Locally Convex Spaces with Toeplitz Decompositions

Abstract

A Toeplitz decomposition of a locally convex space E into subspaces (E k ) with continuous projections (/\) is a decomposition of every x e E as x = J2k Pk x where ordinary summability has been replaced by summability with respect to an infinite and row-finite matrix. We extend to the setting of Toeplitz decompositions a number of results about the locally convex structure of a space with a Schauder decomposition. Namely, we give some necessary or sufficient conditions for being reflexive, a Montel space or a Schwartz space. Roughly speaking, each of these locally convex properties is linked to a property of the convergence of the decomposition. We apply these results to study some structural questions in projective tensor products and spaces with Cesaro bases.1991 Mathematics subject classification (Amer. Math. Soc): primary 46A11,46A25,46A35, 46A45.