Summary: A new version of the Hahn--Banach theorem
This is an extract from a paper titled ``Hahn--Banach theorems and maximal monotonic
ity'' that will appear in the volume ``Variational analysis and Applications'' edited by F.
Giannessi and A. Maugeri. In it, 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, and convex analysis. All vector spaces in this
paper will be real.
The main result appears in Theorem 2.8, which is bootstrapped from the special case
contained in Lemma 2.4.
In Section 3, we sketch how Theorem 2.8 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 2.8 to convex
analysis, pointing the reader to  for further details in two of these cases. One noteworthy
property of proofs using Theorem 2.8 is that they allow us to avoid the problem of the
In Section 4, we show how Theorem 2.8 can be used to obtain considerable insight on
the existence of Lagrange multipliers for constrained convex minimization problems. The