# MathSciDoc: An Archive for Mathematician ∫

#### TBDmathscidoc: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$B$_{$X$}.
@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.