Abstract

The following results is proved: Let AA be a convex bounded non weakly relatively compact subset of a Banach space EE. We consider a convex weakly compact subset DD of EE which does not contain the origin. Then there is a sequence {xn∗}n≥1\left\{x_n^*\right\}_{n\ge 1} in BE∗B_{E^*} and g0∗∈coσ{xn∗:n≥1}g_0^*\in \hbox{co}_{\sigma}\{x_n^*:n\ge 1\} such that for all h∈ℓ∞(A)h\in \ell_\infty (A) satisfying that for all a∈A,a\in A, lim inf⁡n≥1xn∗(a)≤h(a)≤lim sup⁡n≥1xn∗(a),\liminf_{n\ge 1}x_n^*(a) \le h(a) \le\limsup_{n\ge 1}x_n^*(a), we have that g0∗−hg_0^*- h does not attain its supremum on AA and (g0∗−h)(d)>0( g_0^*- h)(d)>0 for every d∈Dd\in D.

Author information

Contact details are reproduced from the original publication and may be historical.

José Orihuela

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

joseori@um.es

Keywords

  • James' compactness theorem
  • weakly compact set

Mathematics Subject Classification

46A50, 46B50

References

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Suggested citation

J. Orihuela. “Conic James' Compactness Theorem.” Journal of Convex Analysis 25 (2018), No. 4, 1335–1344. https://doi.org/10.68381/jca25081

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Published by Heldermann Verlag, 2018. Rights now held by Banach Press LLC.