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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 \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.

Let be a set and σ be a positive function on . We introduce and study a locally convex topology β 1 ( , σ ) on the space 1 ( , σ ) such that the strong dual of ( 1 ( , σ ), β 1 ( , σ )) can be identified with the Banach space (c 0 ( , 1/σ ), · ∞,σ ). We also show that, except for the case where is finite, there are infinitely many such locally convex topologies on 1 ( , σ ). Finally, we investigate some other properties of the locally convex space ( 1 ( , σ ), β 1 ( , 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 L 1 (G) with a locally convex topology', Michigan Math. J. 46 (1999), 143-150].