Dual of an extended locally convex space

TL;DR

This paper investigates the dual space of extended locally convex spaces, examining various topologies and applying findings to function spaces, particularly the space of continuous functions with specific convergence topologies.

Contribution

It characterizes the dual and topological properties of extended locally convex spaces and identifies conditions under which certain topologies coincide in function spaces.

Findings

Weak topology on C(X) coincides with the finest locally convex topology only for finite bornologies.

Analyzes the weak and weak* topologies in extended locally convex spaces.

Provides conditions for the topology of uniform convergence on bounded sets.

Abstract

This paper aims to study the dual of an extended locally convex space. In particular, we study the weak and weak* topologies as well as the topology of uniform convergence on bounded subsets of an extended locally convex space. As an application to function spaces, we show that the weak topology for the space C(X) of all real-valued continuous functions on a metric space (X,d) endowed with the topology of strong uniform convergence on bornology coincides with its finest locally convex topology if and only if the bornology is of all finite subsets of X.