MathSciDoc: An Archive for Mathematician ∫

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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