1 Norms and Vector Spaces

008.10.07.01 1 Norms and Vector Spaces Suppose we have a complex vector space V. A norm is a function f : V R which satisfies (i) f(x) 0 for all x V (ii) f(x + y) f(x) + f(y) for all x,y V (iii) f(λx)

### Lebesgue Measure on R n

8 CHAPTER 2 Lebesgue Measure on R n Our goal is to construct a notion of the volume, or Lebesgue measure, of rather general subsets of R n that reduces to the usual volume of elementary geometrical sets

### Notes V General Equilibrium: Positive Theory. 1 Walrasian Equilibrium and Excess Demand

Notes V General Equilibrium: Positive Theory In this lecture we go on considering a general equilibrium model of a private ownership economy. In contrast to the Notes IV, we focus on positive issues such

### Research Article Stability Analysis for Higher-Order Adjacent Derivative in Parametrized Vector Optimization

Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 510838, 15 pages doi:10.1155/2010/510838 Research Article Stability Analysis for Higher-Order Adjacent Derivative

### Analytic cohomology groups in top degrees of Zariski open sets in P n

Analytic cohomology groups in top degrees of Zariski open sets in P n Gabriel Chiriacescu, Mihnea Colţoiu, Cezar Joiţa Dedicated to Professor Cabiria Andreian Cazacu on her 80 th birthday 1 Introduction

### Vector Spaces II: Finite Dimensional Linear Algebra 1

John Nachbar September 2, 2014 Vector Spaces II: Finite Dimensional Linear Algebra 1 1 Definitions and Basic Theorems. For basic properties and notation for R N, see the notes Vector Spaces I. Definition

### FIXED POINT SETS OF FIBER-PRESERVING MAPS

FIXED POINT SETS OF FIBER-PRESERVING MAPS Robert F. Brown Department of Mathematics University of California Los Angeles, CA 90095 e-mail: rfb@math.ucla.edu Christina L. Soderlund Department of Mathematics

### SOME RESULTS ON HYPERCYCLICITY OF TUPLE OF OPERATORS. Abdelaziz Tajmouati Mohammed El berrag. 1. Introduction

italian journal of pure and applied mathematics n. 35 2015 (487 492) 487 SOME RESULTS ON HYPERCYCLICITY OF TUPLE OF OPERATORS Abdelaziz Tajmouati Mohammed El berrag Sidi Mohamed Ben Abdellah University

### ON GENERALIZED RELATIVE COMMUTATIVITY DEGREE OF A FINITE GROUP. A. K. Das and R. K. Nath

International Electronic Journal of Algebra Volume 7 (2010) 140-151 ON GENERALIZED RELATIVE COMMUTATIVITY DEGREE OF A FINITE GROUP A. K. Das and R. K. Nath Received: 12 October 2009; Revised: 15 December

### LINEAR ALGEBRA W W L CHEN

LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,

### 4: SINGLE-PERIOD MARKET MODELS

4: SINGLE-PERIOD MARKET MODELS Ben Goldys and Marek Rutkowski School of Mathematics and Statistics University of Sydney Semester 2, 2015 B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market

### 1. Let X and Y be normed spaces and let T B(X, Y ).

Uppsala Universitet Matematiska Institutionen Andreas Strömbergsson Prov i matematik Funktionalanalys Kurs: NVP, Frist. 2005-03-14 Skrivtid: 9 11.30 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok

### Topics in weak convergence of probability measures

This version: February 24, 1999 To be cited as: M. Merkle, Topics in weak convergence of probability measures, Zb. radova Mat. Inst. Beograd, 9(17) (2000), 235-274 Topics in weak convergence of probability

### Lecture Notes on Measure Theory and Functional Analysis

Lecture Notes on Measure Theory and Functional Analysis P. Cannarsa & T. D Aprile Dipartimento di Matematica Università di Roma Tor Vergata cannarsa@mat.uniroma2.it daprile@mat.uniroma2.it aa 2006/07 Contents

### A survey on multivalued integration 1

Atti Sem. Mat. Fis. Univ. Modena Vol. L (2002) 53-63 A survey on multivalued integration 1 Anna Rita SAMBUCINI Dipartimento di Matematica e Informatica 1, Via Vanvitelli - 06123 Perugia (ITALY) Abstract

### FIBER PRODUCTS AND ZARISKI SHEAVES

FIBER PRODUCTS AND ZARISKI SHEAVES BRIAN OSSERMAN 1. Fiber products and Zariski sheaves We recall the definition of a fiber product: Definition 1.1. Let C be a category, and X, Y, Z objects of C. Fix also

### NOTES ON CATEGORIES AND FUNCTORS

NOTES ON CATEGORIES AND FUNCTORS These notes collect basic definitions and facts about categories and functors that have been mentioned in the Homological Algebra course. For further reading about category

### RIGIDITY OF HOLOMORPHIC MAPS BETWEEN FIBER SPACES

RIGIDITY OF HOLOMORPHIC MAPS BETWEEN FIBER SPACES GAUTAM BHARALI AND INDRANIL BISWAS Abstract. In the study of holomorphic maps, the term rigidity refers to certain types of results that give us very specific

### 1 Sets and Set Notation.

LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most

### 3. INNER PRODUCT SPACES

. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.

### Convex analysis and profit/cost/support functions

CALIFORNIA INSTITUTE OF TECHNOLOGY Division of the Humanities and Social Sciences Convex analysis and profit/cost/support functions KC Border October 2004 Revised January 2009 Let A be a subset of R m

### (Basic definitions and properties; Separation theorems; Characterizations) 1.1 Definition, examples, inner description, algebraic properties

Lecture 1 Convex Sets (Basic definitions and properties; Separation theorems; Characterizations) 1.1 Definition, examples, inner description, algebraic properties 1.1.1 A convex set In the school geometry

### About the inverse football pool problem for 9 games 1

Seventh International Workshop on Optimal Codes and Related Topics September 6-1, 013, Albena, Bulgaria pp. 15-133 About the inverse football pool problem for 9 games 1 Emil Kolev Tsonka Baicheva Institute

### 4.5 Linear Dependence and Linear Independence

4.5 Linear Dependence and Linear Independence 267 32. {v 1, v 2 }, where v 1, v 2 are collinear vectors in R 3. 33. Prove that if S and S are subsets of a vector space V such that S is a subset of S, then

### Linear Codes. In the V[n,q] setting, the terms word and vector are interchangeable.

Linear Codes Linear Codes In the V[n,q] setting, an important class of codes are the linear codes, these codes are the ones whose code words form a sub-vector space of V[n,q]. If the subspace of V[n,q]

### 1 Local Brouwer degree

1 Local Brouwer degree Let D R n be an open set and f : S R n be continuous, D S and c R n. Suppose that the set f 1 (c) D is compact. (1) Then the local Brouwer degree of f at c in the set D is defined.

### and s n (x) f(x) for all x and s.t. s n is measurable if f is. REAL ANALYSIS Measures. A (positive) measure on a measurable space

RAL ANALYSIS A survey of MA 641-643, UAB 1999-2000 M. Griesemer Throughout these notes m denotes Lebesgue measure. 1. Abstract Integration σ-algebras. A σ-algebra in X is a non-empty collection of subsets

### NOTES ON LINEAR TRANSFORMATIONS

NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all

### A CHARACTERIZATION OF C k (X) FOR X NORMAL AND REALCOMPACT

Bol. Soc. Mat. Mexicana (3) Vol. 12, 2006 A CHARACTERIZATION OF C k (X) FOR X NORMAL AND REALCOMPACT F. MONTALVO, A. A. PULGARÍN AND B. REQUEJO Abstract. We present some internal conditions on a locally

### The degree, size and chromatic index of a uniform hypergraph

The degree, size and chromatic index of a uniform hypergraph Noga Alon Jeong Han Kim Abstract Let H be a k-uniform hypergraph in which no two edges share more than t common vertices, and let D denote the

### REAL ANALYSIS LECTURE NOTES: 1.4 OUTER MEASURE

REAL ANALYSIS LECTURE NOTES: 1.4 OUTER MEASURE CHRISTOPHER HEIL 1.4.1 Introduction We will expand on Section 1.4 of Folland s text, which covers abstract outer measures also called exterior measures).

### The Markov-Zariski topology of an infinite group

Mimar Sinan Güzel Sanatlar Üniversitesi Istanbul January 23, 2014 joint work with Daniele Toller and Dmitri Shakhmatov 1. Markov s problem 1 and 2 2. The three topologies on an infinite group 3. Problem

### Recall that two vectors in are perpendicular or orthogonal provided that their dot

Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal

### it is easy to see that α = a

21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UF. Therefore

### 8.1 Examples, definitions, and basic properties

8 De Rham cohomology Last updated: May 21, 211. 8.1 Examples, definitions, and basic properties A k-form ω Ω k (M) is closed if dω =. It is exact if there is a (k 1)-form σ Ω k 1 (M) such that dσ = ω.

Adaptive Online Gradient Descent Peter L Bartlett Division of Computer Science Department of Statistics UC Berkeley Berkeley, CA 94709 bartlett@csberkeleyedu Elad Hazan IBM Almaden Research Center 650

### MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column