The minimal context for local boundedness in topological vector spaces

TL;DR

This paper investigates the conditions under which classes of operators are locally bounded in topological vector spaces, linking these conditions to properties like barreledness and the uniform boundedness principle.

Contribution

It characterizes the topological vector spaces where local boundedness of operators holds, especially relating to Fitzpatrick functions and the uniform boundedness principle.

Findings

Local boundedness of maximal monotone operators on the algebraic interior of their domain convex hull characterizes barreled locally convex spaces.

Provides new characterizations of topological vector spaces based on local boundedness properties.

Connects Fitzpatrick functions with local boundedness and space properties.

Abstract

The local boundedness of classes of operators is analyzed on different subsets directly related to their Fitzpatrick functions and characterizations of the topological vector spaces for which that local boundedness holds is given in terms of the uniform boundedness principle. For example the local boundedness of a maximal monotone operator on the algebraic interior of its domain convex hull is a characteristic of barreled locally convex spaces.