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arXiv · 2309.06704

A non-Archimedean Arens--Eells isometric embedding theorem on valued fields

Yoshito Ishiki
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Abstract

In 1959, Arens and Eells proved that every metric space can be isometrically embedded into a normed linear space as a closed subset. In later years, in the paper on a short proof of the Arens--Eells theorem, Michael implicitly pointed out that the Arens--Eells theorem follows from the statement that every metric space can be isometrically embedded into a normed linear space as a linearly independent subset. In this paper, we prove a non-Archimedean analogue of the Arens--Eells isometric embedding theorem, which states that for every non-Archimedean valued field $K$, every ultrametric space can be isometrically embedded into a non-Archimedean valued field that is a valued field extension of $K$ such that the image of the embedding is algebraically independent over $K$.

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Yoshito Ishiki. 2023-09-13. A non-Archimedean Arens--Eells isometric embedding theorem on valued fields. https://arxiv.org/abs/2309.06704

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