Some compact-like properties in non-archimedean functional analysis

TL;DR

This paper explores new properties related to local compactoidity and c-compactness in non-archimedean functional analysis, providing characterizations, conditions, and alternative approaches, especially over spherically complete fields, with implications for classical theorems.

Contribution

It introduces novel concepts and characterizations of compactoidity in non-archimedean spaces, and offers new perspectives on classical theorems under spherically complete fields.

Findings

Characterization of local compactoidity over spherically complete fields

Necessary and sufficient conditions for c-compactness

New approach to non-complete local compactoids

Abstract

First, we define some concepts similar to the local compactoidity or the c-compactness, and study relationships between these concepts and the original ones. As a result, we find a characterization of the local compactoidity when its coefficient field is spherically complete. Moreover, from the point of view of the minimum principle, we give a necessary and sufficient condition for the c-compactness under a suitable condition. Secondly, we try a new approach to a non-complete local compactoid, which gives us a different perspective than before. Thirdly, we study the non-archimedean Goldstine theorem and Eberlein-Smulian theorem. Consequently, if the coefficient field is spherically complete, we get results completely different from the classical ones. Finally, we give a new result about the closed range theorem by using epicompactness.