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

TL;DR

This paper proves a non-Archimedean analogue of the Arens--Eells isometric embedding theorem, showing ultrametric spaces can be embedded into valued field extensions with algebraic independence.

Contribution

It introduces a non-Archimedean version of the Arens--Eells theorem, extending isometric embedding results to ultrametric spaces over valued fields.

Findings

Ultrametric spaces can be isometrically embedded into valued field extensions.

Embeddings are algebraically independent over the base field.

The theorem generalizes classical isometric embedding results to non-Archimedean contexts.

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 $K$.