## Topological Methods in Nonlinear Analysis

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

#### Abstract

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.

#### Article information

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

Dates
First available in Project Euclid: 30 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1459343999

Digital Object Identifier
doi:10.12775/TMNA.2015.030

Mathematical Reviews number (MathSciNet)
MR3408839

Zentralblatt MATH identifier
1370.46007

#### Citation

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. https://projecteuclid.org/euclid.tmna/1459343999

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