New Characterizations of the Reflexivity in Terms of the Set of Norm Attaining Functionals
Canadian mathematical bulletin, Tome 41 (1998) no. 3, pp. 279-289
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As a consequence of results due to Bourgain and Stegall, on a separable Banach space whose unit ball is not dentable, the set of norm attaining functionals has empty interior (in the norm topology). First we show that any Banach space can be renormed to fail this property. Then, our main positive result can be stated as follows: if a separable Banach space $X$ is very smooth or its bidual satisfies the ${{\mathcal{w}}^{*}}$ -Mazur intersection property, then either $X$ is reflexive or the set of norm attaining functionals has empty interior, hence the same result holds if $X$ has the Mazur intersection property and so, if the norm of $X$ is Fréchet differentiable. However, we prove that smoothness is not a sufficient condition for the same conclusion.
Acosta, Mariá D.; Galán, Manuel Ruiz. New Characterizations of the Reflexivity in Terms of the Set of Norm Attaining Functionals. Canadian mathematical bulletin, Tome 41 (1998) no. 3, pp. 279-289. doi: 10.4153/CMB-1998-040-x
@article{10_4153_CMB_1998_040_x,
author = {Acosta, Mari\'a D. and Gal\'an, Manuel Ruiz},
title = {New {Characterizations} of the {Reflexivity} in {Terms} of the {Set} of {Norm} {Attaining} {Functionals}},
journal = {Canadian mathematical bulletin},
pages = {279--289},
year = {1998},
volume = {41},
number = {3},
doi = {10.4153/CMB-1998-040-x},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1998-040-x/}
}
TY - JOUR AU - Acosta, Mariá D. AU - Galán, Manuel Ruiz TI - New Characterizations of the Reflexivity in Terms of the Set of Norm Attaining Functionals JO - Canadian mathematical bulletin PY - 1998 SP - 279 EP - 289 VL - 41 IS - 3 UR - http://geodesic.mathdoc.fr/articles/10.4153/CMB-1998-040-x/ DO - 10.4153/CMB-1998-040-x ID - 10_4153_CMB_1998_040_x ER -
%0 Journal Article %A Acosta, Mariá D. %A Galán, Manuel Ruiz %T New Characterizations of the Reflexivity in Terms of the Set of Norm Attaining Functionals %J Canadian mathematical bulletin %D 1998 %P 279-289 %V 41 %N 3 %U http://geodesic.mathdoc.fr/articles/10.4153/CMB-1998-040-x/ %R 10.4153/CMB-1998-040-x %F 10_4153_CMB_1998_040_x
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