The unit ball of the Hilbert space in its weak topology

TL;DR

This paper investigates the topological structure of the unit ball in a Hilbert space under the weak topology, revealing its relationship with the Alexandroff compactification and exploring unique lattice properties in nonseparable cases.

Contribution

It demonstrates that the unit ball's weak topology is a continuous image of a countable power of the Alexandroff compactification, highlighting distinctive combinatorial properties.

Findings

The unit ball in weak topology is a continuous image of a countable product of the Alexandroff compactification.

Identifies unique lattice properties of the unit ball in nonseparable Hilbert spaces.

Shows these properties are not shared by balls of other equivalent norms.

Abstract

We show that the unit ball of a Hilbert space in its weak topology is a continuous image of the countable power of the Alexandroff compactification of a discrete set, and we deduce some combinatorial properties of its lattice of open sets which are not shared by the balls of other equivalent norms when the space is nonseparable.