One side James' Compactness Theorem

TL;DR

This paper extends classical dual space results like James' compactness theorem, providing new characterizations of weak compactness and reflexivity in Banach spaces under geometric and topological conditions.

Contribution

It introduces novel conditions involving dual elements that guarantee weak compactness and reflexivity, answering a question posed by F. Delbaen.

Findings

Characterization of weakly compact sets via dual functionals

New criteria for reflexivity in Banach spaces

Results on variational problems in dual Banach spaces

Abstract

We present some extensions of classical results that involve elements of the dual of Banach spaces, such as Bishop-Phelp's theorem and James' compactness theorem, but restricting to sets of functionals determined by geometrical properties. The main result, which answers a question posed by F. Delbaen, is the following: Let $E$ be a Banach space such that $(B_{E^\ast}, \omega^\ast)$ is convex block compact. Let $A$ and $B$ be bounded, closed and convex sets with distance $d(A,B) > 0$. If every $x^\ast \in E^\ast$ with \[ \sup(x^\ast,B) < \inf(x^\ast,A) \] attains its infimum on $A$ and its supremum on $B$, then $A$ and $B$ are both weakly compact. We obtain new characterizations of weakly compact sets and reflexive spaces, as well as a result concerning a variational problem in dual Banach spaces.