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Locally Convex Spaces with Toeplitz Decompositions

Part of: Topological linear spaces and related structures

Published online by Cambridge University Press:  09 April 2009

Pedro J. Paúl ,
Carmen Sáez  and
Juan M. Virués
Juan M. Virués
Affiliation:
Escuela Superior de Ingenieros Camino de los Descubrimientos s/n 41092-SevillaSpain e-mail: piti@cica.es, virues@matinc.us.es
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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(Ek) with continuous projections (Pk) is a decomposition of every x ∈ E as x = ΣkPkx where ordinary summability has been replaced by summability with respect to an infinite and row-finite matrix.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.

Keywords

Decompositions of locally convex spacesreflexivemontel and Schwartz spacessummability and basestensor productssequence spaces

MSC classification

Secondary: 46A11: Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.) 46A25: Reflexivity and semi-reflexivity 46A35: Summability and bases 46A45: Sequence spaces (including Köthe sequence spaces)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 68 , Issue 1 , February 2000 , pp. 19 - 40
Copyright
Copyright © Australian Mathematical Society 2000

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