Two characterizations of super-reflexive Banach spaces by the behaviour of differences of convex functions
A function defined on a Banach space \(X\) is called \(\Delta\)-convex if it can be represented as a difference of two continuous convex functions. In the present paper, the author shows that super-reflexivity is characterized in terms of \(\Delta\)-convex functions. The author proves the following theorems: Theorem: Let \((X,\|\cdot\|)\) be a real Banach space. Then the following statements are equivalent: (1) \(X\) is super-reflexive. (2) For every sequence \((x_n)_{n\in \mathbb{N}}\) of uniformly separated points in \(X\) (that is, \(\inf_{n\neq m} \|x_n-x_m \|>0)\) and for every bounded sequence of real numbers \((\beta_n)_{n\in \mathbb{N}}\) there exists a pair of continuous convex functions, \(c\) and \(d\), which are bounded on bounded sets and so that for any \(n\in\mathbb{N}\) we have \((c-d)(x_n)= \beta_n\). Theorem: For a real Banach space \(X\) the following properties are equivalent: (1) \(X\) is superreflexive. (2) For any nonempty norm-closed subset \(F\) of \(X\) there exists a pair of continuous convex functions, \(c\) and \(d\), that are bounded on bounded sets and so that \(F\) is the strong minimum of \(c-d\). (3) For any nonempty norm-closed subset \(F\) of \(X\) there exist two continuous convex functions which are bounded on bounded sets and coincide in a strong way on \(F\).
scientific article; zbMATH DE number 1797582
scientific article