One side James' Compactness Theorem
Abstract
We present some extensions of classical results that involve elements of the dual of Banach spaces, such as BishopPhelp's theorem and James' compactness theorem, but restricting to sets of functionals determined by geometrical properties. The main result, which answers a question posed by F. Delbaen, is the following: Let $E$ be a Banach space such that $(B_{E^\ast}, \omega^\ast)$ is convex block compact. Let $A$ and $B$ be bounded, closed and convex sets with distance $d(A,B) > 0$. If every $x^\ast \in E^\ast$ with \[ \sup(x^\ast,B) < \inf(x^\ast,A) \] attains its infimum on $A$ and its supremum on $B$, then $A$ and $B$ are both weakly compact. We obtain new characterizations of weakly compact sets and reflexive spaces, as well as a result concerning a variational problem in dual Banach spaces.
 Publication:

arXiv eprints
 Pub Date:
 August 2015
 DOI:
 10.48550/arXiv.1508.00496
 arXiv:
 arXiv:1508.00496
 Bibcode:
 2015arXiv150800496C
 Keywords:

 Mathematics  Functional Analysis;
 46A50;
 46B50
 EPrint:
 18 pages