Some compact-like properties in non-archimedean functional analysis
Kosuke Ishizuka
TL;DR
This work develops a cohesive framework for compact-like phenomena in non-archimedean functional analysis by introducing and relating $c$-precompactness and $cf$-compactness to local compactoidity. It shows that, under spherical completeness, $c$-precompactness characterizes local compactoids and that $cf$-compactness coincides with $c$-compactness in metrizable spaces, while highlighting distinctions in general settings. The paper extends to non-complete local compactoids, and advances non-archimedean analogues of Goldstine, Eberlein–Šmulian, and closed-range theorems, including stability results for complete metrizable local compactoids and epicompactness-based arguments. These results yield new insights into dualities, operator theory, and compact-like structures in non-archimedean spaces, with implications for the structure of norm-polar spaces and their subspaces. Overall, it provides a rigorous, interconnected treatment of compactness analogues and their consequences in non-archimedean functional analysis.
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.