On$M$-structure, the asymptotic-norming property and locally uniformly rotund renormings

Eduardo Nieto Departamento de Análisis Matemático Facultad de Ciencias, Universidad de Granada Migdalia Rivas Departamento de Análisis Matemático Facultad de Ciencias, Universidad de Granada

TBD mathscidoc:1701.332987

Arkiv for Matematik, 40, (2), 323-333, 2000.8
Let$r, s$∈ [0, 1], and let$X$be a Banach space satisfying the$M(r, s)$-inequality, that is, $$\parallel x^{***} \parallel \geqslant r\parallel \pi _X x^{***} \parallel + s\parallel x^{***} - \pi _X x^{***} \parallel for x^{***} \in X^{***} ,$$ where π_{$X$}is the canonical projection from$X$^{***}onto$X$^{*}. We show some examples of Banach spaces not containing$c$_{0}, having the point of continuity property and satisfying the above inequality for$r$not necessarily equal to one. On the other hand, we prove that a Banach space$X$satisfying the above inequality for$s$=1 admits an equivalent locally uniformly rotund norm whose dual norm is also locally uniformly rotund. If, in addition,$X$satisfies $$\mathop {\lim \sup }\limits_\alpha \parallel u^* + sx_\alpha ^* \parallel \leqslant \mathop {\lim \sup }\limits_\alpha \parallel v^* + x_\alpha ^* \parallel $$ whenever$u$^{*},$v$^{*}∈$X$^{*}with ‖$u$^{*}‖≤‖$v$^{*}‖ and ($x$_{α}^{*}) is a bounded weak^{*}null net in$X$^{*}, then$X$can be renormed to satisfy the,$M(r, 1)$and the$M(1, s)$-inequality such that$X$^{*}has the weak^{*}asymptotic-norming property I with respect to$B$_{$X$}.
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@inproceedings{eduardo2000on$m$-structure,,
  title={On$M$-structure, the asymptotic-norming property and locally uniformly rotund renormings},
  author={Eduardo Nieto, and Migdalia Rivas},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203605152890796},
  booktitle={Arkiv for Matematik},
  volume={40},
  number={2},
  pages={323-333},
  year={2000},
}
Eduardo Nieto, and Migdalia Rivas. On$M$-structure, the asymptotic-norming property and locally uniformly rotund renormings. 2000. Vol. 40. In Arkiv for Matematik. pp.323-333. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203605152890796.
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