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On Asplund Functions Comment.Math.Univ.Carolin. 40,1 (1999)121–132 121 On Asplund functions Wee-Kee Tang Abstract. A class of convex functions where the sets of subdifferentials behave like the unit ball of the dual space of an Asplund space is found. These functions, which we called Asplund functions also possess some stability properties. We also give a sufficient condition for a function to be an Asplund function in terms of the upper-semicontinuity of the subdifferential map. Keywords: Fr´echet differentiability, convex functions, Asplund spaces Classification: 46B03 Introduction It is known that a Banach space is an Asplund space if and only if BX∗ is dentable ∗ if and only if (BX∗ , w ) is fragmentable by norm, and if and only if every separable subspace Y of X has separable dual Y ∗. It is also known that being an Asplund space is a three-space property. The purpose of this note is to present a functional version of this theory. We study these equivalent conditions in a certain class of functions which may be defined on a non-Asplund space. For instance, suppose g is a continuous convex function defined on an Asplund space Y and T : X → Y is a bounded linear map. Then regardless of X, the function defined by f = g ◦ T is a generically Fr´echet differentiable convex function. The function f and all convex functions bounded above by f belong to the class which we want to consider. These functions exhibit properties similar to those of continuous convex functions defined on an Asplund space. In Section 1, we present a theorem that consists of several equivalent condi- tions that are well known in the Asplund space version. We call a continuous convex function an Asplund function if it satisfies any of these conditions. As a consequence, a Banach space is an Asplund space if and only if its norm is an Asplund function. In Section 2, we show that the property of being an Asplund function is stable under restriction and taking quotient to a subspace. However, it does not enjoy a three-space like property. Nevertheless, a condition of a subspace is given to ensure a function is Asplund whenever its restriction to that subspace is Asplund. Further, we modify the proof of a theorem in [C-P] to yield a sufficient condition for a function to be Asplund. 122 Wee-Kee Tang Some related problems were studied in [T1] and [T2]. In contrast to these two articles, we do not attempt to establish any approximation theorem in this paper, for we know that Asplund spaces in general may not even admit a Gˆateaux differentiable norm. Notation and preliminaries We will use the standard notation in the theory of convex functions on a Banach space and Banach space theory. Given a continuous convex function f on a Banach space X, the subdifferential of f at a point x is defined by ∂f(x) = {x∗ ∈ X∗ : x∗(y − x) ≤ f(y) − f(x) for all y ∈ X}. The Legendre-Fenchel conjugate f ∗ of f is defined by f ∗(x∗) = sup{(x∗, x) − f(x) : x ∈ X} for all x ∈ X. The function f ∗ is w∗-lower semicontinuous, i.e., {x∗ ∈ X∗ : f ∗(x∗) ≤ r} is w∗-closed for all r ∈ R. A continuous convex function is said to be generically Fr´echet differentiable if it is Fr´echet differentiable on a dense Gδ set. Given a ∗ bounded convex set A ⊂ X , the indicator function δA(·) is a convex function ∗ that takes values zero on A and +∞ elsewhere. The function δA(·) is w -lower semicontinuous if and only if A is w∗-closed. A slice of A ⊂ X∗ is a set of the form S(A, x∗∗, α) = {x∗ ∈ X∗ : x∗∗(x∗) > sup x∗∗(A) − α}, for some x∗∗ ∈ X∗∗ and α > 0. If x∗∗ ∈ X, then S(A, x∗∗, α) is called a w∗-slice. We say that a set F is dentable (w∗-dentable) if for every ε > 0 every bounded subset of F has slices (w∗-slices) of diameter less than ε. Let (Z, τ) be a topological space and ρ be a metric on Z that is not necessarily related to the topology of Z. The space Z is said to be fragmentable by ρ if every non-empty subset of Z admits relatively open sets of arbitrarily small ρ-diameter. We refer the readers to [J-N-R] for the theory of fragmentability. An infinite tree in a set A ⊂ X is a 1 sequence {xn} in A such that xn = 2 (x2n + x2n+1) for each n. An infinite tree such that kx2n −x2n+1k > 2ε for all n is called an infinite ε-tree. Unless otherwise stated, all topological notions in the dual space refer to the norm topology. We refer to [Ph] and [D-G-Z] for all other unexplained notions and results. We also refer the reader to [Y] for an excellent introduction to the theory of Asplund spaces. 1. Asplund functions In this section, we establish using known techniques some equivalent definitions of an Asplund function. These properties will be used in the subsequent sections. We find the following fact useful: Lemma 1. If f is a continuous convex function defined on X, then we have the k·k following inclusions: ∂f(X) ⊂ dom f ∗ ⊂ ∂f(X) . The first inclusion is clear. For the second inclusion, we apply Ekeland’s vari- ational principle as in [F] (see also the proof of (iii) ⇒ (i) in [T1, Theorem 1]). The main result in this section is the following: On Asplund functions 123 Theorem 2. Let f be a continuous convex function defined on a Banach space X. Then the following assertions are equivalent. (1) If h is a continuous convex function on X such that h ≤ f on X, then h is generically Fr´echet differentiable on X. ∗ (2) For each positive integer n, every bounded subset of (Cn, w ), where ∗ ∗ ∗ ∗ Cn = {x ∈ X : f (x ) ≤ n}, is fragmentable by the norm. ∗ (3) For each separable subspace Y of X, the set dom (f↾Y ) is separable. ∗ ∗ (4a) Every compact subset of (dom f , w ) is fragmentable by the norm. ∗ ∗ ∗ (4b) Every compact subset of (dom f , w ) is w -dentable. (5) For every ε> 0, no w∗-compact subset of dom f ∗ contains an ε-tree. (6) Every compact subset of (dom f ∗, w∗) is dentable. If moreover, f is bounded on bounded sets, then the above conditions are also equivalent to: (7) let Y be a separable subspace of X. If h is a continuous convex function on X such that h ≤ f then there is a selector s for ∂h↾Y such that s(Y )= {s(y) : y ∈ Y } is separable. Proof: (1) ⇒ (2). Indeed, for otherwise, there exists a bounded w∗-closed subset A of Cn that is not fragmentable by the norm. The function h(·) = ∗ ((δA(·)+ n) )↾X is bounded above by f and it can be checked that h is nowhere Fr´echet differentiable (cf., for instance, [Ph, 2.18]). (2) ⇒ (3). Let Y be a separable subspace of X. By Lemma 1, to show the ∗ separability of dom (f↾Y ) , it suffices to establish the separability of ∂f↾Y (Y ). Let R : X∗ → Y ∗ be the restriction map. The map R is w∗ to w∗ continuous. By Hahn-Banach theorem, we have (1) R∂f(y)= ∂f↾Y (y) for all y ∈ Y. Y For each positive integer n, we define the sets Cn and Hn as follows: Y ∗ ∗ ∗ ∗ Cn = y ∈ Y : f↾Y (y ) ≤ n , and n Y o Hn = ∂f↾Y (Y ) ∩ Cn . We note that ∞ ∞ ∞ Y Y Hn = ∂f↾Y (Y ) ∩ Cn = ∂f↾Y (Y ) ∩ Cn (2) n=1 n=1 n=1 [ [ ∗ [ = ∂f↾Y (Y ) ∩ dom f↾Y = ∂f↾Y (Y ). ∗ We claim that Hn ⊂ R(Cn) for each n ∈ N. Indeed, let y ∈ Hn, then ∗ ∗ y ∈ ∂f↾Y (y) for some y ∈ Y . According to (1), we can find ay ˆ ∈ ∂f(y) such that R(ˆy∗)= y∗. 124 Wee-Kee Tang We have f ∗(ˆy∗)=(ˆy∗,y) − f(y) ∗ = (y ,y) − f↾Y (y) ∗ ∗ = f↾Y (y ) ≤ n. ∗ ∗ ∗ Thereforey ˆ ∈ Cn, and hence y = R(ˆy ) ∈ R(Cn). From the claim and (2) ∞ we have ∂f↾Y (Y ) ⊂ n=1 R(Cn). Suppose ∂f↾Y (Y ) is not separable, then there exists an integer N such that R(CN ) is not separable. Therefore there exists S a k ∈ N such that R(CN ∩ kBX∗ ) is not separable. For simplicity, we write C = CN ∩ kBX∗ . Note that R(C) is a w∗-compact subset of Y ∗. Since Y is separable and R(C) is a nonseparable subset of Y ∗, by the arguments as in [Ph, 2.19], we obtain an uncountable set A ⊂ R(C) and ε> 0 such that any w∗-open subset of A contains two distinct points x∗ and y∗ such that kx∗ − y∗k >ε. Now we follow the proof of [Ph, 5.4]; let A ⊂ C be a minimal w∗-compact set ∗ 1 w ∗ such that R(A1) = A . If U is a non-empty relatively w -open subset of A1, then A \U is w∗-compact and A = R(A \U) is a proper w∗-compact subset of ∗ 1 2 1 w ∗ A (since A1 is minimal). Thus A\A2 is a non-empty w -open subset of A and it contains two distinct points which are at least ε far apart. Therefore there exist x∗ and y∗ in U such that kx∗ − y∗k >ε, contradicting the assumption that C is norm fragmentable.
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