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Journal of Convex Analysis 25 (2018), No. 4, 1335--1344
Copyright Heldermann Verlag 2018

Conic James' Compactness Theorem

José Orihuela
Dep. de Matemáticas, Universidad de Murcia, 30100 Espinardo-Murcia, Spain


The following results is proved:\par Let $A$ be a convex bounded non weakly relatively compact subset of a Banach space $E$. We consider a convex weakly compact subset $D$ of $E$ which does not contain the origin.\par Then there is a sequence $\left\{x_n^*\right\}_{n\ge 1}$ in $B_{E^*}$ and $g_0^*\in \hbox{co}_{\sigma}\{x_n^*:n\ge 1\}$ such that for all $h\in \ell_\infty (A)$ satisfying that for all $a\in A,$ $$ \liminf_{n\ge 1}x_n^*(a) \le h(a) \le\limsup_{n\ge 1}x_n^*(a), $$ we have that\ \ $g_0^*- h$\ \ does not attain its supremum on $A$ and\ \ $( g_0^*- h)(d)>0$\ \ for every $d\in D$.

Keywords: James' compactness theorem, weakly compact set.

MSC: 46A50, 46B50

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