An isomorphism theorem for valuated vector spaces

Abstract

The following isomorphism theorem is proved for valuated vector spaces. Let N N and N ′ N’ be nice subspaces of free valuated vector spaces F F and F ′ F’ , respectively. If N N and N ′ N’ have isomorphic basic subspaces and if the quotient spaces F / N F/N and F ′ / N ′ F’/N’ are isomorphic, there exists an isomorphism from F F onto F ′ F’ that maps N N onto N ′ N’ and induces the given isomorphism on the quotient spaces. In particular, N N and N ′ N’ are isomorphic.