Lattice Copies of ℓ2 in L1 of a Vector Measure and Strongly Orthogonal Sequences
| Autor: | E. Jiménez Fernández, E. A. Sánchez Pérez |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2012 |
| Předmět: | |
| Zdroj: | Journal of Function Spaces and Applications, Vol 2012 (2012) |
| Druh dokumentu: | article |
| ISSN: | 0972-6802 1758-4965 |
| DOI: | 10.1155/2012/357210 |
| Popis: | Let m be an ℓ2-valued (countably additive) vector measure and consider the space L2(m) of square integrable functions with respect to m. The integral with respect to m allows to define several notions of orthogonal sequence in these spaces. In this paper, we center our attention in the existence of strongly m-orthonormal sequences. Combining the use of the Kadec-Pelczyński dichotomy in the domain space and the Bessaga-Pelczyński principle in the range space, we construct a two-sided disjointification method that allows to prove several structure theorems for the spaces L1(m) and L2(m). Under certain requirements, our main result establishes that a normalized sequence in L2(m) with a weakly null sequence of integrals has a subsequence that is strongly m-orthonormal in L2(m∗), where m∗ is another ℓ2-valued vector measure that satisfies L2(m) = L2(m∗). As an application of our technique, we give a complete characterization of when a space of integrable functions with respect to an ℓ2-valued positive vector measure contains a lattice copy of ℓ2. |
| Databáze: | Directory of Open Access Journals |
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