The space of vector-valued integrable functions under certain locally convex topologies

Authors

  • Saeid Maghsoudi

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    1. Department of Mathematics, University of Zanjan, Zanjan 45195-313, Iran
    • Department of Mathematics, University of Zanjan, Zanjan 45195-313, Iran. Phone: +981 241 5154047, Fax: +981 241 5154047
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Abstract

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 β1 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 L1(μ, E) can be identified with a Banach space. We also show that the topology β1 is a metrizable, barrelled or bornological space if and only if Ω is compact. Finally, we consider the generalized group algebra equation image under certain natural locally convex topologies. As an application of our results, we prove that equation image under the topology β1 is a complete semi-topological algebra.

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