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Locally Convex Spaces with Toeplitz Decompositions

Published online by Cambridge University Press:  09 April 2009

Juan M. Virués
Affiliation:
Escuela Superior de Ingenieros Camino de los Descubrimientos s/n 41092-SevillaSpain e-mail: [email protected], [email protected]
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Abstract

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A Toeplitz decomposition of a locally convez space E into subspaces (Ek) with continuous projections (Pk) is a decomposition of every x ∈ E as x = ΣkPkx where ordinary summability has been replaced by summability with respect to an infinite and row-finite matrix.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 Cesàro bases.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2000

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