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THE STRICT TOPOLOGY ON THE DISCRETE LEBESGUE SPACES

Part of: Topological algebras, normed rings and algebras, Banach algebras Topological linear spaces and related structures

Published online by Cambridge University Press:  06 December 2010

SAEID MAGHSOUDI  and
RASOUL NASR-ISFAHANI
SAEID MAGHSOUDI*
Affiliation:
Department of Mathematics, Zanjan University, Zanjan, 313, Iran Research Institute for Fundamental Science, Tabriz, Iran (email: s_maghsodi@znu.ac.ir)
RASOUL NASR-ISFAHANI
Affiliation:
Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, 84156-83111, Iran (email: isfahani@cc.iut.ac.ir)
*
✉For correspondence; e-mail: s˙maghsodi@znu.ac.ir
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Abstract

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Let Σ be a set and σ be a positive function on Σ. We introduce and study a locally convex topology β1(Σ,σ) on the space ℓ1(Σ,σ) such that the strong dual of (ℓ1(Σ,σ),β1(Σ,σ)) can be identified with the Banach space . We also show that, except for the case where Σ is finite, there are infinitely many such locally convex topologies on ℓ1(Σ,σ). Finally, we investigate some other properties of the locally convex space (ℓ1(Σ,σ),β1(Σ,σ)) , and as an application, we answer partially a question raised by A. I. Singh [‘L∞0(G)* as the second dual of the group algebra L1 (G) with a locally convex topology’, Michigan Math. J.46 (1999), 143–150].

Keywords

discrete Lebesgue spacelocally convex spacestrict topologystrong dual

MSC classification

Secondary: 46A03: General theory of locally convex spaces 46H05: General theory of topological algebras
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 83 , Issue 2 , April 2011 , pp. 241 - 255
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

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