On the concept of non-ultrametric non-Archimedean analysis

Javier Cabello Sánchez; Francisco J. Carmona Fuertes

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arXiv · 2108.12400

On the concept of non-ultrametric non-Archimedean analysis

Javier Cabello Sánchez·Francisco J. Carmona Fuertes

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Abstract

Given some non-Archimedean field $\mathbb{K}$ and some $\mathbb{K}$-linear space $X$, the usual way to define a norm over $X$ involves the {\em ultrametric inequality} $\|x+y\|\leq\max\{\|x\|,\|y\|\}$. In this note we will try to analyse the convenience of considering a wider variety of norms. The main result of the present note is a characterisation of the isometries between finite-dimensional linear spaces over some valued field endowed with the norm $\|\,\cdot\,\|_1$, a result that can be seen as the closest to a Mazur--Ulam Theorem in non-Archimedean analysis.

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BibTeX RIS

Javier Cabello Sánchez, Francisco J. Carmona Fuertes. 2021-08-27. On the concept of non-ultrametric non-Archimedean analysis. https://arxiv.org/abs/2108.12400

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