Strongly embedded subspaces of p-convex Banach function spaces

Calabuig, J. M.; Rodríguez, José; Pérez, Enrique Alfonso Sánchez

doi:10.1007/s11117-012-0204-6

Cited by 6 publications

(9 citation statements)

References 15 publications

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“…Finally, in Section 5 we show how integration with respect to vector measures can be used to obtain concrete representations of positively norming sets for order continuous Banach lattices. These results are related to the recent developments on the spaces of integrable functions with respect to a vector measure given in [3,11].…”

Section: Introductionmentioning

confidence: 84%

“…Let us comment that for positive norming sets of the form R(m) that were considered in [3] (see Section 5) equi-integrability is related to an important property of the integration operator. Recall that for a vector measure m : Σ → E the integration operator I m :…”

Section: Geometry Of Banach Lattices and Positively Norming Setsmentioning

confidence: 99%

“…This set is positively norming for L 1 (m) = X(µ) (see the third example of positively norming set in Section 1 of [3]). Our interest is to show that this class of sets provides a standard procedure for defining positively norming sets.…”

Section: Representation Of Positively Norming Sets and Vector Measuresmentioning

confidence: 99%

“…This tool provides a characterization of those subspaces of Banach function spaces that are strongly embedded in L 1 -spaces (see [3]). Actually, this technique has shown that in the setting of spaces of integrable functions the notion of positively norming set seems to be more natural and useful than that of norming set.…”

Section: Introductionmentioning

confidence: 99%

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Positively Norming Sets in Banach Function Spaces

Pérez

Tradacete

2013

The Quarterly Journal of Mathematics

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Abstract. The notion of positively norming set, a specific definition of norming type sets for Banach lattices, is analyzed. We show that the size of positively norming sets (in terms of compactness and order boundedness) is directly related to the existence of lattice copies of L 1 -spaces. As an application, we provide a version of Kadec-Pelczynski's dichotomy for order continuous Banach function spaces. A general description of positively norming sets using vector measure integration is also given.

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Section: Introductionmentioning

confidence: 84%

Section: Geometry Of Banach Lattices and Positively Norming Setsmentioning

confidence: 99%

Section: Representation Of Positively Norming Sets and Vector Measuresmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Positively Norming Sets in Banach Function Spaces

Pérez

Tradacete

2013

The Quarterly Journal of Mathematics

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“…is the L 1 space of some 2 -valued measure, see [4,Example 3.7]. This space is WSC and does not embed isomorphically into any SWCG space (see [22,Corollary 2.29]).…”

Section: When Is L 1 (M) a Swcg Space?mentioning

confidence: 99%

Compactness in L¹of a vector measure

et al. 2014

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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 the convergence of the 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.

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