Direct Limit Constructions in Infinite
Dimensional Lie Theory
Dem Institut f¨
ur Mathematik
der Universit¨
at Paderborn
zur Erlangung des Grades eines
Doktors der Naturwissenschaften
(Dr. rer. nat.)
vorgelegte Dissertation
von
Rafael Dahmen
March 1, 2011
Contents
Acknowledgement 5
Zusammenfassung 7
Introduction 9
1 Preliminaries 11
1.1 Infinite dimensional differential calculus . . . . . . . . . . . . . . . . . . . 11
1.1.1 Ckand FC kmappings . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.1.2 Polynomials between normed spaces . . . . . . . . . . . . . . . . . 14
1.1.3 Analytic mappings between locally convex spaces . . . . . . . . . . 19
1.1.4 Ordinary differential equations in Banach spaces . . . . . . . . . . 22
1.1.5 Composition operators . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.2 Locally convex direct limits of normed spaces . . . . . . . . . . . . . . . . 27
1.2.1 Compact regularity and bounded regularity . . . . . . . . . . . . . 28
1.2.2 Curves in direct limits . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.3 Lie Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.3.1 Lie groups and regularity . . . . . . . . . . . . . . . . . . . . . . . 31
1.3.2 Local Lie groups and regularity . . . . . . . . . . . . . . . . . . . . 33
2 Analytic maps on (LB)-spaces 39
3 Germs of diffeomorphisms around a compact set in a Banach space 43
3.1 Construction of DiffGerm(K, X) . . . . . . . . . . . . . . . . . . . . . . . 43
3.1.1 The modelling space . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.1.2 The monoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.1.3 The group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2 Regularity of DiffGerm(K, X) . . . . . . . . . . . . . . . . . . . . . . . . . 56
4 Ascending unions of Banach Lie groups 67
4.1 Construction of the Lie group structure . . . . . . . . . . . . . . . . . . . 67
4.2 Regularity of local Banach Lie groups . . . . . . . . . . . . . . . . . . . . 70
4.3 Regularity of (local and global) (LB)-Lie groups . . . . . . . . . . . . . . 75
5 Examples of ascending unions of Banach Lie groups 79
5.1 Groups of germs of Lie group-valued mappings . . . . . . . . . . . . . . . 79
5.2 Lie groups associated to Dirichlet series . . . . . . . . . . . . . . . . . . . 80
3
Contents
5.2.1 Banach spaces of Dirichlet series . . . . . . . . . . . . . . . . . . . 80
5.2.2 (LB)-spaces of Dirichlet series . . . . . . . . . . . . . . . . . . . . . 81
5.2.3 Lie groups associated with Dirichlet series . . . . . . . . . . . . . . 82
5.3 Lie groups associated to H¨
older continuous functions . . . . . . . . . . . . 85
5.3.1 Spaces of H¨
older Continuous Functions . . . . . . . . . . . . . . . 85
5.3.2 Inclusion Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.3.3 Completeness of the H¨
older Spaces . . . . . . . . . . . . . . . . . . 89
5.3.4 Products of H¨
older Continuous Functions . . . . . . . . . . . . . . 91
5.3.5 Directed Unions of H¨
older Spaces . . . . . . . . . . . . . . . . . . . 93
5.3.6 Lie groups associated to H¨
older continuous functions . . . . . . . . 95
5.4 Lie groups associated to ℓp-Spaces . . . . . . . . . . . . . . . . . . . . . . 97
References 101
Index 104
Notation 106
4
Acknowledgement
This thesis would not have been possible without the support and assistance of some
people, whom I would like to thank here.
First of all, I thank my advisor Professor Dr. Helge Gl¨
ockner. In the past three years,
he was consistently at my disposal. He was a reliable partner and a big help for all my
questions. This project would not have been possible without his extreme dedication.
Furthermore, I am grateful for the support of the German Research Foundation (DFG),
who financed my work within the framework of the research project GL 357/7-1 and
allowed me to participate in many interesting conferences in Germany and abroad.
I also want to express my sincerest thanks to the members of the research group
“Unendlich-dimensionale Analysis und Geometrie” at the mathematics department of
the University of Paderborn. They provided a pleasant atmosphere and many interest-
ing discussions. We could always converse on high scientific level and mutually support
our research work.
Further thanks go to my family and my girlfriend for their support all this time. And last
but not least, I would like to thank my friends at the Technische Universit¨
at Darmstadt,
who accompanied and encouraged me the entire time. They always had an open ear for
my troubles and assisted me both with mathematical and technical help.
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