On the concept of non-ultrametric non-Archimedean analysis

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} $\Vert x+y\Vert \le \max \{\Vert x\Vert ,\Vert y\Vert \}$. 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 $\Vert \cdot {\Vert }_1$, a result that can be seen as the closest to a Mazur--Ulam Theorem in non-Archimedean analysis.

8 pages

References in corpus (2)

• Every 2-dimensional Banach space has the Mazur-Ulam property
• A precision on the concept of strict convexity in non-Archimedean analysis