Some remarks on stronger versions of the Boundary Problem for Banach spaces

TL;DR

This paper discusses stronger variants of the Boundary Problem in Banach spaces, exploring conditions under which certain compactness properties imply weak compactness, with results tailored to specific Banach space types.

Contribution

It presents new stronger versions of the Boundary Problem, extending Pfitzner's positive solution to particular classes of Banach spaces.

Findings

Certain compactness conditions imply weak compactness in specific Banach spaces.

The paper extends known results to special types of Banach spaces.

Utilizes techniques from Pfitzner, Cascales et al., and Moors.

Abstract

Let $X$ be a real Banach space. A subset $B$ of the dual unit sphere of $X$ is said to be a boundary for $X$, if every element of $X$ attains its norm on some functional in $B$. The well-known Boundary Problem originally posed by Godefroy asks whether a bounded subset of $X$ which is compact in the topology of pointwise convergence on $B$ is already weakly compact. This problem was recently solved by H.Pfitzner in the positive. In this note we collect some stronger versions of the solution to the Boundary Problem, most of which are restricted to special types of Banach spaces. We shall use the results and techniques of Pfitzner, Cascales et al., Moors and others.