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Hardcover ISBN:  9780821807804 
Product Code:  SURV/53 
List Price:  $129.00 
MAA Member Price:  $116.10 
AMS Member Price:  $103.20 
eBook ISBN:  9780821833964 
Product Code:  SURV/53.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
Hardcover ISBN:  9780821807804 
eBook ISBN:  9780821833964 
Product Code:  SURV/53.B 
List Price:  $254.00 $191.50 
MAA Member Price:  $228.60 $172.35 
AMS Member Price:  $203.20 $153.20 

Book DetailsMathematical Surveys and MonographsVolume: 53; 1997; 618 ppMSC: Primary 22; 26; 46; 58
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.
ReadershipGraduate students and research mathematicians interested in global analysis and analysis on manifolds.

Table of Contents

Chapters

I. Calculus of smooth mappings

II. Calculus of holomorphic and real analytic mappings

III. Partitions of unity

IV. Smoothly realcompact spaces

V. Extensions and liftings of mappings

VI. Infinite dimensional manifolds

VII. Calculus on infinite dimensional manifolds

VIII. Infinite dimensional differential geometry

IX. Manifolds of mappings

X. Further applications


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


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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.
Graduate students and research mathematicians interested in global analysis and analysis on manifolds.

Chapters

I. Calculus of smooth mappings

II. Calculus of holomorphic and real analytic mappings

III. Partitions of unity

IV. Smoothly realcompact spaces

V. Extensions and liftings of mappings

VI. Infinite dimensional manifolds

VII. Calculus on infinite dimensional manifolds

VIII. Infinite dimensional differential geometry

IX. Manifolds of mappings

X. Further applications

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