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ON $M$-STRUCTURE AND WEAKLY COMPACTLY GENERATED BANACH SPACES

Published online by Cambridge University Press:  10 December 2003

Eduardo Nieto
Eduardo Nieto
Affiliation:
Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain ([email protected])
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Abstract

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It is well known that every non-reflexive $M$-ideal is weakly compactly generated (in short, WCG). We present a family of Banach spaces $\{V_{s}:0 \lt s \lt 1\}$ which are not WCG and such that every $V_{s}$ satisfies the inequality

$$ \|\f\|\geq\|\pi\f\|+s\|\f-\pi\f\|\quad\forall\f\in V_{s}^{\ast\ast\ast}, $$

where $\pi$ is the canonical projection from $V_{s}^{\ast\ast\ast}$ onto $V_{s}^{\ast}$. In particular, no $V_{s}$ can be renormed to be an $M$-ideal.

AMS 2000 Mathematics subject classification: Primary 46B20

Keywords

$M$-ideal$M(r,s)$-inequalityweakly compactly generated (WCG) Banach space
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 46 , Issue 3 , October 2003 , pp. 679 - 686
Copyright
Copyright © Edinburgh Mathematical Society 2003