Boundaries for Banach spaces determine weak compactness

TL;DR

This paper proves that for Banach spaces, the topology induced by a boundary on the dual space coincides with the weak topology regarding bounded compact sets, resolving Godefroy's Boundary Problem.

Contribution

It provides a positive answer to Godefroy's Boundary Problem, establishing the equivalence of certain topologies on Banach spaces.

Findings

Boundaries determine the weak compactness in Banach spaces

Pointwise convergence on boundaries matches weak topology for bounded sets

Resolves a long-standing open problem in functional analysis

Abstract

A boundary for a Banach space is a subset of the dual unit sphere with the property that each element of the Banach space attains its norm on an element of that subset. Trivially, the pointwise convergence with respect to such a boundary is coarser than the weak topology on the Banach space. Godefroy's Boundary Problem asks whether nevertheless both topologies have the same bounded compact sets. This paper contains the answer in the positive.