2015/08/03 by Cascales, Bernardo, Orihuela, José, Pérez, Antonio
#46A50 #46B50 #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.1508.00496
We present some extensions of classical results that involve elements of the dual of Banach spaces, such as Bishop-Phelp'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 (BE^∗, ω^∗) is convex block compact. Let A and B be bounded, closed and convex sets with distance d(A,B) > 0. If every x^∗ ∈ E^∗ with sup(x^∗,B) lt; inf(x^∗,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.