On non-archimedean Gurarii spaces

Abstract

Let U F N A be the class of all non-archimedean finite-dimensional Banach spaces. A non-archimedean Gurari ǐ 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 F N A . This means: for every isometry g : X → Y , where Y ∈ U F N A and X is a subspace of G, and every ε ∈ (0, 1) there exists an ε-isometry f : Y → G such that f (g(x)) = x for all x ∈ X. We show that all non-archimedean Banach spaces of almost universal disposition for the class U F N A are ε-isometric. Furthermore, all non-archimedean Banach spaces of almost universal disposition for the class U F N A are isometrically isomorphic if and only if K is spherically complete and {|λ| : λ ∈ K\ {0}} = (0, ∞).