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

[摘要] 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$$ |f|geq|pif|+s|f-pif|quadforallfin V_{s}^{astastast}, $$where $pi$ is the canonical projection from $V_{s}^{astastast}$ onto $V_{s}^{ast}$. In particular, no $V_{s}$ can be renormed to be an $M$-ideal.AMS 2000 Mathematics subject classification: Primary 46B20

[学科分类] 数学（综合）

[关键词] $M$-ideal;$M(r;s)$-inequality;weakly compactly generated (WCG) Banach space