This volume is intended as an essentially self-contained exposition of portions of the theory of nonlinear analysis. It grew out of lecture notes for graduate courses by the author at National Tsing Hua University, Hsinchu, Taiwan.
　For many decades, great mathematical interest has focused on problems associated with linear operators and the extension of the results of linear algebra to an in¯nite-dimensional context. When one drops the assumption of linearity, the associated operator theory and the many concrete problems associated with such a theory represent a frontier of mathematical research. Just as in the linear case, these results were inspired by and are highly relevant to concrete problems in mathematical analysis. We shall take up a variety of analytic and topological techniques for the study of nonlinear problems and their applicability to a variety of concrete problems taken
from various ¯elds of mathematical analysis.
　This volume contains: function analysis; ¯xed point theorems in Banach spaces; calculus in Banach spaces; bifurcation theory for operators on Banach spaces; Sobolev spaces in R and in RN; N > 1; the theory of compact operators, Fredholm operators, and monotone operators; the variational methods for nonlinear equations; the topological degree of the compactness erturbation of the identity map; and the applications of topological degrees to ODEs and PDEs.

Hwai-Chiuan Wang
July 2003