A counterexample for a suptheorem in locally convex spaces

TL;DR

This paper presents a counterexample in the theory of locally convex spaces, showing that certain bounded sets are not compact despite all continuous linear functionals attaining their supremum over them.

Contribution

It provides the first known example of a non-complete locally convex space with a bounded set where all linear functionals attain their supremum but the set is not compact.

Findings

Counterexample in non-complete locally convex space

Bounded set not compact despite all functionals attaining supremum

Highlights limitations of a classical theorem in locally convex spaces

Abstract

In this brief note, we provide an example of non complete locally convex space $E$ with a $\sigma(E, E^*)$ closed bounded subset $C\subset E$, which is not $\sigma(E, E^*)$-compact, even if every $\varphi\in E^*$ attains its sup over $C$.