Available in electronic format
Available in print format		 Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826 (e) ISSN 0002-9939 (p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

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

Author(s): Antonio Aizpuru; Francisco Javier Garcia-Pacheco
Journal: Proc. Amer. Math. Soc. 134 (2006), 2109-2115.
MSC (2000): Primary 46B20, 46C05, 46B04
Posted: January 17, 2006
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

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.

References:

1.
A. Aizpuru, F.J. García-Pacheco, and F. Rambla, Isometric reflection vectors in Banach spaces, J. Math. Anal. Appl. 299 (2004) 40-48. MR 2091268 (2005g:46051)

2.
E. Behrends et al., $ \mathsf{L}^{\mathtt{p}}$-structure in real Banach spaces, Lecture Notes in Mathematics 613, Berlin-Heidelberg-New York, Springer-Verlag, 1977. MR 0626051 (58:30082)

3.
E. Behrends, $ \mathsf{L}^{\mathtt{p}}$-Struktur in Banachräumen, Studia Math. 55 (1976) 71-85. MR 0402466 (53:6286)

4.
P. Bandyopadhyay, D. Huang, B.-L. Lin, and S.L. Troyanski, Some Generalizations of Locally Uniform Rotundity, J. Math. Anal. Appl. 252 (2000) 906-916. MR 1801249 (2001j:46011)

5.
J. Becerra Guerrero and A. Rodríguez Palacios, Transitivity of the norm on Banach spaces, Extracta Math. 17 1 (2002) 1-58. MR 1914238 (2003i:46022)

6.
J.W. Carlson and T.L. Hicks, A characterization of inner product spaces, Math. Japonica 23 4 (1978) 371-373. MR 0524986 (80b:46034)

7.
R. Deville, G. Godefroy, and V. Zizler, Smoothness and Renormings in Banach Spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics 64, Longman, Brunt Mill, 1993. MR 1211634 (94d:46012)

8.
R.E. Megginson, An Introduction to Banach Space Theory, Graduate Texts in Mathematics 183, New York, Springer-Verlag, 1998. MR 1650235 (99k:46002)

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 46B20, 46C05, 46B04

Retrieve articles in all Journals with MSC (2000): 46B20, 46C05, 46B04

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: 10.1090/S0002-9939-06-08243-8
PII: S 0002-9939(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.
Posted: January 17, 2006
Communicated by: N. Tomczak-Jaegermann
Copyright of article: Copyright 2006, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2009, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google