Topological Methods in Nonlinear Analysis

Compactness in spaces of $p$-integrable functions with respect to a vector measure

Pilar Rueda and Enrique A. Sánchez-Pérez

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We prove that, under some reasonable requirements, the unit balls of the spaces $L^p(m)$ and $L^\infty(m)$ of a vector measure of compact range $m$ are compact with respect to the topology $\tau_m$ of pointwise convergence of the integrals. This result can be considered as a generalization of the classical Alaoglu Theorem to spaces of $p$-integrable functions with respect to vector measures with relatively compact range. Some applications to the analysis of the Saks spaces defined by the norm topology and $\tau_m$ are given.

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Topol. Methods Nonlinear Anal., Volume 45, Number 2 (2015), 641-653.

First available in Project Euclid: 30 March 2016

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Rueda, Pilar; Sánchez-Pérez, Enrique A. Compactness in spaces of $p$-integrable functions with respect to a vector measure. Topol. Methods Nonlinear Anal. 45 (2015), no. 2, 641--653. doi:10.12775/TMNA.2015.030.

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