A Banach space characterization of (sequentially) Ascoli spaces

TL;DR

This paper characterizes Ascoli and sequentially Ascoli spaces using Banach space theory, linking topological properties to the continuity of certain classes of maps into Banach duals.

Contribution

It provides a Banach space characterization of (sequentially) Ascoli spaces via the continuity of specific classes of maps into Banach duals.

Findings

Characterization of Ascoli spaces through Banach space properties.

Extension of the characterization to sequentially Ascoli spaces.

Establishment of equivalence between topological and Banach space conditions.

Abstract

We prove that a Tychonoff space $X$ is an Ascoli space (resp., a sequentially Ascoli space) if and only if for each Banach space $E$, every $k$-continuous and almost $k$-compact (resp., almost $k$-sequential) map $T$ form $X$ into the Banach dual $E'$ of $E$ is continuous.