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 self-study 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