Generalized compactness in linear spaces and its applications

TL;DR

This paper introduces the concept of μ-compact sets in locally convex spaces, generalizing many properties of compact sets and extending key theorems like Vesterstrom-O'Brien to broader contexts with applications in quantum information theory.

Contribution

It generalizes the Vesterstrom-O'Brien theorem to μ-compact convex sets, broadening the scope of known properties of compact sets in locally convex spaces.

Findings

Many properties of compact sets are extended to μ-compact sets.

The Vesterstrom-O'Brien theorem is generalized to μ-compact convex sets.

Applications to quantum information theory are demonstrated.

Abstract

The class of subsets of locally convex spaces called $\mu$-compact sets is considered. This class contains all compact sets as well as several noncompact sets widely used in applications. It is shown that many results well known for compact sets can be generalized to $\mu$-compact sets. Several examples are considered. The main result of the paper is a generalization to $\mu$-compact convex sets of the Vesterstrom-O'Brien theorem showing equivalence of the particular properties of a compact convex set (s.t. openness of the mixture map, openness of the barycenter map and of its restriction to maximal measures, continuity of a convex hull of any continuous function, continuity of a convex hull of any concave continuous function). It is shown that the Vesterstrom-O'Brien theorem does not hold for pointwise $\mu$-compact convex sets defined by the slight relaxing of the $\mu$-compactness condition. Applications of the obtained results to quantum information theory are considered.