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The space of vector‐valued integrable functions under certain locally convex topologies
Author(s) -
Maghsoudi Saeid
Publication year - 2013
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201200013
Subject(s) - mathematics , metrization theorem , weak topology (polar topology) , locally convex topological vector space , banach space , topology (electrical circuits) , space (punctuation) , dual space , topological group , radon measure , lebesgue integration , regular polygon , locally compact space , reflexive space , pure mathematics , general topology , topological space , combinatorics , mathematical analysis , interpolation space , extension topology , functional analysis , separable space , geometry , biochemistry , chemistry , gene , linguistics , philosophy
Let E be a Banach space, Ω a locally compact space, and μ a positive Radon measure on Ω. In this paper, through extending to Lebesgue‐Bochner spaces, we show that the topology β 1 introduced by Singh is a type of strict topology. We then investigate various properties of this locally convex topology. In particular, we show that the strong dual of L 1 (μ, E ) can be identified with a Banach space. We also show that the topology β 1 is a metrizable, barrelled or bornological space if and only if Ω is compact. Finally, we consider the generalized group algebra \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$L^1(G, \mathbf {A})$\end{document} under certain natural locally convex topologies. As an application of our results, we prove that \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$L^1(G,\mathbf {A})$\end{document} under the topology β 1 is a complete semi‐topological algebra.