**Mathematical Surveys and Monographs**

Volume: 53;
1997;
618 pp;
Hardcover

MSC: Primary 22; 26; 46; 58;
Secondary 57

Print ISBN: 978-0-8218-0780-4

Product Code: SURV/53

List Price: $84.00

Individual Member Price: $67.20

**Electronic ISBN: 978-0-8218-3396-4
Product Code: SURV/53.E**

List Price: $84.00

Individual Member Price: $67.20

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# The Convenient Setting of Global Analysis

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*Andreas Kriegl; Peter W. Michor*

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.

#### Table of Contents

# Table of Contents

## The Convenient Setting of Global Analysis

- Table of Contents ix10 free
- Introduction 112 free
- Chapter I: Calculus of Smooth Mappings 718 free
- Chapter II: Calculus of Holomorphic and Real Analytic Mappings 7990
- Chapter III: Partitions of Unity 117128
- Chapter IV: Smoothly Realcompact Spaces 183194
- Chapter V: Extensions and Liftings of Mappings 219230
- 21. Extension and Lifting Properties 220231
- 22. Whitney's Extension Theorem Revisited 226237
- 23. Frölicher Spaces and Free Convenient Vector Spaces 238249
- 24. Smooth Mappings on Non-Open Domains 247258
- 25. Real Analytic Mappings on Non-Open Domains 254265
- 26. Holomorphic Mappings on Non-Open Domains 261272

- Chapter VI: Infinite Dimensional Manifolds 263274
- Chapter VII: Calculus on Infinite Dimensional Manifolds 321332
- Chapter VIII: Infinite Dimensional Differential Geometry 369380
- Chapter IX: Manifolds of Mappings 429440
- 41. Jets and Whitney Topologies 431442
- 42. Manifolds of Mappings 439450
- 43. Diffeomorphism Groups 454465
- 44. Principal Bundles with Structure Group a Diffeomorphism Group 474485
- 45. Manifolds of Riemannian Metrics 487498
- 46. The Korteweg – De Vrieß Equation as a Geodesic Equation 498509
- Complements to Manifolds of Mappings 510521

- Chapter X: Further Applications 511522
- 47. Manifolds for Algebraic Topology 512523
- 48. Weak Symplectic Manifolds 522533
- 49. Applications to Representations of Lie Groups 528539
- 50. Applications to Perturbation Theory of Operators 536547
- 51. The Nash-Moser Inverse Function Theorem 553564
- 52. Appendix: Functional Analysis 575586
- 53. Appendix: Projective Resolutions of Identity on Banach spaces 582593

- References 597608
- Index 611622

#### 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