An isomorphism theorem for valuated vector spaces

Abstract:

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