Table Of ContentMathematical
Surveys
and
Monographs
Volume 53
The Convenient
Setting of
Global Analysis
Andreas Kriegl
Peter W.Michor
ERMIACAN MΑΓΕΩΜΕAΤTΡHΗEΤΟMΣ AΜTΗICΕΙΣΙΤΩALSOYCTIE American Mathematical Society
FOUNDED 1888
Editorial Board
HowardA. 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.Theaimofthisbookistolayfoundationsofdifferentialcalculusininfinitedimensions
andtodiscussthoseapplicationsininfinitedimensionaldifferentialgeometryandglobalanalysis
which do not involve Sobolev completions 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,
powerfulsmoothuniformlyboundednesstheoremsareavailable. UptoFr´echetspacesthisnotion
ofsmoothnesscoincides withallknownreasonableconcepts. Inthe samespirit calculus ofholo-
morphicmappings(includingHartogs’theoremandholomorphicuniformboundednesstheorems)
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
andderivations,anddifferentialformsarediscussedthoroughly. Specialemphasisisgiventothe
notionofregularinfinite dimensionalLie groups. Many applications ofthis theoryareincluded:
manifolds of smooth mappings, groups of diffeomorphisms, geodesics on spaces of Riemannian
metrics,directlimitmanifolds,perturbationtheoryofoperators,anddifferentiabilityquestionsof
infinitedimensionalrepresentations.
Corrections and complements to this book will be posted on the internet at the URL
http://www.mat.univie.ac.at/~michor/apbook.ps
Library of Congress Cataloging-in-Publication Data
Kriegl,Andreas.
Theconvenientsettingofglobalanalysis/AndreasKriegl,PeterW.Michor.
p.cm. —(Mathematicalsurveysandmonographs,ISSN0076-5376;v.53)
Includesbibliographicalreferences(p. – )andindex.
ISBN0-8218-0780-3(alk.paper)
1.Globalanalysis(Mathematics) I.Michor,PeterW.,1949–. II.Title. III.Series: Math-
ematicalsurveysandmonographs;no.53.
QA614.K75 1997
514(cid:1).74—dc21 97-25857
CIP
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(including abstracts) is permitted only under license from the American Mathematical Society.
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To Elli, who made working on this
book into a culinary experience.
iv
v
Table of Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Chapter I
Calculus of Smooth Mappings . . . . . . . . . . . . . . . . . . . . . 7
1. Smooth Curves . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2. Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3. Smooth Mappings and the Exponential Law . . . . . . . . . . . . . 22
∞
4. The c -Topology . . . . . . . . . . . . . . . . . . . . . . . . 34
5. Uniform Boundedness Principles and Multilinearity . . . . . . . . . 52
6. Some Spaces of Smooth Functions . . . . . . . . . . . . . . . . . 66
Historical Remarks on Smooth Calculus . . . . . . . . . . . . . . . . 73
Chapter II
Calculus of Holomorphic and Real Analytic Mappings . . . . . . . . . 79
7. Calculus of Holomorphic Mappings . . . . . . . . . . . . . . . . . 80
8. Spaces of Holomorphic Mappings and Germs . . . . . . . . . . . . 91
9. Real Analytic Curves . . . . . . . . . . . . . . . . . . . . . . . 97
10. Real Analytic Mappings . . . . . . . . . . . . . . . . . . . . . 101
11. The Real Analytic Exponential Law . . . . . . . . . . . . . . . . 105
Historical Remarks on Holomorphic and Real Analytic Calculus . . . . . 116
Chapter III
Partitions of Unity . . . . . . . . . . . . . . . . . . . . . . . . . 117
12. Differentiability of Finite Order . . . . . . . . . . . . . . . . . . 118
13. Differentiability of Seminorms . . . . . . . . . . . . . . . . . . 127
14. Smooth Bump Functions . . . . . . . . . . . . . . . . . . . . . 152
15. Functions with Globally Bounded Derivatives . . . . . . . . . . . 159
16. Smooth Partitions of Unity and Smooth Normality . . . . . . . . . 165
Chapter IV
Smoothly Realcompact Spaces . . . . . . . . . . . . . . . . . . . . 183
17. Basic Concepts and Topological Realcompactness . . . . . . . . . . 184
18. Evaluation Properties of Homomorphisms . . . . . . . . . . . . . 188
19. Stability of Smoothly Realcompact Spaces . . . . . . . . . . . . . 203
20. Sets on which all Functions are Bounded . . . . . . . . . . . . . . 217
Chapter V
Extensions and Liftings of Mappings . . . . . . . . . . . . . . . . . 219
21. Extension and Lifting Properties . . . . . . . . . . . . . . . . . 220
22. Whitney’s Extension Theorem Revisited . . . . . . . . . . . . . . 226
23. Fro¨licher Spaces and Free Convenient Vector Spaces . . . . . . . . . 238
24. Smooth Mappings on Non-Open Domains . . . . . . . . . . . . . 247
25. Real Analytic Mappings on Non-Open Domains . . . . . . . . . . 254
26. Holomorphic Mappings on Non-Open Domains . . . . . . . . . . . 261
vi
Chapter VI
Infinite Dimensional Manifolds . . . . . . . . . . . . . . . . . . . . 263
27. Differentiable Manifolds . . . . . . . . . . . . . . . . . . . . . 264
28. Tangent Vectors . . . . . . . . . . . . . . . . . . . . . . . . 276
29. Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . 287
30. Spaces of Sections of Vector Bundles . . . . . . . . . . . . . . . 293
31. Product Preserving Functors on Manifolds . . . . . . . . . . . . . 305
Chapter VII
Calculus on Infinite Dimensional Manifolds . . . . . . . . . . . . . . 321
32. Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 321
33. Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . 336
34. De Rham Cohomology . . . . . . . . . . . . . . . . . . . . . . 353
35. Derivations on Differential Forms and the Fro¨licher-Nijenhuis Bracket . 358
Chapter VIII
Infinite Dimensional Differential Geometry . . . . . . . . . . . . . . . 369
36. Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 369
37. Bundles and Connections . . . . . . . . . . . . . . . . . . . . . 375
38. Regular Lie Groups . . . . . . . . . . . . . . . . . . . . . . . 404
39. Bundles with Regular Structure Groups . . . . . . . . . . . . . . 422
40. Rudiments of Lie Theory for Regular Lie Groups . . . . . . . . . . 426
Chapter IX
Manifolds of Mappings . . . . . . . . . . . . . . . . . . . . . . . 429
41. Jets and Whitney Topologies . . . . . . . . . . . . . . . . . . . 431
42. Manifolds of Mappings . . . . . . . . . . . . . . . . . . . . . 439
43. Diffeomorphism Groups . . . . . . . . . . . . . . . . . . . . . 454
44. Principal Bundles with Structure Group a Diffeomorphism Group . . . 474
45. Manifolds of Riemannian Metrics . . . . . . . . . . . . . . . . . 487
46. The Korteweg – De Vries Equation as a Geodesic Equation . . . . . 498
Complements to Manifolds of Mappings . . . . . . . . . . . . . . . . 510
Chapter X
Further Applications . . . . . . . . . . . . . . . . . . . . . . . . 511
47. Manifolds for Algebraic Topology . . . . . . . . . . . . . . . . . 512
48. Weak Symplectic Manifolds . . . . . . . . . . . . . . . . . . . 522
49. Applications to Representations of Lie Groups . . . . . . . . . . . 528
50. Applications to Perturbation Theory of Operators . . . . . . . . . 536
51. The Nash-Moser Inverse Function Theorem . . . . . . . . . . . . 553
52. Appendix: Functional Analysis . . . . . . . . . . . . . . . . . . 575
53. Appendix: Projective Resolutions of Identity on Banach spaces . . . . 582
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611
1
Introduction
At the very conception of the notion of manifolds, in the Habilitationsschrift [Rie-
mann, 1854], infinite dimensional manifolds were mentioned explicitly:
“EsgiebtindessauchMannigfaltigkeiten,inwelchendieOrtsbestimmungnichteineendliche
Zahl, sondern entweder eine unendliche Reihe oder eine stetige Mannigfaltigkeit von Gr¨os-
senbestimmungen erfordert. Solche Mannigfaltigkeiten bilden z.B. die m¨oglichen Bestim-
mungen einer Function fu¨r ein gegebenes Gebiet, die m¨oglichen Gestalten einer r¨aumlichen
Figur u.s.w.”
The purpose of this book is to lay the foundationsof infinite dimensionaldifferential
geometry. The book [Palais, 1968] and review article [Eells, 1966] have similar titles
and treat global analysis mainly on manifolds modeled on Banach spaces. Indeed
classical calculus works quite well up to and including Banach spaces: Existence
and uniqueness hold for solutions of smooth ordinary differential equations (even
Lipschitz ones), but not existence for all continuous ordinary differential equations.
The inverse function theorem works well, but the theorem of constant rank presents
problems, and the implicit function theorem requires additional assumptions about
existence of complementary subspaces. There are also problems with partitions of
unity, with the Whitney extension theorem, and with Morse theory and transver-
sality.
Further development has shown that Banach manifolds are not suitable for many
questions of Global Analysis, as shown by the following result, which is due to
[Omori, de la Harpe, 1972], see also [Omori, 1978b]: If a Banach Lie group acts
effectively on a finite dimensional compact smooth manifold it must be finite di-
mensional itself. The study of Banach manifolds per se is not very interesting, since
they turn out to be open subsets of the modeling space for many modeling spaces,
see [Eells, Elworthy, 1970].
Our aim in this book is to treat manifolds which are modeled on locally convex
spaces, and which are smooth, holomorphic, or real analytic in an appropriate
sense. To do this we start with a careful exposition of smooth, holomorphic, and
real analytic calculus in infinite dimensions. Differential calculus in infinite dimen-
sions has already quite a long history; in fact it goes back to Bernoulli and Euler,
to the beginnings of variational calculus. During the 20-th century the urge to dif-
ferentiate in spaces which are more general than Banach spaces became stronger,
and many different approaches and definitions were attempted. The main difficulty
encountered was that composition of (continuous) linear mappings ceases to be a
jointly continuous operation exactly at the level of Banach spaces, for any suitable
topology on spaces of linear mappings. This can easily be explained in a somewhat
simpler example:
2 Introduction
Consider the evaluation ev : E ×E∗ → R, where E is a locally convex space and
∗
E is its dual of continuous linear functionals equipped with any locally convex
topology. Let us assume that the evaluation is jointly continuous. Then there are
neighborhoods U ⊆ E and V ⊆ E∗ of zero such that ev(U×V) ⊆ [−1,1]. But then
U is contained in the polar of V, so it is bounded in E, and so E admits a bounded
neighborhood and is thus normable.
The difficulty described here was the original motivation for the development of
a whole new field within general topology, convergence spaces. Fortunately it is
no longer necessary to delve into this, because [Fr¨olicher, 1981] and [Kriegl, 1982],
[Kriegl, 1983] presented independently the solution to the question for the right
differential calculus in infinite dimensions, see the monograph [Fr¨olicher, Kriegl,
1988]. The smooth calculus which we present here is the same as in this book, but
our exposition is based on functional analysis rather than on category theory.
Let us try to describe the basic ideas of smooth calculus: One can say that it is
a (more or less unique) consequence of taking variational calculus seriously. We
start by looking at the space of smooth curves C∞(R,E) with values in a locally
convex space E and note that it does not depend on the topology of E, only on
the underlying system of bounded sets. This is due to the fact, that for a smooth
curve difference quotients converge to the derivative much better than arbitrary
converging nets or filters. Smooth curves have integrals in E if and only if a
weak completeness condition is satisfied: it appeared as ‘bornologically complete’
∞
or ‘locally complete’ in the literature; we call it c -complete. Surprisingly, this is
equivalent to the condition that scalarwise smooth curves are smooth. All calculus
in this book will be done on convenient vector spaces. These are locally convex
∞
vector spaces which are c -complete. Note that the locally convex topology on a
convenient vector space can vary in some range – only the system of bounded set
mustremainthesame. Thenextstepsaretheneasy: amappingbetweenconvenient
vector spaces is called smooth if it maps smooth curves to smooth curves, and
everything else is a theorem – existence, smoothness, and linearity of derivatives,
the chain rule, and also the most important feature, cartesian closedness
(1) C∞(E ×F,G) ∼= C∞(E,C∞(F,G))
holds without any restriction, for a natural convenient vector space structure on
∞
C (F,G): So the old dream of variational calculus becomes true in a concise way.
If one wants (1) and some other mild properties of calculus, then smooth calculus
as described here is unique. Let us point out that on some convenient vector spaces
therearesmoothfunctionswhicharenotcontinuousforthelocallyconvextopology.
This is not so horrible as it sounds, and is unavoidable if we want the chain rule,
sinceev : E×E∗ → RisalwayssmoothbutcontinuousonlyifE isnormable, bythe
discussion above. This just tells us that locally convex topology is not appropriate
∞
for non-linear questions in infinite dimensions. We will, however, introduce the c -
topology on any convenient vector space, which survives as the fittest for non-linear
questions.
Introduction 3
An eminent mathematician once said that for infinite dimensional calculus each
seriousapplicationneedsitsownfoundation. Byaseriousapplicationoneobviously
means some application of a hard inverse function theorem. These theorems can
be proved, if by assuming enough a priori estimates one creates enough Banach
space situation for some modified iteration procedure to converge. Many authors
try to build their platonic idea of an a priori estimate into their differential calculus.
We think that this makes the calculus inapplicable and hides the origin of the a
priori estimates. We believe that the calculus itself should be as easy to use as
possible, and that all further assumptions (which most often come from ellipticity
ofsome nonlinearpartialdifferentialequationofgeometric origin)shouldbe treated
separately, in a setting depending on the specific problem. We are sure that in this
sense the setting presented here (and the setting in [Fr¨olicher, Kriegl, 1988]) is
useful for most applications. To give a basis to this statement we present also the
hard implicit function theorem of Nash and Moser, in the approach of [Hamilton,
1982] adapted to convenient calculus, but we give none of its serious applications.
A surprising and very satisfying feature of the notion of convenient vector spaces
is that it is also the right setting for holomorphic calculus as shown in [Kriegl, Nel,
1985], for real analytic calculus as shown by [Kriegl, Michor, 1990], and also for
multilinear algebra.
In chapter III we investigate the existence of smooth bump functions and smooth
partitions of unity. This is tied intimately to special properties of the locally convex
spaces in question. There is also a section on differentiability of finite order, based
on Lipschitz conditions, whereas the rest of the book is devoted to differentiability
of infinite order. Chapter IV answers the question whether real valued algebra
homomorphisms on algebras of smooth functions are point evaluations. Germs,
extension results like (22.17), and liftings are the topic of chapter V. Here we also
treat Fr¨olicher spaces (i.e. spaces with a fairly general smooth structure) and free
convenient vector spaces over them.
Chapters VI to VIII are devoted to the theory of infinite dimensional manifolds and
Lie groups and some of their applications. We treat here only manifolds described
by charts although this limits cartesian closedness of the category of manifolds
drastically, see (42.14) and section (23) for more thorough discussions. Then we
investigate tangent vectors seen as derivations or kinematically (via curves): these
concepts differ, and there are some surprises even on Hilbert spaces, see (28.4).
Accordingly, we have different kinds of tangent bundles, vector fields, differential
forms, which we list in a somewhat systematic manner. The theorem of De Rham
is proved, and a (small) version of the Fro¨licher-Nijenhuis bracket in infinite di-
mensions is treated. Finally, we discuss Weil functors (certain product preserving
functors of manifolds) as generalized tangent bundles. The theory of infinite di-
mensional Lie groups can be pushed surprisingly far: Exponential mappings are
unique if they exist. A stronger requirement (leading to regular Lie groups) is that
one assumes that smooth curves in the Lie algebra integrate to smooth curves in
the group in a smooth way (an ‘evolution operator’ exists). This is due to [Milnor,
1984] who weakened the concept of [Omori, Maeda, Yoshioka, 1982]. It turns out
4 Introduction
that regular Lie groups have strong permanence properties. Up to now (April 1997)
no non-regular Lie group is known. Connections on smooth principal bundles with
a regular Lie group as structure group have parallel transport (39.1), and for flat
connections the horizontal distribution is integrable (39.2). So some (equivariant)
partial differential equations in infinite dimensions are very well behaved, although
in general there are counter-examples in every possible direction. As consequence
we obtain in (40.3) that a bounded homomorphism from the Lie algebra of simply
connected Lie group into the Lie algebra of a regular Lie group integrates to a
smooth homomorphism of Lie groups.
The rest of the book describes applications: In chapter IX we treat manifolds of
mappings between finite dimensional manifolds. We show that the group of all
diffeomorphisms of a finite dimensional manifold is a regular Lie group, also the
group of all real analytic diffeomorphisms, and some subgroups of diffeomorphism
groups, namely those consisting of symplectic diffeomorphisms, volume preserving
diffeomorphism, and contact diffeomorphisms. Then we treat principal bundles
with structure group a diffeomorphism group. The first example is the space of all
embeddings between two manifolds, a sort of nonlinear Grassmann manifold, which
leads to a smooth manifold which is a classifying space for the diffeomorphism
group of a compact manifold. Another example is the nonlinear frame bundle
of a fiber bundle with compact fiber, for which we investigate the action of the
gauge group on the space of generalized connections and show that there are no
slices. In section (45) we compute explicitly all geodesics for some natural (pseudo)
Riemannian metrics on the space of all Riemannian metrics. Section (46) is devoted
to the Korteweg–De Vrieß equation which is shown to be the geodesic equation of
a certain right invariant Riemannian metric on the Virasoro group.
∞
Chapter X start with section (47) on direct limit manifolds like the sphere S
or the Grassmannian G(k,∞) and shows that they are real analytic regular Lie
groups or associated homogeneous spaces. This put some constructions of alge-
braic topology directly into differential geometry. Section (48) is devoted to weak
symplectic manifolds (where the symplectic form is injective but not surjective as
a mapping from the tangent bundle into the cotangent bundle). Here we describe
precisely the space of smooth functions for which the Poisson bracket makes sense.
In section (49) on representation theory we show how easily the spaces of smooth
(real analytic) vectors can be treated with the help of the calculus developed in this
book. The results (49.3) – (49.5) and their real analytic analogues (49.8) – (49.10)
should convince the reader who has seen the classical proofs that convenient anal-
ysis is worthwhile to use. We included also some material on the moment mapping
for unitary representations. This mapping is defined on the space of smooth (real
analytic) vectors. Section (50) is devoted to the preparations and the proof of the-
orem (50.16) which says that a smooth curve of unbounded selfadjoint operators
on Hilbert space with compact resolvent admits smooth parameterizations of its
eigenvalues and eigenvectors, under some condition. The real analytic version of
this is due to [Rellich, 1940]; we also give a new and simpler proof of this result.
In our view, the best advantage of our approach is the natural and easy way to
