An open mapping theorem for basis separating maps

As a consequence of the open mapping theorem, a continuous linear bijection H: X --> Y between Banach spaces X and Y must be a linear homeomorphism. The main result of this article (Theorem 9) is similar in form but makes no continuity assumptions on H: If X and Y have symmetric Schauder bases (see before Theorem 9 for the definition), then a "basis separating" linear bijection H is a linear homeomorphism. Given Banach spaces X and Y with Schauder bases {x(n)} and {y(n)}, respectively, we say that H: X --> Y, H(Sigma(nis an element ofN) x(n)x(n)) = Sigma(nis an element ofN) Hx(n)y(n), is basis separating if for all elements x = Sigma(nis an element ofN) x(n)x(n) and y = Sigma(nis an element ofN) y(n)x(n) of X, x(n)y(n) = 0 for all n is an element of N implies that Hx(n)Hy(n) = 0 for all n is an element of N. We show that associated with a linear basis separating map H, there is a support map h: N --> N-infinity. The support map enables us to develop a canonical form (Theorem 4) for basis separating maps that plays a crucial role in the development of the main results. (C) 2003 Elsevier B.V. All rights reserved.