A Note on the Completeness of Cc(X,Y)

TL;DR

This paper investigates conditions under which the space of continuous linear mappings between convergence vector spaces is complete, providing sufficient conditions that include all complete Hausdorff topological vector spaces.

Contribution

It establishes sufficient conditions on the target space Y ensuring the completeness of Cc(X,Y) for any convergence space X, extending known results.

Findings

Cc(X,Y) is complete when Y is a complete Hausdorff topological vector space

Provides conditions ensuring completeness of continuous linear mappings spaces

Extends understanding of convergence vector spaces and their mapping spaces

Abstract

It is known that there are complete, Hausdorff and regular convergence vector spaces X and Y such that Lc(X,Y), the space of continuous linear mappings from X into Y equipped with the continuous convergence structure, is not complete. In this paper, we give sufficient conditions on a convergence vector space Y such that Cc(X,Y) is complete for any convergence space X. In particular, we show that this is true for every complete and Hausdorff topological vector space Y.