Let $G$ be a locally compact group with a fixed left Haar measure. In this paper, given a strictly positive Young function ${\rm\Phi}$, we consider $L^{{\rm\Phi}}(G)$ as a Banach left $L^{1}(G)$-module. Then we equip $L^{{\rm\Phi}}(G)$ with the strict topology induced by $L^{1}(G)$ in the sense of Sentilles and Taylor. Some properties of this locally convex topology and a comparison with weak$^{\ast }$, bounded weak$^{\ast }$ and norm topologies are presented.

Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}K$ be a locally compact hypergroup endowed with a left Haar measure and let $L^1(K)$ be the usual Lebesgue space of $K$ with respect to the left Haar measure. We investigate some properties of $L^1(K)$ under a locally convex topology $\beta ^1$. Among other things, the semireflexivity of $(L^1(K), \beta ^1)$ and of sequentially$\beta ^1$-continuous functionals is studied. We also show that $(L^1(K), \beta ^1)$ with the convolution multiplication is always a complete semitopological algebra, whereas it is a topological algebra if and only if $K$ is compact.

For a locally compact group $ \mathcal{G} $, we introduce and study a class of locally convex topologies $\tau $ on the measure algebra $M( \mathcal{G} )$ of $ \mathcal{G} $. In particular, we show that the strong dual of $(M( \mathcal{G} ), \tau )$ can be identified with a closed subspace of the Banach space $M\mathop{( \mathcal{G} )}\nolimits ^{\ast } $; we also investigate some properties of the locally convex space $(M( \mathcal{G} ), \tau )$.

Let X be a locally compact space, and 𝔏∞0(X,ι) be the space of all essentially bounded ι-measurable functions f on X vanishing at infinity. We introduce and study a locally convex topology β1(X,ι) on the Lebesgue space 𝔏1(X,ι) such that the strong dual of (𝔏1(X,ι),β1(X,ι)) can be identified with . Next, by showing that β1(X,ι) can be considered as a natural mixed topology, we deduce some of its basic properties. Finally, as an application, we prove that L1 (G) , the group algebra of a locally compact Hausdorff topological group G, equipped with the convolution multiplication is a complete semitopological algebra under the β1 (G) topology.

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].