Conic James' Compactness Theorem
José Orihuela · Journal of convex analysis · 2018
The following results is proved: Let A A be a convex bounded non weakly relatively compact subset of a Banach space E E . We consider a convex weakly compact subset D D of E E which does not contain the origin. Then there is a sequence \left\{x_n^*\right\}_{n\ge 1} { x n ∗ } n ≥ 1 in B_{E^*} B E ∗ and g_0^*\in \hbox{co}_{\sigma}\{x_n^*:n\ge 1\} g 0 ∗ ∈ co σ { x n ∗ : n ≥ 1 } such that for all h\in \ell_\infty (A) h ∈ ℓ ∞ ( A ) satisfying that for all a\in A, a ∈ A , \liminf_{n\ge 1}x_n^*(a) \le h(a) \le\limsup_{n\ge 1}x_n^*(a), lim inf n ≥ 1 x n ∗ ( a ) ≤ h ( a ) ≤ lim sup n ≥ 1 x n ∗ (