The $\kappa$-topologies on the spaces $\mathscr{D}_{L^p}$, $L^p$ and $\mathscr{M}^1$ are defined by a neighbourhood basis consisting of polars of absolutely convex and compact subsets of their (pre-)dual spaces. In many cases it is more convenient to work with a description of the topology by means of a family of semi-norms defined by multiplication and/or convolution with functions and by classical norms. We give such families of semi-norms generating the $\kappa$-topologies on the above spaces of functions and measures defined by integrability properties. In addition, we present a sequence-space representation of the spaces $\mathscr{D}_{L^p}$ equipped with the $\kappa$-topology, which complements a result of J.~Bonet and M.~Maestre. As a byproduct, we give a characterisation of the compact subsets of the spaces $\mathscr{D}'_{L^p}$, $L^p$ and $\mathscr{M}^1$.

中文翻译：

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空间 DLp,Lp,M1 上 κ 拓扑的简单描述

空间 $\mathscr{D}_{L^p}$、$L^p$ 和 $\mathscr{M}^1$ 上的 $\kappa$-拓扑由绝对其（前）对偶空间的凸和紧子集。在许多情况下，通过由乘法和/或卷积函数和经典范数定义的半范数族来处理拓扑描述更方便。我们给出了在由可积性属性定义的函数和测度的上述空间上生成 $\kappa$-拓扑的半范数族。此外，我们提出了配备 $\kappa$ 拓扑的空间 $\mathscr{D}_{L^p}$ 的序列空间表示，它补充了 J.~Bonet 和 M.~Maestre 的结果. 作为副产品，我们给出了空间 $\mathscr{D}'_{L^p}$ 的紧子集的特征，