Automatic norm continuity of weak* homeomorphisms

TL;DR

This paper demonstrates that in a specific class of nonseparable Banach spaces, weak* homeomorphisms between dual balls are inherently norm-continuous due to the definability of the norm topology from the weak* topology.

Contribution

It establishes that in a certain class of nonseparable Banach spaces, weak* homeomorphisms are automatically norm-continuous, revealing a new connection between topologies.

Findings

Weak* homeomorphisms are norm-continuous in class E.

Norm topology is definable from weak* topology in these spaces.

Results apply to nonseparable Banach spaces in class E.

Abstract

We prove that in a certain class E of nonseparable Banach spaces the norm topology of the dual ball is definable in terms of its weak* topology. Thus, any weak* homeomorphism between duals balls of spaces in E is automatically norm-continuous.