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arXiv · 2412.17385

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

Kosuke Ishizuka
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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.

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Kosuke Ishizuka. 2024-12-23. An application of the spherical completion to finite-dimensional normed spaces. https://doi.org/10.36045/j.bbms.241223

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