Riesz type theorem in locally convex vector spaces

TL;DR

This paper generalizes the Riesz representation theorem from Banach spaces to complete Hausdorff locally convex vector spaces, characterizing weakly compact linear mappings as integrals involving functions with weakly compact semivariation.

Contribution

It extends the classical Riesz type theorem to locally convex vector spaces, providing a new integral representation for weakly compact mappings.

Findings

Weakly compact mappings are representable as integrals over functions with weakly compact semivariation.

The theorem generalizes classical results from Banach spaces to locally convex vector spaces.

Provides a framework for understanding linear mappings in more general topological vector spaces.

Abstract

The present paper is concerned with some representatons of linear mappings of continuous functions into locally convex vector spaces, namely: If X is a complete Hausdorff locally convex vector space, then a general form of weakly compact mapping T:C{[a,b]}\to X is of the form Tg=\int_a^bg(t)dx(t), where the function $x(\cdot):[a,b] \to X$ has a weakly compact semivariation on $[a,b]$. This theorem is a generalization of the result from Banach spaces to locally convex vector spaces.