OnM-structure, the asymptotic-norming property and locally uniformly rotund renormings
AbstractLetr, s ∈ [0, 1], and letX be a Banach space satisfying theM(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 πX is the canonical projection fromX*** ontoX*. We show some examples of Banach spaces not containingc0, having the point of continuity property and satisfying the above inequality forr not necessarily equal to one. On the other hand, we prove that a Banach spaceX satisfying the above inequality fors=1 admits an equivalent locally uniformly rotund norm whose dual norm is also locally uniformly rotund. If, in addition,X satisfies
$$\mathop {\lim \sup }\limits_\alpha \parallel u^* + sx_\alpha ^* \parallel \leqslant \mathop {\lim \sup }\limits_\alpha \parallel v^* + x_\alpha ^* \parallel $$
 wheneveru*,v* ∈X* with ‖u*‖≤‖v*‖ and (xα*) is a bounded weak* null net inX*, thenX can be renormed to satisfy the,M(r, 1) and theM(1, s)-inequality such thatX* has the weak* asymptotic-norming property I with respect toBX.