Surveys

and

Monographs

Volume 53

The Convenient

Setting of

Global Analysis

Andreas Kriegl

Peter W. Michor

HEMATIC

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AMERICAN

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American Mathematical Society

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Editorial Board

Howard A. Masur Michael Renardy

Tudor Stefan Ratiu, Chair

1991 Mathematics Subject Classi¬cation. 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 di¬erential calculus in in¬nite dimensions

and to discuss those applications in in¬nite dimensional di¬erential geometry and global analysis

which do not involve Sobolev completions and ¬xed 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´chet spaces this notion

e

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 in¬nite dimensions, the relation between tangent vectors

and derivations, and di¬erential forms are discussed thoroughly. Special emphasis is given to the

notion of regular in¬nite dimensional Lie groups. Many applications of this theory are included:

manifolds of smooth mappings, groups of di¬eomorphisms, geodesics on spaces of Riemannian

metrics, direct limit manifolds, perturbation theory of operators, and di¬erentiability questions of

in¬nite dimensional representations.

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.

The convenient setting of global analysis / Andreas Kriegl, Peter W. Michor.

p. cm. ” (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 53)

Includes bibliographical references (p. “ ) and index.

ISBN 0-8218-0780-3 (alk. paper)

1. Global analysis (Mathematics) I. Michor, Peter W., 1949“ . II. Title. III. Series: Math-

ematical surveys and monographs ; no. 53.

QA614.K75 1997

514 .74”dc21 97-25857

CIP

Copying and reprinting. Individual readers of this publication, and nonpro¬t libraries acting

for them, are permitted to make fair use of the material, such as to copy a chapter for use

in teaching or research. Permission is granted to quote brief passages from this publication in

reviews, provided the customary acknowledgment of the source is given.

Republication, systematic copying, or multiple reproduction of any material in this publication

(including abstracts) is permitted only under license from the American Mathematical Society.

Requests for such permission should be addressed to the Assistant to the Publisher, American

Mathematical Society, P. O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also

be made by e-mail to reprint-permission@ams.org.

c 1997 by the American Mathematical Society. All rights reserved.

The American Mathematical Society retains all rights

except those granted to the United States Government.

Printed in the United States of America.

∞ The paper used in this book is acid-free and falls within the guidelines

established to ensure permanence and durability.

Visit the AMS homepage at URL: http://www.ams.org/

10 9 8 7 6 5 4 3 2 1 02 01 00 99 98 97

iii

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. Di¬erentiability of Finite Order . . . . . . . . . . . . . . . . . . 118

13. Di¬erentiability 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. Fr¨licher Spaces and Free Convenient Vector Spaces

o . . . . . . . . . 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

In¬nite Dimensional Manifolds . . . . . . . . . . . . . . . . . . . . 263

27. Di¬erentiable 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 In¬nite Dimensional Manifolds . . . . . . . . . . . . . . 321

32. Vector Fields . . . . . . . . . . . . ............. . 321

33. Di¬erential Forms . . . . . . . . . . ............. . 336

34. De Rham Cohomology . . . . . . . . ............. . 353

35. Derivations on Di¬erential Forms and the Fr¨licher-Nijenhuis Bracket

o . 358

Chapter VIII

In¬nite Dimensional Di¬erential 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. Di¬eomorphism Groups . . . . . . . . . . . . . . . . . . . . . 454

44. Principal Bundles with Structure Group a Di¬eomorphism 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], in¬nite dimensional manifolds were mentioned explicitly:

“Es giebt indess auch Mannigfaltigkeiten, in welchen die Ortsbestimmung nicht eine endliche

Zahl, sondern entweder eine unendliche Reihe oder eine stetige Mannigfaltigkeit von Gr¨s-

o

senbestimmungen erfordert. Solche Mannigfaltigkeiten bilden z.B. die m¨glichen Bestim-

o

mungen einer Function f¨r ein gegebenes Gebiet, die m¨glichen Gestalten einer r¨umlichen

u o a

Figur u.s.w.”

The purpose of this book is to lay the foundations of in¬nite dimensional di¬erential

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 di¬erential equations (even

Lipschitz ones), but not existence for all continuous ordinary di¬erential 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

e¬ectively on a ¬nite dimensional compact smooth manifold it must be ¬nite 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 in¬nite dimensions. Di¬erential calculus in in¬nite 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 di¬erent approaches and de¬nitions were attempted. The main di¬culty

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 di¬culty described here was the original motivation for the development of

a whole new ¬eld within general topology, convergence spaces. Fortunately it is

no longer necessary to delve into this, because [Fr¨licher, 1981] and [Kriegl, 1982],

o

[Kriegl, 1983] presented independently the solution to the question for the right

di¬erential calculus in in¬nite dimensions, see the monograph [Fr¨licher, Kriegl,

o

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 di¬erence quotients converge to the derivative much better than arbitrary

converging nets or ¬lters. Smooth curves have integrals in E if and only if a

weak completeness condition is satis¬ed: 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

must remain the same. The next steps are then easy: a mapping between convenient

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

C ∞ (E — F, G) ∼ C ∞ (E, C ∞ (F, G))

(1) =

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

there are smooth functions which are not continuous for the locally convex topology.

This is not so horrible as it sounds, and is unavoidable if we want the chain rule,

since ev : E —E — ’ R is always smooth but continuous only if E is normable, by the

discussion above. This just tells us that locally convex topology is not appropriate

for non-linear questions in in¬nite dimensions. We will, however, introduce the c∞ -

topology on any convenient vector space, which survives as the ¬ttest for non-linear

questions.

Introduction 3

An eminent mathematician once said that for in¬nite dimensional calculus each

serious application needs its own foundation. By a serious application one obviously

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 modi¬ed iteration procedure to converge. Many authors

try to build their platonic idea of an a priori estimate into their di¬erential 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

of some nonlinear partial di¬erential equation of geometric origin) should be treated

separately, in a setting depending on the speci¬c problem. We are sure that in this

sense the setting presented here (and the setting in [Fr¨licher, Kriegl, 1988]) is

o

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 di¬erentiability of ¬nite order, based

on Lipschitz conditions, whereas the rest of the book is devoted to di¬erentiability

of in¬nite 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¨licher spaces (i.e. spaces with a fairly general smooth structure) and free

o

convenient vector spaces over them.

Chapters VI to VIII are devoted to the theory of in¬nite 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 di¬er, and there are some surprises even on Hilbert spaces, see (28.4).

Accordingly, we have di¬erent kinds of tangent bundles, vector ¬elds, di¬erential

forms, which we list in a somewhat systematic manner. The theorem of De Rham

is proved, and a (small) version of the Fr¨licher-Nijenhuis bracket in in¬nite di-

o

mensions is treated. Finally, we discuss Weil functors (certain product preserving

functors of manifolds) as generalized tangent bundles. The theory of in¬nite 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 ¬‚at

connections the horizontal distribution is integrable (39.2). So some (equivariant)

partial di¬erential equations in in¬nite 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 ¬nite dimensional manifolds. We show that the group of all

di¬eomorphisms of a ¬nite dimensional manifold is a regular Lie group, also the

group of all real analytic di¬eomorphisms, and some subgroups of di¬eomorphism

groups, namely those consisting of symplectic di¬eomorphisms, volume preserving

di¬eomorphism, and contact di¬eomorphisms. Then we treat principal bundles

with structure group a di¬eomorphism group. The ¬rst 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 di¬eomorphism

group of a compact manifold. Another example is the nonlinear frame bundle

of a ¬ber bundle with compact ¬ber, 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 di¬erential 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 de¬ned 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

Introduction 5

express what a smooth or real analytic curve of unbounded operators really is.

Hints for the reader. The numbering of subsections is done extensively and

consecutively, the number valid at the bottom of each page can be found in the

running head, opposite to the page number. Concepts which are not central are

usually de¬ned after the formulation of the result, before the proof, and sometimes

even in the proof. So please look ahead rather than behind (which is advisable in

everyday life also). Related materials from the literature are listed under the name