An application of the spherical completion to finite-dimensional normed spaces

TL;DR

This paper introduces a method using spherical completion to classify finite-dimensional normed spaces over non-spherically complete fields, providing new insights into their structure and applications to open problems.

Contribution

It develops a general approach employing spherical completion for classifying finite-dimensional normed spaces over non-spherically complete fields.

Findings

Classification of 3- and 4-dimensional normed spaces achieved.

Embedding of finite-dimensional spaces into simple spaces via spherical completion.

Characterization of strictly epicompact sets provided.

Abstract

In this paper, we will establish a general method of studying finite-dimensional normed spaces, and apply this method to classifying $3$-dimensional and $4$-dimensional normed spaces over a non-spherically complete field. For this purpose, we will use the spherical completion. From the perspective of the spherical completion, each finite-dimensional normed space can be embedded into a simple space. In order to study simple spaces, the orthogonality is important. The orthogonality allows us to find a classification of finite-dimensional normed spaces. As an application of our study, we can get a characterization of strictly epicompact sets, which is an open problem.