×

Criteria for weak compactness of vector-valued integration maps. (English) Zbl 0805.46040

Summary: Criteria are given for determining the weak compactness, or otherwise, of the integration map associated with a vector measure. For instance, the space of integrable functions of a weakly compact integration map is necessarily normable for the mean convergence topology. Results are presented which relate weak compactness of the integration map with the property of being a bicontinuous isomorphism onto its range. Finally, a detailed description is given of the compactness properties for the integration maps of a class of measures taking their values in \(\ell^ 1\), equipped with various weak topologies.

MSC:

46G10 Vector-valued measures and integration
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
47B07 Linear operators defined by compactness properties
46A03 General theory of locally convex spaces
PDFBibTeX XMLCite
Full Text: EuDML