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 {xn} and {yn}, respectively, we say that H :X→Y, H(∑n∈Nx(n)xn)=∑n∈NHx(n)yn, is basis separating if for all elements x=∑n∈Nx(n)xn and y=∑n∈Ny(n)xn of X, x(n)y(n)=0 for all n∈N implies that Hx(n)Hy(n)=0 for all n∈N. We show that associated with a linear basis separating map H, there is a support map h :N→N∞. 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.