This monograph is based in part on a series of lectures given by the first author at Cleveland State University and at Texas A & M University since 1980, and in part on the lectures given by the second author at Dalhousie University and the Australian National University since 1981. It concerns the development of certain topics in differential nonlinear analysis in infinite dimensional real Banach spaces. Our motivation derives from the rich and elegant theory of nonlinear analysis in finite dimensional setting and from the fact that the powerful theorems of Stone-Weierstrass, Whitney and Bernstein do not extend to infinite dimensional Banach spaces in general. There seems to be no comprehensive discussion available in the literature on the topics dealt with in these notes. During the last three decades many mathematicians have contributed to various problems on nonlinear analysis in Banach spaces which lie scattered in various journals. A substantial part of these contributions concern the unravelling of the geometric structure of infinite dimensional Banach spaces and the progress that has been made in applying these results to solve problems in nonlinear analysis on such spaces.
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