Lattice Copies of ℓ2 in L1 of a Vector Measure and Strongly Orthogonal Sequences
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...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2012-01-01
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| Series: | Journal of Function Spaces and Applications |
| Online Access: | http://dx.doi.org/10.1155/2012/357210 |
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| Summary: | 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. |
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| ISSN: | 0972-6802 1758-4965 |