A non-Archimedean Arens--Eells isometric embedding theorem on valued fields
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909990488375296 |
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| author | Ishiki, Yoshito |
| author_facet | Ishiki, Yoshito |
| contents | 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 $K$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_06704 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A non-Archimedean Arens--Eells isometric embedding theorem on valued fields Ishiki, Yoshito Metric Geometry Commutative Algebra General Topology 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 $K$. |
| title | A non-Archimedean Arens--Eells isometric embedding theorem on valued fields |
| topic | Metric Geometry Commutative Algebra General Topology |
| url | https://arxiv.org/abs/2309.06704 |