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DUAL DIFFERENTIATION SPACES

Part of: Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  28 March 2019

WARREN B. MOORS  and
NEŞET ÖZKAN TAN
WARREN B. MOORS*
Affiliation:
Department of Mathematics, The University of Auckland, Auckland 1142, New Zealand email moors@math.auckland.ac.nz
NEŞET ÖZKAN TAN
Affiliation:
Department of Mathematics, The University of Auckland, Auckland 1142, New Zealand email neset.tan@auckland.ac.nz
*
✉moors@math.auckland.ac.nz
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Abstract

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We show that if $(X,\Vert \cdot \Vert )$ is a Banach space that admits an equivalent locally uniformly rotund norm and the set of all norm-attaining functionals is residual then the dual norm $\Vert \cdot \Vert ^{\ast }$ on $X^{\ast }$ is Fréchet at the points of a dense subset of $X^{\ast }$. This answers the main open problem in a paper by Guirao, Montesinos and Zizler [‘Remarks on the set of norm-attaining functionals and differentiability’, Studia Math.241 (2018), 71–86].

Keywords

norm-attaining functionalslocally uniformly rotund normdual differentiation space

MSC classification

Primary: 46B20: Geometry and structure of normed linear spaces
Secondary: 46B26: Nonseparable Banach spaces
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 99 , Issue 3 , June 2019 , pp. 467 - 472
Copyright
© 2019 Australian Mathematical Publishing Association Inc. 

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