Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Banach spaces that have normal structure and are isomorphic to a Hilbert space


Authors: Javier Bernal and Francis Sullivan
Journal: Proc. Amer. Math. Soc. 90 (1984), 550-554
MSC: Primary 46B20; Secondary 46C05
DOI: https://doi.org/10.1090/S0002-9939-1984-0733404-9
MathSciNet review: 733404
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that given a Hilbert space $ \left( {E,\vert\vert \cdot \vert\vert} \right)$, and $ \vert \cdot \vert$ a norm on $ E$ such that for all $ x \in E$, $ 1/\beta \left\vert x \right\vert \leqslant \left\Vert x \right\Vert \leqslant \left\vert x \right\vert$ for some $ \beta $, if $ 1 \leqslant \beta < \sqrt 2 $, then $ \left( {E,\vert \cdot \vert} \right)$ satisfies a convexity property from which normal structure follows.


References [Enhancements On Off] (What's this?)

  • [1] J. B. Baillon and R. Schöneberg, Asymptotic normal structure and fixed points of nonexpansive mappings, Proc. Amer. Math. Soc. 81 (1981), 257-264. MR 593469 (82c:47068)
  • [2] J. Bernal, Behavior of $ k$-dimensional convexity moduli, Thesis, Catholic University of America, Washington, D.C., 1980.
  • [3] J. Bernal and F. Sullivan, Multi-dimensional volumes, super-reflexivity and normal structure in Banach space, Illinois J. Math. 27 (1983), 501-513. MR 698311 (84g:46021)
  • [4] W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004-1006. MR 0189009 (32:6436)
  • [5] F. Sullivan, A generalization of uniformly rotund Banach spaces, Canad. J. Math. 31 (1979), 628-636. MR 536368 (80h:46023)

Similar Articles

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

Retrieve articles in all journals with MSC: 46B20, 46C05


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1984-0733404-9
Article copyright: © Copyright 1984 American Mathematical Society

American Mathematical Society