Uniform boundedness deciding sets, and a problem of M. Valdivia

TL;DR

The paper proves that unions of certain sets in Banach spaces preserve non-uniform boundedness deciding properties, answering a question posed by M. Valdivia.

Contribution

It establishes a new result on the behavior of uniform boundedness deciding sets under countable unions in Banach spaces.

Findings

Unions of non-uniform boundedness deciding sets are also non-uniform boundedness deciding.

Provides a positive answer to M. Valdivia's question about these sets.

Advances understanding of the structure of boundedness in Banach spaces.

Abstract

We prove that if a set $B$ in a Banach space $X$ can be written as an increasing, countable union $B=\cup_n B_n$ of sets $B_n$ such that no $B_n$ is uniform boundedness deciding, then also $B$ is not uniform boundedness deciding. From this we can give a positive answer to a question of M. Valdivia.