Mathematical Surveys and Monographs 1997; 618 pp; hardcover Volume: 53 ISBN10: 0821807803 ISBN13: 9780821807804 List Price: US$80 Member Price: US$64 Order Code: SURV/53
 This book lays the foundations of differential calculus in infinite dimensions and discusses those applications in infinite dimensional differential geometry and global analysis not involving Sobolev completions and fixed point theory. The approach is simple: a mapping is called smooth if it maps smooth curves to smooth curves. Up to Fréchet spaces, this notion of smoothness coincides with all known reasonable concepts. In the same spirit, calculus of holomorphic mappings (including Hartogs' theorem and holomorphic uniform boundedness theorems) and calculus of real analytic mappings are developed. Existence of smooth partitions of unity, the foundations of manifold theory in infinite dimensions, the relation between tangent vectors and derivations, and differential forms are discussed thoroughly. Special emphasis is given to the notion of regular infinite dimensional Lie groups. Many applications of this theory are included: manifolds of smooth mappings, groups of diffeomorphisms, geodesics on spaces of Riemannian metrics, direct limit manifolds, perturbation theory of operators, and differentiability questions of infinite dimensional representations. Readership Graduate students and research mathematicians interested in global analysis and analysis on manifolds. Reviews "Very interesting ... covers many topics that are difficult to find elsewhere in book form ... a valuable tool for selfstudy as well as an excellent reference."  Mathematical Reviews Table of Contents  Introduction
 Calculus of smooth mappings
 Calculus of holomorphic and real analytic mappings
 Partitions of unity
 Smoothly realcompact spaces
 Extensions and liftings of mappings
 Infinite dimensional manifolds
 Calculus on infinite dimensional manifolds
 Infinite dimensional differential geometry
 Manifolds of mappings
 Further applications
 References
 Index
