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A Continuity Characterization of Asplund Spaces

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A Continuity Characterization of Asplund Spaces Bull. Aust. Math. Soc. 83 (2011), 450–455 doi:10.1017/S0004972710001978 A CONTINUITY CHARACTERIZATION OF ASPLUND SPACES J. R. GILES (Received 5 August 2010) Abstract A Banach space is an Asplund space if every continuous gauge has a point where the subdifferential mapping is Hausdorff weak upper semi-continuous with weakly compact image. This contributes towards the solution of a problem posed by Godefroy, Montesinos and Zizler. 2010 Mathematics subject classification: primary 46B22; secondary 58C20. Keywords and phrases: Fréchet differentiability, subdifferential mapping, weak upper semi-continuity, gauge, extreme point, polar, Radon–Nikodým property. 1. Introduction A continuous convex function φ on a nonempty open convex subset A of a Banach space .X; k · k/ is said to be strongly subdifferentiable at x 2 A if, given > 0, there exists δ > 0 such that 0 0 ≤ φ.x C y/ − φ.x/ − φC.x/.y/ ≤ kyk for all kyk < δ; and was the subject of an interesting study in T11U. The function φ is Fréchet 0 differentiable at x if also φC.x/.y/ is linear in y. A Banach space X is an Asplund space if every continuous convex function φ on a nonempty open convex subset A of X is Fréchet differentiable at the points of a dense Gδ subset of A. It is known that X is an Asplund space if X possesses an equivalent strongly subdifferentiable norm; see T5, Theorem 5.1, p. 68U and T9, Proposition 8, p. 64U. Further, if X is separable then it is an Asplund space if every equivalent norm has a nonzero point where it is strongly subdifferentiable T12, Theorem 1, p. 494U. It has remained an open question whether this last result can be extended to nonseparable spaces T12, Problem 6(v), p. 501U. Our aim here is to work towards such an extension. Given a continuous convex function φ on a nonempty open convex subset A of a Banach space X, the subdifferential of φ at x 2 A is the nonempty weak∗ compact convex subset ∗ 0 @φ.x/ ≡ f f 2 X V f .y/ ≤ φC.x/.y/ for all y 2 Xg: c 2011 Australian Mathematical Publishing Association Inc. 0004-9727/2011 $16.00 450 Downloaded from https://www.cambridge.org/core. IP address: 170.106.34.90, on 29 Sep 2021 at 05:36:56, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972710001978 [2] A continuity characterization of Asplund spaces 451 The set-valued subdifferential mapping x 7! @φ.x/ is always Hausdorff weak∗ upper semi-continuous on A; that is, given x 2 A and weak∗ open neighbourhood W of 0 in X ∗, there exists δ > 0 such that @φ.B.xI δ// ⊆ @φ.x/ C W by T13, Proposition 2.5, p. 19U. Further, φ is strongly subdifferentiable at x 2 A if and only if, given > 0, there exists δ > 0 such that @φ.B.xI δ// ⊆ @φ.x/ C B.X ∗/ by T8, Theorem 3.2, p. 28U. We work with the slightly weaker property; we say that the subdifferential mapping x 7! @φ.x/ is Hausdorff weak upper semi-continuous at x 2 A if, given a weak open neighbourhood V of 0 in X ∗, there exists δ > 0 such that @φ.B.xI δ// ⊆ @φ.x/ C V: This has been studied in T1U and for subdifferentials of the norm in T10U. Contreras and Payá showed that X is an Asplund space if it possesses an equivalent norm whose subdifferential mapping is Hausdorff weak upper semi-continuous on its unit sphere T3, Theorem 1.2, p. 453U. Here we show that X is an Asplund space if every continuous gauge p on X has a point in its domain where the subdifferential mapping x 7! @p.x/ is Hausdorff weak upper semi-continuous with weakly compact image. (We note that every continuous gauge p on a Banach space X is always strongly differentiable at 0 so is always Hausdorff weak upper semi-continuous there but the subdifferential @p.0/ is not in general weakly compact.) 2. The density property We explore the effect of Hausdorff weak upper semi-continuity on higher dual spaces. Consider a bounded closed convex set K with 0 2 int K in a Banach space X. The gauge p of X defined by p.x/ ≡ inffλ > 0 V x 2 λK g is a continuous positive sublinear functional on X. The polar of K is the subset K 0 of X ∗ defined by K 0 ≡ f f 2 X ∗ V f .x/ ≤ 1 for all x 2 K g and is weak∗ compact convex and 0 2 int K 0. The gauge p∗ of K 0 on X ∗ is continuous and weak∗ lower semi-continuous. We denote by K 00 the polar of K 0 !∗ ∗∗ ∗∗ 00 ∗∗ 00 D O ∗∗j D in X and by p the gauge of K on X , and note that K K and p XO p, T7, Lemma 3.1(i), p. 255U. We now characterize Hausdorff weak upper semi-continuity by a density property. Such a characterization for subdifferentials of norms was given in T6, Theorem 3.1, p. 103U and was proved more generally for subdifferentials of proper lower semi- continuous convex functions in T1, Theorem 3.1, p. 98U. Since our theorem in Section3 concerns gauges of bounded closed convex sets we include a direct proof for subdifferentials of gauges. Downloaded from https://www.cambridge.org/core. IP address: 170.106.34.90, on 29 Sep 2021 at 05:36:56, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972710001978 452 J. R. Giles [3] THEOREM 2.1. Consider a Banach space X and a bounded closed convex subset K with 0 2 int K . The continuous gauge p of K has subdifferential mapping x 7! @p.x/ Hausdorff weak upper semi-continuous at x 2 X if and only if @\p.x/ is weak∗ dense in @p∗∗.xO/: PROOF. Suppose that the density property does not hold. Then there exists F0 2 !∗ @p∗∗.xO/ which we can strongly separate from @\p.x/ by a weak∗ continuous linear ∗ functional F 2 S.X ∗∗/. So there exists a weak∗ neighbourhood N of 0 in X ∗∗∗ 0 F0 generated by F0 such that ∗ F 2= @\p.x/ C N : 0 F0 O 0 ∗ 000 0 Since K is weak dense in K , for each n 2 N there exists fn 2 K such that 1 1 j. f − F /.xO/j < and j. f − F /.F /j < : n 0 n n 0 0 n ∗∗ ∗∗ ∗∗ ∗∗ Since F0 2 @p .xO/ then F0.xO/ D p .xO/ and F0.F/ ≤ p .F/ for all F 2 X . Then ∗∗ 0 j fn.xO/ − p .xO/j < 1=n and, since fn 2 K , fn.y/ ≤ p.y/ for all y 2 X. So fn.xO/ ≥ ∗∗ p .xO/ − 1=n D p.x/ − 1=n and fn.y − x/ ≤ p.y/ − p.x/ C 1=n for all y 2 X. By 2 2 2 the Brøndsted–Rockafellar theorem for each n N there exist yn X and fyn @p.yn/ such that 1 1 k fn − fy k ≤ p and kx − ynk ≤ p n n n by T13, Theorem 3.17, p. 48U. But if the subdifferential mapping x 7! @p.x/ is Hausdorff weak upper semi-continuous at x, then, for sufficiently large n 2 N, 2 C fyn @p.x/ NF0 ∗ where N is the weak neighbourhood of 0 in X ∗, the restriction of N in X ∗∗∗. F0 F0 However, 1 1 j. fy − F /.F /j ≤ k fy − fnk C j. fn − F /.F /j ≤ p C ; n 0 0 n 0 0 n n which contradicts our original separation. Conversely, suppose that @\p.x/ is weak∗ dense in @p∗∗.xO/: Consider a weak neighbourhood V of 0 in X ∗. Now V is the restriction of a weak∗ neighbourhood V ∗ of 0 in X ∗∗∗. Since the subdifferential mapping F 7! @p∗∗.F/ is Hausdorff weak∗ upper semi-continuous at xO 2 X ∗∗, there exists δ > 0 such that ∗∗ O ⊆ ∗∗ O C 1 ∗ ∗∗ O ⊆ C 1 ∗ @p .B.x; δ// @p .x/ 2 V but @p .x/ @\p.x/ 2 V so @p.B.x; δ// ⊆ @p.x/ C V . 2 It is instructive to see how the density property has implications for the density of extreme points. Downloaded from https://www.cambridge.org/core. IP address: 170.106.34.90, on 29 Sep 2021 at 05:36:56, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972710001978 [4] A continuity characterization of Asplund spaces 453 COROLLARY 2.2. A continuous gauge p on a Banach space X with subdifferential mapping x 7! @p.x/ Hausdorff weak upper semi-continuous at x 2 X satisfies: !∗ (i) @p∗∗.xO/ D co ext @\p.x/ I !∗ (ii) ext @p∗∗.xO/ ⊆ ext @\p.x/ : ∗∗ PROOF. Consider F0 an extreme point of @p .xO/. Choquet’s theorem T4, p. 77U, gives us that weak∗ slices of @p∗∗.xO/, sets of the form Sl.@p∗∗.xO/; F; δ/ ≡ fF 2 @p∗∗.xO/ V F.F/ > sup @p∗∗.xO/.F/ − δg; ∗ containing F0 form a weak neighbourhood base for F0. It then follows from Theorem 2.1 that there exists an element of @\p.x/ in the slice and, moreover, an extreme point of @\p.x/ .
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