Operator and Dual Operator Bases in Linear Topological Spaces

Abstract

A net $\{ {S_d}:d \in D\}$ of continuous linear projections of finite range on a Hausdorff linear topological space V is said to be a Schauder operator basis—S.O.B. —(resp. Schauder dual operator basis—S.D.O.B.) provided it is pointwise bounded and converges pointwise to the identity operator on V, and ${S_e}{S_d} = {S_d}$ (resp. ${S_d}{S_e} = {S_d}$) whenever $e \geqq d$. S.O.B.’s and S.D.O.B.’s are natural generalizations of finite dimensional Schauder bases of subspaces. In fact, a sequence of operators is both a S.O.B. and S.D.O.B. iff it is the sequence of partial sum operators associated with a finite dimensional Schauder basis of subspaces. We show that many duality-theory results concerning Schauder bases can be extended to S.O.B.’s or S.D.O.B.’s. In particular, a space with a S.D.O.B. is semi-reflexive if and only if the S.D.O.B. is shrinking and boundedly complete. Several results on S.O.B.’s and S.D.O.B.’s were previously unknown even in the case of Schauder bases. For example, Corollary IV.2 implies that the strong dual of an evaluable space which admits a shrinking Schauder basis is a complete barrelled space.