From Hahn-Banach to Monotonicity

ISBN-13: 9781402069185 / Angielski / Miękka / 2008 / 248 str.

Stephen Simons

Opis

This new edition of LNM 1693 aims to reduce questions on monotone multifunctions to questions on convex functions. However, rather than 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 Fitzpatrick function is used. The journey begins with the Hahn-Banach theorem and culminates in a survey of current results on monotone multifunctions on a Banach space.

Recenzje

From the reviews of the second edition:

"Like the first edition it is cleanly, indeed elegantly, written and singularly free of even minor infelicities. ... The book has seven core chapters ... . In conclusion, I highly recommend this book as a resource to anyone interested in learning, teaching or applying the modern abstract theory of monotone multifunctions." (J. Borwein, Mathematical Reviews, Issue 2008 k)

Spis treści

The Hahn-Banach-Lagrange theorem and some consequences.- Fenchel duality.- Multifunctions, SSD spaces, monotonicity and Fitzpatrick functions.- Monotone multifunctions on general Banach spaces.- Monotone multifunctions on reflexive Banach spaces.- Special maximally monotone multifunctions.- The sum problem for general Banach spaces.- Open problems.- Glossary of classes of multifunctions.- A selection of results.

Blurb

In this new edition of LNM 1693 the essential idea is to reduce questions on monotone multifunctions to questions on convex functions. However, rather than 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 Fitzpatrick function is used. The journey begins with a generalization of the Hahn-Banach theorem uniting classical functional analysis, minimax theory, Lagrange multiplier theory and convex analysis and culminates in a survey of current results on monotone multifunctions on a Banach space.

The first two chapters are aimed at students interested in the development of the basic theorems of functional analysis, which leads painlessly to the theory of minimax theorems, convex Lagrange multiplier theory and convex analysis. The remaining five chapters are useful for those who wish to learn about the current research on monotone multifunctions on (possibly non reflexive) Banach space.