A note on norm attaining functionals

Authors:
M. Jiménez Sevilla and J. P. Moreno

Journal:
Proc. Amer. Math. Soc. **126** (1998), 1989-1997

MSC (1991):
Primary 46B20

DOI:
https://doi.org/10.1090/S0002-9939-98-04739-X

MathSciNet review:
1485482

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Abstract | References | Similar Articles | Additional Information

Abstract: We are concerned in this paper with the density of functionals which do not attain their norms in Banach spaces. Some earlier results given for separable spaces are extended to the nonseparable case. We obtain that a Banach space is reflexive if and only if it satisfies any of the following properties: (i) admits a norm with the Mazur Intersection Property and the set of all norm attaining functionals of contains an open set, (ii) the set of all norm one elements of contains a (relative) weak* open set of the unit sphere, (iii) has and contains a (relative) weak open set of the unit sphere, (iv) is , has and contains a (relative) weak open set of the unit sphere. Finally, if is separable, then is reflexive if and only if contains a (relative) weak open set of the unit sphere.

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Additional Information

**M. Jiménez Sevilla**

Affiliation:
Departamento de Análisis Matemático, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, Madrid, 28040, Spain

Email:
marjim@sunam1.mat.ucm.es

**J. P. Moreno**

Affiliation:
Departamento de Matemáticas C–XV, Universidad Autónoma, Madrid, 28049, Spain

Email:
moreno@sunam1.mat.ucm.es

DOI:
https://doi.org/10.1090/S0002-9939-98-04739-X

Keywords:
Reflexive spaces,
Mazur Intersection Property,
(Weak*) Convex Point of Continuity Property

Received by editor(s):
December 2, 1996

Additional Notes:
Partially supported by DGICYT PB 96-0607.

Communicated by:
Palle E. T. Jorgensen

Article copyright:
© Copyright 1998
American Mathematical Society