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

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

2010-11-10

arxiv.org/abs/1011.2372v2

Mathematics

Functional Analysis

15 pages, version 2, references added, two remarks added, some arguments slightly changed, revised version accepted for public

Scientific paper

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.

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