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$ \mathsf{L}^{\mathtt{2}}$-summand vectors in Banach spaces


Authors: Antonio Aizpuru and Francisco Javier Garcia-Pacheco
Journal: Proc. Amer. Math. Soc. 134 (2006), 2109-2115
MSC (2000): Primary 46B20, 46C05, 46B04
DOI: https://doi.org/10.1090/S0002-9939-06-08243-8
Published electronically: January 17, 2006
MathSciNet review: 2215781
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Abstract: The aim of this paper is to study the set $ \mathsf{L} _{X}^{\mathtt{2}}$ of all $ \mathsf{L}^{\mathtt{2}}$-summand vectors of a real Banach space $ X$. We provide a characterization of $ \mathsf{L}^{\mathtt{ 2}}$-summand vectors in smooth real Banach spaces and a general decomposition theorem which shows that every real Banach space can be decomposed as an $ \mathsf{L}^{\mathtt{2}}$-sum of a Hilbert space and a Banach space without nontrivial $ \mathsf{L}^{\mathtt{2}}$-summand vectors. As a consequence, we generalize some results and we obtain intrinsic characterizations of real Hilbert spaces.


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Additional Information

Antonio Aizpuru
Affiliation: Departamento de Matemáticas, Universidad de Cádiz, Puerto Real, Cádiz, 11510, Spain
Email: antonio.aizpuru@uca.es

Francisco Javier Garcia-Pacheco
Affiliation: Department of Mathematical Sciences, Kent State University, Kent, Ohio 44242
Email: fgarcia@math.kent.edu

DOI: https://doi.org/10.1090/S0002-9939-06-08243-8
Keywords: $\mathsf{L}^{\mathtt{2}}$-summand vector, smoothness, Hilbert space
Received by editor(s): December 1, 2004
Received by editor(s) in revised form: January 31, 2005, and February 18, 2005
Published electronically: January 17, 2006
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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