Two characterizations of super-reflexive Banach spaces by the behaviour of differences of convex functions
type: Item
instance of: scholarly article
title: Two characterizations of super-reflexive Banach spaces by the behaviour of differences of convex functions
zbMATH Open document ID: 1010.46010
author: Manuel Cepedello Boiso
published in: Journal of Functional Analysis
publication date: 2002-09-08T00:00:00Z
review text: 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 \(F\).
Mathematics Subject Classification ID: 46B10
zbMATH DE Number: 1797582
zbMATH Keywords: superreflexivity
zbMATH Keywords: Delta-convex function
reviewed by: Shozo Koshi
MaRDI profile type: Publication
full work available at URL: jfan.2001.3854
OpenAlex ID: W2071058995
cites work: Approximation of Lipschitz functions by \(\Delta\)-convex functions in Banach spaces
cites work: On regularization in superreflexive Banach spaces by infimal convolution formulas
cites work: Q3134552
cites work: Banach spaces which can be given an equivalent uniformly convex norm
cites work: Q3720308
cites work: Optimization of Globally Convex Functions
cites work: Q5641646
cites work: Q4137438
cites work: Q4764769
cites work: Martingales with values in uniformly convex spaces
cites work: Q4881711
cites work: On the differentiation of convex functions in finite and infinite dimensional spaces
DOI: 10.1006/JFAN.2001.3854
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description: scientific article; zbMATH DE number 1797582