On the range of the subdifferential in non reflexive Banach spaces
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.
scientific article; zbMATH DE number 7336930
scientific article