1 Introduction ................................................. 6
2 Preliminaries and notation ................................... 8
2.1 Notation ................................................ 8
2.2 Differential calculus of maps between locally convex
spaces .................................................. 8
2.3 Frechet differentiability ............................... 9
3 Weighted function spaces .................................... 10
3.1 Definition and examples ................................ 10
3.2 Topological and uniform structure ...................... 12
3.2.1 Reduction to lower order ........................ 12
3.2.2 Projective limits and the topology of C∞w(U,Y) ... 13
3.2.3 A completeness criterion ........................ 14
3.3 Composition on weighted functions and superposition
operators .............................................. 17
3.3.1 Composition with a multilinear map .............. 17
3.3.2 Composition of weighted functions with bounded
functions ....................................... 21
3.3.3 Composition of weighted functions with an
analytic map .................................... 26
3.4 Weighted maps into locally convex spaces ............... 30
3.4.1 Definition and topological structure ............ 30
3.4.2 Weighted decreasing maps ........................ 33
3.4.3 Composition and superposition ................... 35
4 Lie groups of weighted diffeomorphisms ...................... 41
4.1 Weighted diffeomorphisms and endomorphisms ............. 41
4.1.1 Composition of weighted endomorphisms in
charts .......................................... 42
4.1.2 Smooth monoids of weighted endomorphisms ........ 47
4.2 Lie group structures on weighted diffeomorphisms ....... 48
4.2.1 The Lie group structure of Diffw(X) ............. 48
4.2.2 On decreasing weighted diffeomorphisms and
dense subgroups ................................. 54
4.2.3 On diffeomorphisms that are weighted
endomorphisms ................................... 55
4.3 Regularity ............................................. 57
4.3.1 The tangent group and the regularity
differential equation of Diffw(X) ............... 57
4.3.2 Conclusion and calculation of one-parameter
groups .......................................... 61
5 Integration of certain Lie algebras of vector fields ........ 62
5.1 On the smoothness of the conjugation action on
Diffw(X)0 .............................................. 63
5.1.1 Contravariant composition on weighted
functions ....................................... 64
5.2 Conclusion and examples ................................ 67
6 Lie group structures on weighted mapping groups ............. 69
6.1. Weighted maps into Banach Lie groups ................... 70
6.1.1 Construction of the Lie group ................... 70
6.1.2 Regularity ...................................... 74
6.1.3 Semidirect products with weighted
diffeomorphisms ................................. 77
6.2 Weighted maps into locally convex Lie groups ........... 78
6.2.1 Construction of the Lie group ................... 78
6.2.2 A larger Lie group of weighted mappings ......... 81
A Differential calculus ....................................... 95
A.l Differential calculus of maps between locally convex
spaces ................................................. 95
A.1.1 Curves and integrals ............................ 95
A.1.2 Differentiable maps ............................. 98
A.2 Frechet differentiability ............................. 104
A.3 Relation between the differential calculi ............. 108
A.4 Some facts concerning ordinary differential
equations ............................................. 1ll
A.4.1 Maximal solutions of ODEs ...................... 1ll
A.4.2 Flows and dependence on parameters and
initial values ................................. 113
B Locally convex Lie groups .................................. 116
B.l Locally convex manifolds .............................. 116
B.2. Lie groups ............................................ 117
B.2.1 Generation of Lie groups ....................... 118
B.2.2 Regularity ..................................... 118
B.2.3 Group actions .................................. 119
C Quasi-inversion in algebras ................................ 120
C.l Definition ............................................ 121
C.2 Topological monoids and algebras with continuous
quasi-inversion ....................................... 121
References .................................................... 124
Notation ...................................................... 126
Index ......................................................... 128
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