An Open Mapping Theorem for Basis Separating Maps
by
Lawrence Narici
St. John's University, Jamaica, NY 11439, USA
Coauthors: Edward Beckenstein (St. John's University, Staten Island, New York)

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 is similar in form but makes no continuity assumptions on H: If X and Y have symmetric Schauder bases, and H is a ``basis separating'' linear bijection, then 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,

is basis separating if for all elements x=\sum_{n in N}x(n) x_n and y=\sum_{n in N}y(n) x_n of X,  x(n) y(n) = 0 for all n in N implies that Hx(n) Hy(n) = 0 for all n in N . We show that associated with a linear basis separating map H, there is a support map  h: N --> N_\infty. The support map plays a crucial role in the development of the main results.

Date received: February 19, 2001