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Compactness in $L^1$ of a vector measure

Volume 225 / 2014

J. M. Calabuig, S. Lajara, J. Rodríguez, E. A. Sánchez-Pérez Studia Mathematica 225 (2014), 259-282 MSC: 46E30, 46B50, 46G10. DOI: 10.4064/sm225-3-6

Abstract

We study compactness and related topological properties in the space $L^1(m)$ of a Banach space valued measure $m$ when the natural topologies associated to convergence of vector valued integrals are considered. The resulting topological spaces are shown to be angelic and the relationship of compactness and equi-integrability is explored. A natural norming subset of the dual unit ball of $L^1(m)$ appears in our discussion and we study when it is a boundary. The (almost) complete continuity of the integration operator is analyzed in relation with the positive Schur property of $L^1(m)$. The strong weakly compact generation of $L^1(m)$ is discussed as well.

Authors

  • J. M. CalabuigInstituto Universitario de Matemática Pura y Aplicada
    Universitat Politècnica de València
    Camino de Vera s/n
    46022 Valencia, Spain
    e-mail
  • S. LajaraDepartamento de Matemáticas
    Escuela de Ingenieros Industriales
    Universidad de Castilla-La Mancha
    02071 Albacete, Spain
    e-mail
  • J. RodríguezDepartamento de Matemática Aplicada
    Facultad de Informática
    Universidad de Murcia
    30100 Espinardo (Murcia), Spain
    e-mail
  • E. A. Sánchez-PérezInstituto Universitario de Matemática Pura y Aplicada
    Universitat Politècnica de València
    Camino de Vera s/n
    46022 Valencia, Spain
    e-mail

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