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Hahn--Banach theorems and maximal monotonicity 0. Introduction

Summary: Hahn--Banach theorems and maximal monotonicity
S. Simons
0. Introduction
In this paper, we discuss new versions of the Hahn--Banach theorem that have 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. All vector
spaces in this paper will be real.
The main result appears in Theorem 1.7, which is bootstrapped from the special case
contained in Lemma 1.4.
In Section 2, we sketch how Theorem 1.7 can be used to give the main existence
theorems for linear functionals in functional analysis, and also how it gives a result that
leads to a minimax theorem. We also discuss three applications of Theorem 1.7 to convex
analysis, pointing the reader to [26] for further details in two of these cases. One noteworthy
property of proofs using Theorem 1.7 is that they allow us to avoid the problem of the
``vertical hyperplane''.
In Section 3, we show how Theorem 1.7 can be used to obtain considerable insight on
the existence of Lagrange multipliers for constrained convex minimization problems. The
usual su#cient condition for the existence of such multipliers is normally found using the
Eidelheit separation theorem. In Theorem 3.5, we use Theorem 1.7 to derive this su#cient
condition, with the added bonus that we obtain a bound on the norm of the multiplier.


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


Collections: Mathematics