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

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 $K$.

I have fixed some gaps. The part concerning the algebraic structures of the universal space is being prepared as an independent new paper. If you wish to refer to it, please consult version 1. 30pages