2023/09/13 by Yoshito Ishiki, Ishiki, Yoshito
Mathematics · #Mathematical and Theoretical Analysis #advanced mathematical theories #Functional Equations Stability Results
paper · pdf · doi:10.48550/arxiv.2309.06704
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.