On the range of the subdifferential in non reflexive Banach spaces
type: Item
instance of: scholarly article
title: On the range of the subdifferential in non reflexive Banach spaces
zbMATH Open document ID: 1471.46012
published in: Journal of Functional Analysis
publication date: 2021-04-23T00:00:00Z
review text: The paper is concerned with non-linear versions of James's theorem on weak compactness and applications. One considers a function \(f:E\to\mathbb{R}\cup\{\infty\}\), where \(E\) is a real Banach space, and one looks for conditions ensuring the weak compactness of a subset \(A\) of \(E\) provided that \(x^*-f\) attains its supremum on \(A\) for every \(x^*\in E^*\) (the dual of \(E\)). Another version is that all sublevel sets \(f^{-1}((-\infty,c])\) are weakly compact if this supremum is attained on \(E\). These results have found substantial applications to various problems in optimization theory and to mathematical finance. The introductory part of the paper contains a survey of various results obtained along these lines. It was previously proved that, under the above hypotheses, the weak compactness of the sublevel sets of \(f\) follows if its subdifferential \(\partial f:E\to2^{E^*}\) is surjective, i.e., \(\partial f(E)=E^*\). In the present paper, the authors show that the surjectivity condition can be weakened to the nonemptiness of the interior of \(\partial f(E)\) with respect to the Mackey topology \(\tau(E^*,E)\) on \(E^*\). If, in addition, the function \(f\) has a domain with non-empty norm interior, the Banach space \(E\) must be reflexive. As a consequence, one obtains a version of the James theorem in this vein: a closed, convex and bounded subset \(A\) of the Banach space \(E\) is weakly compact if, and only if, there is a nonvoid \(\tau(E^*,E)\)-open subset \(V\) of \( E^*\) such that every \(x^*\in V\) attains its supremum on \(A\) (Corollary~14). The paper also contains other results of this kind, based on the nonemptiness of the interior of some subset of \(E^*\) with respect to the Mackey topology. Some applications to risk measures and to cyclically monotone operators \(\Phi:E\to 2^{E^*}\) are given.
Mathematics Subject Classification ID: 46B10
Mathematics Subject Classification ID: 46A50
Mathematics Subject Classification ID: 46B50
zbMATH DE Number: 7336930
zbMATH Keywords: weak compactness
zbMATH Keywords: reflexivity
zbMATH Keywords: norm attaining linear functionals
zbMATH Keywords: Mackey topology
author: Freddy Delbaen
author: José Orihuela
reviewed by: Stefan Cobzaş
MaRDI profile type: Publication
full work available at URL: j.jfa.2020.108915
OpenAlex ID: W3144490350
cites work: In a nonreflexive space the subdifferential is not onto
cites work: One-sided James' compactness theorem
cites work: Compactness, Optimality, and Risk
cites work: Mackey constraints for James's compactness theorem and risk measures
cites work: Banach space theory. The basis for linear and nonlinear analysis
cites work: Weakly Compact Sets
cites work: Law invariant risk measures have the Fatou property
cites work: On regularly convex sets in the space conjugate to a Banach space
cites work: Q5569793
cites work: Weak compactness of sublevel sets
cites work: Q5204161
cites work: A coercive James's weak compactness theorem and nonlinear variational problems
cites work: Lebesgue property for convex risk measures on Orlicz spaces
cites work: Weak compactness of sublevel sets in complete locally convex spaces
cites work: Weak Compactness in Locally Convex Spaces
cites work: Characterization of the subdifferentials of convex functions
cites work: A new minimax theorem and a perturbed James's theorem
cites work: Weak compactness and variational characterization of the convexity
DOI: 10.1016/J.JFA.2020.108915
description: scientific article; zbMATH DE number 7336930