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A new version of the Hahn--Banach theorem 0. Introduction

Summary: A new version of the Hahn--Banach theorem
S. Simons
0. Introduction
In this paper, we discuss a new version of the Hahn--Banach theorem that has a number of
applications in di#erent 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 ``a#ne'' for ``S--convex''. This change shortens the proof of Lemma 1.4 by a few
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 situa­
tion when the classical Fenchel--Moreau formula is valid, the proof of it given here using
Theorem 1.5 allows us to avoid the problem of the ``vertical hyperplane''. The second
application is a short proof of a fundamental result on dual problems and Lagrangians due


Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara


Collections: Mathematics