On non-archimedean Gurarii spaces

Abstract

Let $U_{FNA}$ be the class of all non-archimedean finite-dimensional Banach spaces. A non-archimedean Gurarii Banach space $G$ over a non-archimedean valued field $K$ is constructed, i.e. a non-archimedean Banach space $G$ of countable type which is of almost universal disposition for the class $U_{FNA}$. This means: for every isometry $g:X\to Y$ , where $Y\in U_{FNA}$ and $X$ is a subspace of $G$, and every $\epsilon \in (0,1)$ there exists an $\epsilon$-isometry $f:Y\to G$ such that $f(g(x))=x$ for all $x\in X$. We show that all non-archimedean Banach spaces of almost universal disposition for the class $U_{FNA}$ are $\epsilon$-isometric. Furthermore, all non-archimedean Banach spaces of almost universal disposition for the class $U_{FNA}$ are isometrically isomorphic if and only if $K$ is spherically complete and $\{|\lambda |:\lambda \in K\setminus \{0\}\}=(0,\infty )$.

13 pages

References in corpus (2)

• A proof of uniqueness of the Gurarii space
• Banach spaces of universal disposition