1993/03/04 by Charles Stegall, Stegall, Charles P.
Mathematics · #Advanced Banach Space Theory #Advanced Topology and Set Theory #Functional Equations Stability Results
paper · pdf · doi:10.48550/arxiv.math/9303201
A remarkable theorem of R. C. James is the following: suppose that X is a Banach space and C ⊆ X is a norm bounded, closed and convex set such that every linear functional x^* ∈ X^* attains its supremum on C; then C is a weakly compact set. Actually, this result is significantly stronger than this statement; indeed, the proof can be used to obtain other surprising results. For example, suppose that X is a separable Banach space and S is a norm separable subset of the unit ball of X^* such that for each x ∈ X there exists x^* ∈ S such that x^*(x) = ‖x‖ then X^* is itself norm separable . If we call S a support set, in this case, with respect to the entire space X, one can ask questions about the size and structure of a support set, a support set not only with respect to X itself but perhaps with respect to some other subset of X@. We analyze one particular case of this as well as give some applications.