A new version of the Hahn - Banach theorem

We discuss a new version of the Hahn-Banach theorem, with applications to linear and nonlinear functional analysis, convex analysis, and the theory of monotone multifunc-tions. We show how our result can be used to prove a " localized " version of the Fenchel-Moreau formula — even when the classical Fenchel-Moreau formula is valid, the proof of it given here avoids the problem of the " vertical hyperplane ". We give a short proof of Rockafellar's fundamental result on dual problems and Lagrangians — obtaining a necessary and sufficient condition instead of the more usual sufficient condition. We show how our result leads to a proof of the (well-known) result that if a monotone multifunction on a normed space has bounded range then it has full domain. We also show how our result leads to generalizations of an existence theorem with no a priori scalar bound that has proved very useful in the investigation of monotone multifunc-tions, and show how the estimates obtained can be applied to Rockafellar's surjectivity theorem for maximal monotone multifunctions in reflexive Banach spaces. Finally, we show how our result leads easily to a result on convex functions that can be used to establish a minimax theorem. 0. Introduction. In this paper, we discuss a new version of the Hahn-Banach theorem that has a number of applications in different fields of analysis. We shall give applications to linear and nonlinear functional analysis, convex analysis, and the theory of monotone multifunctions. After a few preliminaries, the main result appears in Theorem 1.5, which uses the concept of " S-convexity " introduced in Definition 1.3. The full force of this concept will be used only in Theorem 2.4, a result on convex functions with applications to a minimax theorem. For all the other applications of Theorem 1.5 in this paper, the reader can substitute " affine " for " S-convex ". This change shortens the proof of Lemma 1.4 by a few lines. In Section 2, we sketch how Theorem 1.5 can be used to give the main existence theorems for linear functionals in functional analysis, and also how it gives the result referred to above that leads to a minimax theorem. Section 3 contains two applications of Theorem 1.5 to convex analysis. The first, Theorem 3.4, is a " localized " version of the Fenchel-Moreau formula. Even in the situation when the classical Fenchel-Moreau formula is valid, the …