The Bishop-Phelps Theorem in Complex Spaces

Abstract

Recall, first, the statement of the Bishop-Phelps theorem [BP] for a real Banach space E: If C is a nonempty closed convex subset of E, if / £ E* is bounded above on C and if ε > 0, then there exists g £ JE7*, g φ 0, which attains its supremum on C at some point χ of C and which satisfies ||/ — g\\ < ε. (We say that g is a support functional of C and that £ is a support point of C.) Moreover, for any closed convex C the set of support points is dense in the boundary of C.