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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
MathSciNet review: 1485482
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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 $X$ is reflexive if and only if it satisfies any of the following properties: (i) $X$ admits a norm $\|\cdot\|$ with the Mazur Intersection Property and the set $NA_{\,\|\cdot\|}$ of all norm attaining functionals of $X^*$ contains an open set, (ii) the set $NA^1_{\,\|\cdot\|}$ of all norm one elements of $NA_{\,\|\cdot\|}$ contains a (relative) weak* open set of the unit sphere, (iii) $X^*$ has $C^*PCP$ and $NA^1_{\,\|\cdot\|}$ contains a (relative) weak open set of the unit sphere, (iv) $X$ is $WCG$, $X^*$ has $CPCP$ and $NA^1_{\,\|\cdot\|}$ contains a (relative) weak open set of the unit sphere. Finally, if $X$ is separable, then $X$ is reflexive if and only if $NA^1_{\,\|\cdot\|}$ 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

J. P. Moreno
Affiliation: Departamento de Matemáticas C–XV, Universidad Autónoma, Madrid, 28049, Spain

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