The space of vector‐valued integrable functions under certain locally convex topologies
Copyright policy )The space of vector‐valued integrable functions under certain locally convex topologies
AbstractLet 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 \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 β1 is a complete semi‐topological algebra.
- University of Zanjan Iran (Islamic Republic of)
General theory of locally convex spaces, Spaces of vector- and operator-valued functions, Saks spaces and their duals (strict topologies, mixed topologies, two-norm spaces, co-Saks spaces, etc.), spaces of vector-valued integrable functions, generalized group algebra, strict topology, locally compact group, semi-topological algebra, \(L^1\)-algebras on groups, semigroups, etc.
General theory of locally convex spaces, Spaces of vector- and operator-valued functions, Saks spaces and their duals (strict topologies, mixed topologies, two-norm spaces, co-Saks spaces, etc.), spaces of vector-valued integrable functions, generalized group algebra, strict topology, locally compact group, semi-topological algebra, \(L^1\)-algebras on groups, semigroups, etc.
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