On non-archimedean Gurariĭ spaces

Abstract

Let UFNA be the class of all non-archimedean finite-dimensional Banach spaces. A non-archimedean Gurariĭ 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 UFNA. This means: for every isometry g:X→Y, where X,Y∈UFNA 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 countable type and of almost universal disposition for the class UFNA are ε-isometric. Furthermore, all non-archimedean Banach spaces of countable type and of almost universal disposition for the class UFNA are isometrically isomorphic if and only if K is spherically complete and {|λ|:λ∈K\\{0}}=(0,∞).