Automatic continuity of basis separating maps

Let X and Y be non-Archimedean Banach spaces with bases (see Def. 2.1) B-X = {x(s) : s epsilon S} and B-Y = {y(t) : t epsilon T}. We say that an additive map H : X --> Y, Sigma(sepsilonS) x(s)x(s) --> Sigma(tepsilonT) Hx (t) Y-t, is basis separating (with respect to B-X and B-Y) if, for all x, z epsilon X, x (s) y (s) = 0 for all s implies Hx (t) Hz (t) = 0 for all t. We prove (Theorem 4.2(a)) that certain linear injective basis covering maps H are automatically continuous. We also deduce (Theorem 4.2(b)) an open mapping theorem, namely that, with no continuity assumptions on H, a linear bijective basis separating map H : X --> Y is a homeomorphism.