From the Preface: These notes are somewhere between a sequel to and a new edition of [“Minimax monotonicity” (Lect.Notes Math. 1693, Berlin: Springer) (1998; Zbl 0922.47047)]. As in [op.cit.], the essential idea is to reduce questions on monotone multifunctions to questions on convex functions. In [op.cit.], this was achieved using a “big convexification” of the graph of the multifunction and the “minimax technique” for proving the existence of linear functionals satisfying certain conditions. The “big convexification” is a very abstract concept, and the analysis is quite heavy in computation. The Fitzpatrick function gives another, more concrete, way of associating a convex functions with a monotone multifunction. The problem is that many of the questions on convex functions that one obtains require an analysis of the special properties of convex functions on the product of a Banach space with its dual, which is exactly what we do in these notes. It is also worth noting that the minimax theorem is hardly used here.
We envision that these notes could be used for four different possible courses/seminars:
\(\bullet\) An introductory course in functional analysis which would, at the same time, touch on minimax theorems and give a grounding in convex Lagrange multiplier theory and the main theorems in convex analysis.
\(\bullet\) A course in which results on monotonicity on general Banach spaces are established using symmetrically self-dual spaces and Fitzpatrick functions.
\(\bullet\) A course in which results on monotonicity on reflexive Banach spaces are S established using symmetrically self-dual spaces and Fitzpatrick functions.
\(\bullet\) A seminar in which the the more technical properties of maximal monotonicity on general Banach spaces that have been established since 1997 are discussed.