文档介绍：Mathematical
Monographs
Volume 53
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American Mathematical Society
ee .Michor Peter W. Andreas Kriegl Global Analysis Setting of The Convenient
Editorial Board
Howard A. Masur Michael Renardy
Tudor Stefan Ratiu, Chair
1991 Mathematics Subject Classification. Primary 22E65, 26E15, 26E20, 46A17, 46G05,
46G20, 46E25, 46E50, 58B10, 58B12, 58B20, 58B25, 58C20, 46E50, 58D05, 58D10,
58D15, 58D17, 58D19, 58F25; Secondary 22E45, 58C40, 22E67, 46A16, 57N20, 58B05,
58D07, 58D25, 58D27, 58F05, 58F06, 58F07.
Abstract. The aim of this book is to lay foundations of differential calculus in infinite dimensions
and to discuss those applications in infinite dimensional differential geometry and global analysis
which do not involve pletions and fixed point theory. The approach is very simple:
A mapping is called smooth if it maps smooth curves to smooth curves. All other properties
are proved results and not assumptions: Like chain rule,existence and linearity of derivatives,
powerful smooth uniformly boundedness theorems are available. Up to Fr´echet spaces this notion
of smoothness coincides with all known reasonable concepts. In the same spirit calculus of holo-
morphic 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