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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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SOCIETY

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American Mathematical Society

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88

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DED 1

Editorial Board

Howard A. Masur Michael Renardy

Tudor Stefan Ratiu, Chair

1991 Mathematics Subject Classiļ¬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 diļ¬erential calculus in inļ¬nite dimensions

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

which do not involve Sobolev completions and ļ¬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 FrĀ“chet spaces this notion

e

of smoothness coincides with all known reasonable concepts. In the same spirit calculus of holo-

morphic mappings (including Hartogsā™ 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 inļ¬nite dimensions, the relation between tangent vectors

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

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

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

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

inļ¬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. ā” (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 53)

Includes bibliographical references (p. ā“ ) and index.

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

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

ematical surveys and monographs ; no. 53.

QA614.K75 1997

514 .74ā”dc21 97-25857

CIP

Copying and reprinting. Individual readers of this publication, and nonproļ¬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.

ā 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 cā -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. Diļ¬erentiability of Finite Order . . . . . . . . . . . . . . . . . . 118

13. Diļ¬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. Whitneyā™s Extension Theorem Revisited . . . . . . . . . . . . . . 226

23. FrĀØlicher 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

Inļ¬nite Dimensional Manifolds . . . . . . . . . . . . . . . . . . . . 263

27. Diļ¬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 Inļ¬nite Dimensional Manifolds . . . . . . . . . . . . . . 321

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

33. Diļ¬erential Forms . . . . . . . . . . ............. . 336

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

35. Derivations on Diļ¬erential Forms and the FrĀØlicher-Nijenhuis Bracket

o . 358

Chapter VIII

Inļ¬nite Dimensional Diļ¬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. Diļ¬eomorphism Groups . . . . . . . . . . . . . . . . . . . . . 454

44. Principal Bundles with Structure Group a Diļ¬eomorphism Group . . . 474

45. Manifolds of Riemannian Metrics . . . . . . . . . . . . . . . . . 487

46. The Korteweg ā“ 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], inļ¬nite dimensional manifolds were mentioned explicitly:

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

Zahl, sondern entweder eine unendliche Reihe oder eine stetige Mannigfaltigkeit von GrĀØs-

o

senbestimmungen erfordert. Solche Mannigfaltigkeiten bilden z.B. die mĀØglichen Bestim-

o

mungen einer Function fĀØr ein gegebenes Gebiet, die mĀØglichen Gestalten einer rĀØumlichen

u o a

Figur u.s.w.ā

The purpose of this book is to lay the foundations of inļ¬nite dimensional diļ¬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 diļ¬erential equations (even

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

eļ¬ectively on a ļ¬nite dimensional compact smooth manifold it must be ļ¬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 inļ¬nite dimensions. Diļ¬erential calculus in inļ¬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 diļ¬erent approaches and deļ¬nitions were attempted. The main diļ¬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 Ć— E ā— ā’ R, where E is a locally convex space and

E ā— 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 ā E and V ā E ā— of zero such that ev(U Ć— V ) ā [ā’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 diļ¬culty described here was the original motivation for the development of

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

no longer necessary to delve into this, because [FrĀØlicher, 1981] and [Kriegl, 1982],

o

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

diļ¬erential calculus in inļ¬nite dimensions, see the monograph [FrĀØlicher, 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 ā (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 diļ¬erence quotients converge to the derivative much better than arbitrary

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

weak completeness condition is satisļ¬ed: it appeared as ā˜bornologically completeā™

or ā˜locally completeā™ in the literature; we call it cā -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 cā -complete. Note that the locally convex topology on a

convenient vector space can vary in some range ā“ 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 ā“ existence, smoothness, and linearity of derivatives,

the chain rule, and also the most important feature, cartesian closedness

C ā (E Ć— F, G) ā¼ C ā (E, C ā (F, G))

(1) =

holds without any restriction, for a natural convenient vector space structure on

C ā (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 Ć—E ā— ā’ 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 inļ¬nite dimensions. We will, however, introduce the cā -

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

questions.

Introduction 3

An eminent mathematician once said that for inļ¬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 modiļ¬ed iteration procedure to converge. Many authors

try to build their platonic idea of an a priori estimate into their diļ¬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 diļ¬erential equation of geometric origin) should be treated

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

sense the setting presented here (and the setting in [FrĀØlicher, 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 diļ¬erentiability of ļ¬nite order, based

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

of inļ¬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 FrĀØlicher 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 inļ¬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 diļ¬er, and there are some surprises even on Hilbert spaces, see (28.4).

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

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

is proved, and a (small) version of the FrĀØlicher-Nijenhuis bracket in inļ¬nite di-

o

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

functors of manifolds) as generalized tangent bundles. The theory of inļ¬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 ā˜evolution operatorā™ 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 ļ¬‚at

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

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

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

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

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

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

with structure group a diļ¬eomorphism group. The ļ¬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 diļ¬eomorphism

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

of a ļ¬ber bundle with compact ļ¬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 Kortewegā“De VrieĆ 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 ā

or the Grassmannian G(k, ā) and shows that they are real analytic regular Lie

groups or associated homogeneous spaces. This put some constructions of alge-

braic topology directly into diļ¬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) ā“ (49.5) and their real analytic analogues (49.8) ā“ (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 deļ¬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 deļ¬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

Result if we include them without proofs. Appendix (52) collects some background

material from functional analysis in compressed form, and appendix (53) contains

a tool for analyzing non-separable Banach spaces which is used in sections (16) and

(19). A list of symbols has been worked into the index.

Reading map for the cross reader. Most of chapter I is essential. Chapter II

is for readers who also want to know the holomorphic and real analytic calculus,

others may leave it for a second reading. Chapters IIIā“V treat special material

which can be looked up later whenever properties like smooth partitions of unity

in inļ¬nite dimensions are asked for. In chapter VI section (35) can be skipped,

in chapter VII one may omit some proofs in sections (33) and (35). Chapter VIII

contains Lie theory and bundle theory, and is necessary for chapter IX and parts

of chapter X.

Thanks. The work on this book was done from 1989 onwards, most of the material

was presented in our joint seminar and elsewhere several times, which led to a lot of

improvement. We want to thank all participants, who devoted a lot of attention and

energy, in particular our (former) students who presented talks on that subject, also

those who helped with proofreading or gave good advise: Eva Adam, Dmitri Alek-

seevsky, Andreas Cap, Stefan Haller, Ann and Bertram Kostant, Grigori Litvinov,

Mark Losik, Josef Mattes, Martin Neuwirther, Tudor Ratiu, Konstanze Rietsch,

Hermann Schichl, Erhard Siegl, Josef Teichmann, Klaus Wegenkittl. The second

author acknowledges the support of ā˜Fonds zur FĀØrderung der wissenschaftlichen

o

Forschung, Projekt P 10037 PHYā™.

6

7

Chapter I

Calculus of Smooth Mappings

1. Smooth Curves . . . . . . . . . . . . . . . .. . . . . . . . ..8

2. Completeness . . . . . . . . . . . . . . . . .. . . . . . . . . 14

3. Smooth Mappings and the Exponential Law . . . .. . . . . . . . . 22

4. The cā -Topology . . . . . . . . . . . . . . .. . . . . . . . . 34

5. Uniform Boundedness Principles and Multilinearity . . . . . . . . . 52

6. Some Spaces of Smooth Functions . . . . . . . .. . . . . . . . . 66

Historical Remarks on Smooth Calculus . . . . . . .. . . . . . . . . 73

This chapter is devoted to calculus of smooth mappings in inļ¬nite dimensions. The

leading idea of our approach is to base everything on smooth curves in locally

convex spaces, which is a notion without problems, and a mapping between locally

convex spaces will be called smooth if it maps smooth curves to smooth curves.

We start by looking at the set of smooth curves C ā (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, its bornology. This is due to the fact, that

for a smooth curve diļ¬erence quotients converge to the derivative much better (2.1)

than arbitrary converging nets or ļ¬lters: we may multiply it by some unbounded

sequences of scalars without disturbing convergence (or, even better, boundedness).

Then the basic results are proved, like existence, smoothness, and linearity of deriva-

tives, the chain rule (3.18), and also the most important feature, the ā˜exponential

lawā™ (3.12) and (3.13): We have

C ā (E Ć— F, G) ā¼ C ā (E, C ā (F, G)),

=

without any restriction, for a natural structure on C ā (F, G).

Smooth curves have integrals in E if and only if a weak completeness condition

is satisļ¬ed: it appeared as bornological completeness, Mackey completeness, or

local completeness in the literature, we call it cā -complete. This is equivalent to

the condition that weakly smooth curves are smooth (2.14). All calculus in later

chapters in this book will be done on convenient vector spaces: These are locally

convex vector spaces which are cā -complete; note that the locally convex topology

on a convenient vector space can vary in some range, only the system of bounded

sets must remain the same.

Linear or more generally multilinear mappings are smooth if and only if they are

bounded (5.5), and one has corresponding exponential laws (5.2) for them as well.

8 Chapter I. Calculus of smooth mappings 1.2

Furthermore, there is an appropriate tensor product, the bornological tensor prod-

uct (5.7), satisfying

L(E ā—Ī² F, G) ā¼ L(E, F ; G) ā¼ L(E, L(F, G)).

= =

An important tool for convenient vector spaces are uniform boundedness principles

as given in (5.18), (5.24) and (5.26).

It is very natural to consider on E the ļ¬nal topology with respect to all smooth

curves, which we call the cā -topology, since all smooth mappings are continuous

for it: the vector space E, equipped with this topology is denoted by cā E, with

lower case c in analogy to kE for the Kelley-ļ¬cation and in order to avoid any

confusion with any space of smooth functions or sections. The special curve lemma

(2.8) shows that the cā -topology coincides with the usual Mackey closure topology.

The space cā E is not a topological vector space in general. This is related to the

fact that the evaluation E Ć— E ā’ R is jointly continuous only for normable E, but

it is always smooth and hence continuous on cā (E Ć— E ). The cā -open subsets are

the natural domains of deļ¬nitions of locally deļ¬ned functions. For nice spaces (e.g.

FrĀ“chet and strong duals of FrĀ“chet-Schwartz spaces, see (4.11)) the cā -topology

e e

coincides with the given locally convex topology. In general, the cā -topology is

ļ¬ner than any locally convex topology with the same bounded sets.

In the last section of this chapter we discuss the structure of spaces of smooth

functions on ļ¬nite dimensional manifolds and, more generally, of smooth sections

of ļ¬nite dimensional vector bundles. They will become important in chapter IX as

modeling spaces for manifolds of mappings. Furthermore, we give a short account

of reļ¬‚exivity of convenient vector spaces and on (various) approximation properties

for them.

1. Smooth Curves

1.1. Notation. Since we want to have unique derivatives all locally convex spaces

E will be assumed Hausdorļ¬. The family of all bounded sets in E plays an im-

portant rĖle. It is called the bornology of E. A linear mapping is called bounded,

o

sometimes also called bornological, if it maps bounded sets to bounded sets. A

bounded linear bijection with bounded inverse is called bornological isomorphism.

The space of all continuous linear functionals on E will be denoted by E ā— and the

space of all bounded linear functionals on E by E . The adjoint or dual mapping

of a linear mapping , however, will be always denoted by ā— , because of diļ¬erenti-

ation.

See also the appendix (52) for some background on functional analysis.

1.2. Diļ¬erentiable curves. The concept of a smooth curve with values in a

locally convex vector space is easy and without problems. Let E be a locally

convex vector space. A curve c : R ā’ E is called diļ¬erentiable if the derivative

1.2

1.3 1. Smooth curves 9

c (t) := limsā’0 1 (c(t + s) ā’ c(t)) at t exists for all t. A curve c : R ā’ E is called

s

ā

smooth or C if all iterated derivatives exist. It is called C n for some ļ¬nite n if its

iterated derivatives up to order n exist and are continuous.

Likewise, a mapping f : Rn ā’ E is called smooth if all iterated partial derivatives

ā‚ ā‚

ā‚i1 ,...,ip f := ā‚xi1 . . . ā‚xip f exist for all i1 , . . . , ip ā {1, . . . , n}.

A curve c : R ā’ E is called locally Lipschitzian if every point r ā R has a neigh-

borhood U such that the Lipschitz condition is satisļ¬ed on U , i.e., the set

1

c(t) ā’ c(s) : t = s; t, s ā U

tā’s

is bounded. Note that this implies that the curve satisļ¬es the Lipschitz condition

on each bounded interval, since for (ti ) increasing

c(tn ) ā’ c(t0 ) ti+1 ā’ ti c(ti+1 ) ā’ c(ti )

=

tn ā’ t0 tn ā’ t0 ti+1 ā’ ti

is in the absolutely convex hull of a ļ¬nite union of bounded sets.

A curve c : R ā’ E is called Lipk or C (k+1)ā’ if all derivatives up to order k exist and

are locally Lipschitzian. For these properties we have the following implications:

C n+1 =ā’ Lipn =ā’ C n ,

diļ¬erentiable =ā’ C.

In fact, continuity of the derivative implies locally its boundedness, and since this

can be tested by continuous linear functionals (see (52.19)) we conclude from the

one dimensional mean value theorem the boundedness of the diļ¬erence quotient.

See also the lemma (1.3) below.

1.3. Lemma. Continuous linear mappings are smooth. A continuous linear

mapping : E ā’ F between locally convex vector spaces maps Lipk -curves in E to

Lipk -curves in F , for all 0 ā¤ k ā¤ ā, and for k > 0 one has ( ā—¦ c) (t) = (c (t)).

Proof. As a linear map commutes with diļ¬erence quotients, hence the image of

a Lipschitz curve is Lipschitz since is bounded. As a continuous map it commutes

with the formation of the respective limits. Hence ( ā—¦ c) (t) = (c (t)).

Note that a diļ¬erentiable curve is continuous, and that a continuously diļ¬erentiable

curve is locally Lipschitz: For ā E ā— we have

1

c(t) ā’ c(s) ( ā—¦ c)(t) ā’ ( ā—¦ c)(s)

( ā—¦ c) (s + (t ā’ s)r)dr,

= =

tā’s tā’s 0

which is bounded, since ( ā—¦ c) is locally bounded.

Now the rest follows by induction.

1.3

10 Chapter I. Calculus of smooth mappings 1.4

1.4. The mean value theorem. In classical analysis the basic tool for using

the derivative to get statements on the original curve is the mean value theorem.

So we try to generalize it to inļ¬nite dimensions. For this let c : R ā’ E be a

diļ¬erentiable curve. If E = R the classical mean value theorem states, that the

diļ¬erence quotient (c(a)ā’c(b))/(aā’b) equals some intermediate value of c . Already

if E is two dimensional this is no longer true. Take for example a parameterization

of the circle by arclength. However, we will show that (c(a) ā’ c(b))/(a ā’ b) lies

still in the closed convex hull of {c (r) : r}. Having weakened the conclusion, we

can try to weaken the assumption. And in fact c may be not diļ¬erentiable in at

most countably many points. Recall however, that there exist strictly monotone

functions f : R ā’ R, which have vanishing derivative outside a Cantor set (which

is uncountable, but has still measure 0).

Sometimes one uses in one dimensional analysis a generalized version of the mean

value theorem: For an additional diļ¬erentiable function h with non-vanishing deriv-

ative the quotient (c(a)ā’c(b))/(h(a)ā’h(b)) equals some intermediate value of c /h .

A version for vector valued c (for real valued h) is that (c(a) ā’ c(b))/(h(a) ā’ h(b))

lies in the closed convex hull of {c (r)/h (r) : r}. One can replace the assumption

that h vanishes nowhere by the assumption that h has constant sign, or, more gen-

erally, that h is monotone. But then we cannot form the quotients, so we should

assume that c (t) ā h (t) Ā· A, where A is some closed convex set, and we should be

able to conclude that c(b) ā’ c(a) ā (h(b) ā’ h(a)) Ā· A. This is the version of the mean

value theorem that we are going to prove now. However, we will make use of it only

in the case where h = Id and c is everywhere diļ¬erentiable in the interior.

Proposition. Mean value theorem. Let c : [a, b] =: I ā’ E be a continuous

curve, which is diļ¬erentiable except at points in a countable subset D ā I. Let h

be a continuous monotone function h : I ā’ R, which is diļ¬erentiable on I \ D. Let

A be a convex closed subset of E, such that c (t) ā h (t) Ā· A for all t ā D.

/

Then c(b) ā’ c(a) ā (h(b) ā’ h(a)) Ā· A.

Proof. Assume that this is not the case. By the theorem of Hahn Banach (52.16)

there exists a continuous linear functional with (c(b)ā’c(a)) ā ((h(b) ā’ h(a)) Ā· A).

/

But then ā—¦ c and (A) satisfy the same assumptions as c and A, and hence we

may assume that c is real valued and A is just a closed interval [Ī±, Ī²]. We may

furthermore assume that h is monotonely increasing. Then h (t) ā„ 0, and h(b) ā’

h(a) ā„ 0. Thus the assumption says that Ī±h (t) ā¤ c (t) ā¤ Ī²h (t), and we want to

conclude that Ī±(h(b) ā’ h(a)) ā¤ c(b) ā’ c(a) ā¤ Ī²(h(b) ā’ h(a)). If we replace c by

c ā’ Ī²h or by Ī±h ā’ c it is enough to show that c (t) ā¤ 0 implies that c(b) ā’ c(a) ā¤ 0.

For given Īµ > 0 we will show that c(b) ā’ c(a) ā¤ Īµ(b ā’ a + 1). For this let J be

the set {t ā [a, b] : c(s) ā’ c(a) ā¤ Īµ ((s ā’ a) + tn <s 2ā’n ) for a ā¤ s < t}, where

D =: {tn : n ā N}. Obviously, J is a closed interval containing a, say [a, b ]. By

continuity of c we obtain that c(b ) ā’ c(a) ā¤ Īµ ((b ā’ a) + tn <b 2ā’n ). Suppose

b < b. If b ā D, then there exists a subinterval [b , b + Ī“] of [a, b] such that for

/

b ā¤ s < b + Ī“ we have c(s) ā’ c(b ) ā’ c (b )(s ā’ b ) ā¤ Īµ(s ā’ b ). Hence we get

c(s) ā’ c(b ) ā¤ c (b )(s ā’ b ) + Īµ(s ā’ b ) ā¤ Īµ(s ā’ b ),

1.4

1.5 1. Smooth curves 11

and consequently

c(s) ā’ c(a) ā¤ c(s) ā’ c(b ) + c(b ) ā’ c(a)

2ā’n ā¤ Īµ s ā’ a + 2ā’n .

ā¤ Īµ(s ā’ b ) + Īµ b ā’ a +

tn <s

tn <b

On the other hand if b ā D, i.e., b = tm for some m, then by continuity of c we

can ļ¬nd an interval [b , b + Ī“] contained in [a, b] such that for all b ā¤ s < b + Ī“ we

have

c(s) ā’ c(b ) ā¤ Īµ2ā’m .

Again we deduce that

c(s) ā’ c(a) ā¤ Īµ2ā’m + Īµ b ā’ a + 2ā’n ā¤ Īµ s ā’ a + 2ā’n .

tn <s

tn <b

So we reach in both cases a contradiction to the maximality of b .

Warning: One cannot drop the monotonicity assumption. In fact take h(t) := t2 ,

c(t) := t3 and [a, b] = [ā’1, 1]. Then c (t) ā h (t)[ā’2, 2], but c(1) ā’ c(ā’1) = 2 ā

/

{0} = (h(1) ā’ h(ā’1))[ā’2, 2].

1.5. Testing with functionals. Recall that in classical analysis vector valued

curves c : R ā’ Rn are often treated by considering their components ck := prk ā—¦c,

where prk : Rn ā’ R denotes the canonical projection onto the k-th factor R. Since

in general locally convex spaces do not have appropriate bases, we use all continuous

linear functionals instead of the projections prk . We will say that a property of a

curve c : R ā’ E is scalarly true, if ā—¦ c : R ā’ E ā’ R has this property for all

continuous linear functionals on E.

We want to compare scalar diļ¬erentiability with diļ¬erentiability. For ļ¬nite dimen-

sional spaces we know the trivial fact that these to notions coincide. For inļ¬nite

dimensions we ļ¬rst consider Lip-curves c : R ā’ E. Since by (52.19) boundedness

can be tested by the continuous linear functionals we see, that c is Lip if and only if

ā—¦c : R ā’ R is Lip for all ā E ā— . Moreover, if for a bounded interval J ā‚ R we take

B as the absolutely convex hull of the bounded set c(J)āŖ{ c(t)ā’c(s) : t = s; t, s ā J},

tā’s

then we see that c|J : J ā’ EB is a well deļ¬ned Lip-curve into EB . We de-

note by EB the linear span of B in E, equipped with the Minkowski functional

pB (v) := inf{Ī» > 0 : v ā Ī».B}. This is a normed space. Thus we have the following

equivalent characterizations of Lip-curves:

(1) locally c factors over a Lip-curve into some EB ;

(2) c is Lip;

(3) ā—¦ c is Lip for all ā E ā— .

For continuous instead of Lipschitz curves we obviously have the analogous impli-

cations (1 ā’ 2 ā’ 3). However, if we take a non-convergent sequence (xn )n , which

converges weakly (e.g. take an orthonormal base in a separable Hilbert space), and

1

consider an inļ¬nite polygon c through these points xn , say with c( n ) = xn and

1.5

12 Chapter I. Calculus of smooth mappings 1.6

c(0) = 0. Then this curve is obviously not continuous but ā—¦ c is continuous for all

ā Eā—.

Furthermore, the āworstā continuous curve - i.e. c : R ā’ C(R,R) R =: E given

by (c(t))f := f (t) for all t ā R and f ā C(R, R) - cannot be factored locally as

a continuous curve over some EB . Otherwise, c(tn ) would converge into some EB

1

to c(0), where tn is a given sequence converging to 0, say tn := n . So c(tn ) would

converge Mackey to c(0), i.e., there have to be Āµn ā’ ā with {Āµn (c(tn ) ā’ c(0)) : n ā

N} bounded in E. Since a set is bounded in the product if and only if its coordinates

are bounded, we conclude that for all f ā C(R, R) the sequence Āµn (f (tn ) ā’ f (0))

has to be bounded. But we can choose a continuous function f with f (0) = 0 and

ā

f (tn ) = ā1 n and conclude that Āµn (f (tn ) ā’ f (0)) = Āµn is unbounded.

Āµ

Similarly, one shows that the reverse implications do not hold for diļ¬erentiable

curves, for C 1 -curves and for C n -curves. However, if we put instead some Lip-

schitz condition on the derivatives, there should be some chance, since this is a

bornological concept. In order to obtain this result, we should study convergence

of sequences in EB .

1.6. Lemma. Mackey-convergence. Let B be a bounded and absolutely convex

subset of E and let (xĪ³ )Ī³āĪ“ be a net in EB . Then the following two conditions are

equivalent:

(1) xĪ³ converges to 0 in the normed space EB ;

(2) There exists a net ĀµĪ³ ā’ 0 in R, such that xĪ³ ā ĀµĪ³ Ā· B.

In (2) we may assume that ĀµĪ³ ā„ 0 and is decreasing with respect to Ī³, at least for

large Ī³. In the particular case of a sequence (or where we have a coļ¬nal countable

subset of Ī“) we can choose all Āµn > 0 and hence we may divide.

A net (xĪ³ ) for which a bounded absolutely convex B ā E exists, such that xĪ³

converges to x in EB is called Mackey convergent to x or short M -convergent.

Proof. (ā‘) Let xĪ³ = ĀµĪ³ Ā·bĪ³ with bĪ³ ā B and ĀµĪ³ ā’ 0. Then pB (xĪ³ ) = |ĀµĪ³ | pB (bĪ³ ) ā¤

|ĀµĪ³ | ā’ 0, i.e. xĪ³ ā’ x in EB .

x

(ā“) Let Ī“ > 1, and set ĀµĪ³ := Ī“ pB (xĪ³ ). By assumption, ĀµĪ³ ā’ 0 and xĪ³ = ĀµĪ³ ĀµĪ³ , Ī³

xĪ³ xĪ³ xĪ³

1

where ĀµĪ³ := 0 if ĀµĪ³ = 0. Since pB ( ĀµĪ³ ) = Ī“ < 1 or is 0, we conclude that ĀµĪ³ ā B.

For the ļ¬nal assertions, choose Ī³1 such that |ĀµĪ³ | ā¤ 1 for Ī³ ā„ Ī³1 , and for those Ī³ we

replace ĀµĪ³ by sup{|ĀµĪ³ | : Ī³ ā„ Ī³}. Thus we may choose Āµ ā„ 0 and decreasing with

respect to Ī³.

If we have a sequence (Ī³n )nāN which is coļ¬nal in Ī“, i.e. for every Ī³ ā Ī“ there exists

an n ā N with Ī³ ā¤ Ī³n , then we may replace ĀµĪ³ by

1

max({ĀµĪ³ } āŖ {ĀµĪ³m : Ī³m ā„ Ī³} āŖ { : Ī³m ā„ Ī³})

m

to conclude that ĀµĪ³ = 0 for all Ī³.

If Ī“ is the ordered set of all countable ordinals, then it is not possible to ļ¬nd a net

(ĀµĪ³ )Ī³āĪ“ , which is positive everywhere and converges to 0, since a converging net is

ļ¬nally constant.

1.6

1.8 1. Smooth curves 13

1.7. The diļ¬erence quotient converges Mackey. Now we show how to de-

scribe the quality of convergence of the diļ¬erence quotient.

Corollary. Let c : R ā’ E be a Lip1 -curve. Then the curve t ā’ 1 ( 1 (c(t) ā’ c(0)) ā’

tt

c (0)) is bounded on bounded subsets of R \ {0}.

Proof. We apply (1.4) to c and obtain:

c(t) ā’ c(0)

ā’ c (0) ā c (r) : 0 < |r| < |t| ā’ c (0)

t closed, convex

= c (r) ā’ c (0) : 0 < |r| < |t|

closed, convex

c (r) ā’ c (0)

: 0 < |r| < |t|

=r

r closed, convex

Let a > 0. Since { c (r)ā’c (0) : 0 < |r| < a} is bounded and hence contained in a

r

closed absolutely convex and bounded set B, we can conclude that

c(t) ā’ c(0) r c (r) ā’ c (0)

1

ā’ c (0) ā : 0 < |r| < |t| ā B.

t t t r closed, convex

1.8. Corollary. Smoothness of curves is a bornological concept. For

0 ā¤ k < ā a curve c in a locally convex vector space E is Lipk if and only if for

each bounded open interval J ā‚ R there exists an absolutely convex bounded set

B ā E such that c|J is a Lipk -curve in the normed space EB .

Attention: A smooth curve factors locally into some EB as a Lipk -curve for each

ļ¬nite k only, in general. Take the āworstā smooth curve c : R ā’ C ā (R,R) R,

analogously to (1.5), and, using Borelā™s theorem, deduce from c(k) (0) ā EB for all

k ā N a contradiction.

Proof. For k = 0 this was shown before. For k ā„ 1 take a closed absolutely convex

bounded set B ā E containing all derivatives c(i) on J up to order k as well as their

diļ¬erence quotients on {(t, s) : t = s, t, s ā J}. We show ļ¬rst that c is diļ¬erentiable,

say at 0, with derivative c (0). By the proof of the previous corollary (1.7) we have

that the expression 1 ( c(t)ā’c(0) ā’ c (0)) lies in B. So c(t)ā’c(0) ā’ c (0) converges to 0

t t t

in EB . For the higher order derivatives we can now proceed by induction.

The converse follows from lemma (1.3).

A consequence of this is, that smoothness does not depend on the topology but only

on the dual (so all topologies with the same dual have the same smooth curves), and

in fact it depends only on the bounded sets, i.e. the bornology. Since on L(E, F )

there is essentially only one bornology (by the uniform boundedness principle, see

(52.25)) there is only one notion of Lipn -curves into L(E, F ). Furthermore, the

class of Lipn -curves doesnā™t change if we pass from a given locally convex topology

to its bornologiļ¬cation, see (4.2), which by deļ¬nition is the ļ¬nest locally convex

topology having the same bounded sets.

Let us now return to scalar diļ¬erentiability. Corollary (1.7) gives us Lipn -ness

provided we have appropriate candidates for the derivatives.

1.8

14 Chapter I. Calculus of smooth mappings 2.1

1.9. Corollary. Scalar testing of curves. Let ck : R ā’ E for k < n + 1 be

curves such that ā—¦ c0 is Lipn and ( ā—¦ c0 )(k) = ā—¦ ck for all k < n + 1 and all

ā E ā— . Then c0 is Lipn and (c0 )(k) = ck .

Proof. For n = 0 this was shown in (1.5). For n ā„ 1, by (1.7) applied to ā—¦ c we

have that

1 c0 (t) ā’ c0 (0)

ā’ c1 (0)

t t

is locally bounded, and hence by (52.19) the set

c0 (t) ā’ c0 (0)

1

ā’ c1 (0) :tāI

t t

0 0

is bounded. Thus c (t)ā’c (0) converges even Mackey to c1 (0). Now the general

t

statement follows by induction.

2. Completeness

Do we really need the knowledge of a candidate for the derivative, as in (1.9)? In

ļ¬nite dimensional analysis one often uses the Cauchy condition to prove conver-

gence. Here we will replace the Cauchy condition again by a stronger condition,

which provides information about the quality of being Cauchy:

A net (xĪ³ )Ī³āĪ“ in E is called Mackey-Cauchy provided that there exist a bounded

(absolutely convex) set B and a net (ĀµĪ³,Ī³ )(Ī³,Ī³ )āĪ“Ć—Ī“ in R converging to 0, such

that xĪ³ ā’ xĪ³ ā ĀµĪ³,Ī³ B. As in (1.6) one shows that for a net xĪ³ in EB this is

equivalent to the condition that xĪ³ is Cauchy in the normed space EB .

2.1. Lemma. The diļ¬erence quotient is Mackey-Cauchy. Let c : R ā’ E be

scalarly a Lip1 -curve. Then t ā’ c(t)ā’c(0) is a Mackey-Cauchy net for t ā’ 0.

t

Proof. For Lip1 -curves this is a immediate consequence of (1.7) but we only as-

sume it to be scalarly Lip1 . It is enough to show that tā’s c(t)ā’c(0) ā’ c(s)ā’c(0) is

1

t s

bounded on bounded subsets in R \ {0}. We may test this with continuous linear

functionals, and hence may assume that E = R. Then by the fundamental theorem

of calculus we have

1

c(t) ā’ c(0) c(s) ā’ c(0) c (tr) ā’ c (sr)

1

ā’ = dr

tā’s tā’s

t s 0

1

c (tr) ā’ c (sr)

= r dr.

tr ā’ sr

0

Since c (tr)ā’c (sr) is locally bounded by assumption, the same is true for the integral,

trā’sr

and we are done.

2.1

2.4 2. Completeness 15

2.2. Lemma. Mackey Completeness. For a space E the following conditions

are equivalent:

(1) Every Mackey-Cauchy net converges in E;

(2) Every Mackey-Cauchy sequence converges in E;

(3) For every absolutely convex closed bounded set B the space EB is complete;

For every bounded set B there exists an absolutely convex bounded set B ā

(4)

B such that EB is complete.

A space satisfying the equivalent conditions is called Mackey complete. Note that a

sequentially complete space is Mackey complete.

Proof. (1) ā’ (2), and (3) ā’ (4) are trivial.

(2) ā’ (3) Since EB is normed, it is enough to show sequential completeness. So let

(xn ) be a Cauchy sequence in EB . Then (xn ) is Mackey-Cauchy in E and hence

converges in E to some point x. Since pB (xn ā’ xm ) ā’ 0 there exists for every

Īµ > 0 an N ā N such that for all n, m ā„ N we have pB (xn ā’ xm ) < Īµ, and hence

xn ā’ xm ā ĪµB. Taking the limit for m ā’ ā, and using closedness of B we conclude

that xn ā’ x ā ĪµB for all n > N . In particular x ā EB and xn ā’ x in EB .

(4) ā’ (1) Let (xĪ³ )Ī³āĪ“ be a Mackey-Cauchy net in E. So there is some net ĀµĪ³,Ī³ ā’ 0,

such that xĪ³ ā’ xĪ³ ā ĀµĪ³,Ī³ B for some bounded set B. Let Ī³0 be arbitrary. By (4)

we may assume that B is absolutely convex and contains xĪ³0 , and that EB is

complete. For Ī³ ā Ī“ we have that xĪ³ = xĪ³0 + xĪ³ ā’ xĪ³0 ā xĪ³0 + ĀµĪ³,Ī³0 B ā EB , and

pB (xĪ³ ā’ xĪ³ ) ā¤ ĀµĪ³,Ī³ ā’ 0. So (xĪ³ ) is a Cauchy net in EB , hence converges in EB ,

and thus also in E.

2.3. Corollary. Scalar testing of diļ¬erentiable curves. Let E be Mackey

complete and c : R ā’ E be a curve for which ā—¦ c is Lipn for all ā E ā— . Then c

is Lipn .

Proof. For n = 0 this was shown in (1.5) without using any completeness, so let

n ā„ 1. Since we have shown in (2.1) that the diļ¬erence quotient is a Mackey-Cauchy

net we conclude that the derivative c exists, and hence ( ā—¦ c) = ā—¦ c . So we may

apply the induction hypothesis to conclude that c is Lipnā’1 , and consequently c is

Lipn .

Next we turn to integration. For continuous curves c : [0, 1] ā’ E one can show

completely analogously to 1-dimensional analysis that the Riemann sums R(c, Z, Ī¾),

deļ¬ned by k (tk ā’ tkā’1 )c(Ī¾k ), where 0 = t0 < t1 < Ā· Ā· Ā· < tn = 1 is a partition

Z of [0, 1] and Ī¾k ā [tkā’1 , tk ], form a Cauchy net with respect to the partial strict

ordering given by the size of the mesh max{|tk ā’ tkā’1 | : 0 < k < n}. So under

the assumption of sequential completeness we have a Riemann integral of curves.

A second way to see this is the following reduction to the 1-dimensional case.

2.4. Lemma. Let L(Eequi , R) be the space of all linear functionals on E ā— which are

ā—

bounded on equicontinuous sets, equipped with the complete locally convex topology

2.4

16 Chapter I. Calculus of smooth mappings 2.5

of uniform convergence on these sets. There is a natural topological embedding

ā—

Ī“ : E ā’ L(Eequi , R) given by Ī“(x)( ) := (x).

Proof. Let U be a basis of absolutely convex closed 0-neighborhoods in E. Then

the family of polars U o := { ā E ā— : | (x)| ā¤ 1 for all x ā U }, with U ā U form a

basis for the equicontinuous sets, and hence the bipolars U oo := { ā— ā L(Eequi , R) :

ā—

| ā— ( )| ā¤ 1 for all ā U o } form a basis of 0-neighborhoods in L(Eequi , R). By the

ā—

bipolar theorem (52.18) we have U = Ī“ ā’1 (U oo ) for all U ā U. This shows that Ī“ is

a homeomorphism onto its image.

2.5. Lemma. Integral of continuous curves. Let c : R ā’ E be a continuous

curve in a locally convex vector space. Then there is a unique diļ¬erentiable curve

c : R ā’ E in the completion E of E such that ( c)(0) = 0 and ( c) = c.

Proof. We show uniqueness ļ¬rst. Let c1 : R ā’ E be a curve with derivative c and

c1 (0) = 0. For every ā E ā— the composite ā—¦ c1 is an anti-derivative of ā—¦ c with

initial value 0, so it is uniquely determined, and since E ā— separates points c1 is also

uniquely determined.

Now we show the existence. By the previous lemma we have that E is (isomorphic

ā—

to) the closure of E in the obviously complete space L(Eequi , R). We deļ¬ne ( c)(t) :

t

E ā— ā’ R by ā’ 0 ( ā—¦ c)(s)ds. It is a bounded linear functional on Eequi since for

ā—

an equicontinuous subset E ā E ā— the set {( ā—¦ c)(s) : ā E, s ā [0, t]} is bounded.

ā—

So c : R ā’ L(Eequi , R).

c is diļ¬erentiable with derivative Ī“ ā—¦ c.

Now we show that

( c)(t + r) ā’ ( c)(r)

ā’ (Ī“ ā—¦ c)(r) ( ) =

t

t+r r

1

( ā—¦ c)(s)ds ā’ ( ā—¦ c)(s)ds ā’ t( ā—¦ c)(r)

= =

t 0 0

r+t 1

1

( ā—¦ c)(s) ā’ ( ā—¦ c)(r) ds = c(r + ts) ā’ c(r) ds.

=

t r 0

Let E ā E ā— be equicontinuous, and let Īµ > 0. Then there exists a neighborhood U

of 0 such that | (U )| < Īµ for all ā E. For suļ¬ciently small t, all s ā [0, 1] and ļ¬xed

1

r we have c(r + ts) ā’ c(r) ā U . So | 0 (c(r + ts) ā’ c(r))ds| < Īµ. This shows that

the diļ¬erence quotient of c at r converges to Ī“(c(r)) uniformly on equicontinuous

subsets.

It remains to show that ( c)(t) ā E. By the mean value theorem (1.4) the diļ¬erence

ā—

quotient 1 (( c)(t) ā’ ( c)(0)) is contained in the closed convex hull in L(Eequi , R)

t

of the subset {c(s) : 0 < s < t} of E. So it lies in E.

Deļ¬nition of the integral. For continuous curves c : R ā’ E the deļ¬nite integral

b b

c ā E is given by a c = ( c)(b) ā’ ( c)(a).

a

2.5

2.8 2. Completeness 17

2.6. Corollary. Basics on the integral. For a continuous curve c : R ā’ E we

have:

b b

ā Eā—.

(1) ( a c) = a ( ā—¦ c) for all

b d d

(2) a c + b c = a c.

b Ļ•(b)

c for Ļ• ā C 1 (R, R).

(c ā—¦ Ļ•)Ļ•

(3) =

a Ļ•(a)

b

(4) c lies in

the closed convex hull in E of the set

a

{(b ā’ a)c(t) : a < t < b} in E.

b

(5) a : C(R, E) ā’ E is linear.

(6) (Fundamental theorem of calculus.) For each C 1 -curve c : R ā’ E we have

s

c(s) ā’ c(t) = t c .

We are mainly interested in smooth curves and we can test for this by applying linear

functionals if the space is Mackey complete, see (2.3). So let us try to show that

the integral for such curves lies in E if E is Mackey-complete. So let c : [0, 1] ā’ E

be a smooth or just a Lip-curve, and take a partition Z with mesh Āµ(Z) at most

Ī“. If we have a second partition, then we can take the common reļ¬nement. Let

[a, b] be one interval of the original partition with intermediate point t, and let

a = t0 < t1 < Ā· Ā· Ā· < tn = b be the reļ¬nement. Note that |b ā’ a| ā¤ Ī“ and hence

|t ā’ tk | ā¤ Ī“. Then we can estimate as follows.

(b ā’ a) c(t) ā’ (tk ā’ tkā’1 )c(tk ) = (tk ā’ tkā’1 ) (c(t) ā’ c(tk )) = Āµk bk ,

k k k

c(t)ā’c(tk )

where bk := is contained in the absolutely convex Lipschitz bound

Ī“

c(t) ā’ c(s)

: t, s ā [0, 1]

B :=

tā’s abs.conv

of c and Āµk := (tk ā’tkā’1 )Ī“ ā„ 0 and satisļ¬es k Āµk = (bā’a)Ī“. Hence we have for the

Riemann sums with respect to the original partition Z1 and the reļ¬nement Z that

R(c, Z1 ) ā’ R(c, Z ) lies in Ī“ Ā· B. So R(c, Z1 ) ā’ R(c, Z2 ) ā 2Ī“B for any two partitions

Z1 and Z2 of mesh at most Ī“, i.e. the Riemann sums form a Mackey-Cauchy net

with coeļ¬cients ĀµZ1 ,Z2 := 2 max{Āµ(Z1 ), Āµ(Z2 )} and we have proved:

2.7. Proposition. Integral of Lipschitz curves. Let c : [0, 1] ā’ E be a

Lipschitz curve into a Mackey complete space. Then the Riemann integral exists in

E as (Mackey)-limit of the Riemann sums.

2.8. Now we have to discuss the relationship between diļ¬erentiable curves and

Mackey convergent sequences. Recall that a sequence (xn ) converges if and only if

there exists a continuous curve c (e.g. a reparameterization of the inļ¬nite polygon)

and tn 0 with c(tn ) = xn . The corresponding result for smooth curves uses the

following notion.

Deļ¬nition. We say that a sequence xn in a locally convex space E converges fast

to x in E, or falls fast towards x, if for each k ā N the sequence nk (xn ā’ x) is

bounded.

2.8

18 Chapter I. Calculus of smooth mappings 2.10

Special curve lemma. Let xn be a sequence which converges fast to x in E.

Then the inļ¬nite polygon through the xn can be parameterized as a smooth curve

1

c : R ā’ E such that c( n ) = xn and c(0) = x.

Proof. Let Ļ• : R ā’ [0, 1] be a smooth map, which is 0 on {t : t ā¤ 0} and 1 on

{t : t ā„ 1}. The parameterization c is deļ¬ned as follows:

for t ā¤ 0,

ļ£“x

ļ£±

1

ļ£²

tā’ 1 1

c(t) := xn+1 + Ļ• 1 ā’n+1 (xn ā’ xn+1 ) for n+1 ā¤ t ā¤ n , .

1

n n+1

ļ£“

ļ£³

for t ā„ 1

x1

1 1

Obviously, c is smooth on R \ {0}, and the p-th derivative of c for ā¤tā¤ is

n+1 n

given by

1

t ā’ n+1

c(p) (t) = Ļ•(p) 1 (n(n + 1))p (xn ā’ xn+1 ).

1

n ā’ n+1

Since xn converges fast to x, we have that c(p) (t) ā’ 0 for t ā’ 0, because the ļ¬rst

factor is bounded and the second goes to zero. Hence c is smooth on R, by the

following lemma.

2.9. Lemma. Diļ¬erentiable extension to an isolated point. Let c : R ā’ E

be continuous and diļ¬erentiable on R \ {0}, and assume that the derivative c :

R \ {0} ā’ E has a continuous extension to R. Then c is diļ¬erentiable at 0 and

c (0) = limtā’0 c (t).

Proof. Let a := limtā’0 c (t). By the mean value theorem (1.4) we have c(t)ā’c(0) ā t

c (s) : 0 = |s| ā¤ |t| closed, convex . Since c is assumed to be continuously extendable

to 0 we have that for any closed convex 0-neighborhood U there exists a Ī“ > 0 such

that c (t) ā a + U for all 0 < |t| ā¤ Ī“. Hence c(t)ā’c(0) ā’ a ā U , i.e. c (0) = a.

t

The next result shows that we can pass through certain sequences xn ā’ x even

with given velocities vn ā’ 0.

2.10. Corollary. If xn ā’ x fast and vn ā’ 0 fast in E, then there are smoothly

parameterized polygon c : R ā’ E and tn ā’ 0 in R such that c(tn + t) = xn + tvn

for t in a neighborhood of 0 depending on n.

Proof. Consider the sequence yn deļ¬ned by

1 1

y2n+1 := xn ā’

y2n := xn + 4n(2n+1) vn and 4n(2n+1) vn .

It is easy to show that yn converges fast to x, and the parameterization c of the

polygon through the yn (using a function Ļ• which satisļ¬es Ļ•(t) = t for t near 1/2)

has the claimed properties, where

1 1 1

4n+1

tn := = + .

4n(2n+1) 2 2n 2n + 1

As ļ¬rst application (2.10) we can give the following sharpening of (1.3).

2.10

2.13 2. Completeness 19

2.11. Corollary. Bounded linear maps. A linear mapping : E ā’ F between

locally convex vector spaces is bounded (or bornological), i.e. it maps bounded sets

to bounded ones, if and only if it maps smooth curves in E to smooth curves in F .

Proof. As in the proof of (1.3) one shows using (1.7) that a bounded linear map

preserves Lipk -curves. Conversely, assume that a linear map : E ā’ F carries

smooth curves to locally bounded curves. Take a bounded set B, and assume that

f (B) is unbounded. Then there is a sequence (bn ) in B and some Ī» ā F such

that |(Ī» ā—¦ )(bn )| ā„ nn+1 . The sequence (nā’n bn ) converges fast to 0, hence lies on

some compact part of a smooth curve by (2.8). Consequently, (Ī» ā—¦ )(nā’n bn ) =

nā’n (Ī» ā—¦ )(bn ) is bounded, a contradiction.

2.12. Deļ¬nition. The cā -topology on a locally convex space E is the ļ¬nal topol-

ogy with respect to all smooth curves R ā’ E. Its open sets will be called cā -open.

We will treat this topology in more detail in section (4): In general it describes

neither a topological vector space (4.20) and (4.26), nor a uniform structure (4.27).

However, by (4.4) and (4.6) the ļ¬nest locally convex topology coarser than the

cā -topology is the bornologiļ¬cation of the locally convex topology.

Let (Āµn ) be a sequence of real numbers converging to ā. Then a sequence (xn ) in

E is called Āµ-converging to x if the sequence (Āµn (xn ā’ x)) is bounded in E.

2.13. Theorem. cā -open subsets. Let Āµn ā’ ā be a real valued sequence.

Then a subset U ā E is open for the cā -topology if it satisļ¬es any of the following

equivalent conditions:

(1) All inverse images under Lipk -curves are open in R (for ļ¬xed k ā Nā ).

(2) All inverse images under Āµ-converging sequences are open in Nā .

(3) The traces to EB are open in EB for all absolutely convex bounded subsets

B ā E.

Note that for closed subsets an equivalent statement reads as follows: A set A is cā -

closed if and only if for every sequence xn ā A, which is Āµ-converging (respectively

M -converging, resp. fast falling) towards x, the point x belongs to A.

The topology described in (2) is also called Mackey-closure topology. It is not the

Mackey topology discussed in duality theory.

Proof. (1) ā’ (2) Suppose (xn ) is Āµ-converging to x ā U , but xn ā U for inļ¬nitely

/

many n. Then we may choose a subsequence again denoted by (xn ), which is fast

falling to x, hence lies on some compact part of a smooth curve c as described in

1

(2.8). Then c( n ) = xn ā U but c(0) = x ā U . This is a contradiction.

/

(2) ā’ (3) A sequence (xn ), which converges in EB to x with respect to pB , is Mackey

convergent, hence has a Āµ-converging subsequence. Note that EB is normed, and

hence it is enough to consider sequences.

(3) ā’ (2) Suppose (xn ) is Āµ-converging to x. Then the absolutely convex hull B of

{Āµn (xn ā’ x) : n ā N} āŖ {x} is bounded, and xn ā’ x in (EB , pB ), since Āµn (xn ā’ x)

is bounded.

2.13

20 Chapter I. Calculus of smooth mappings 2.14

(2) ā’ (1) Use that for a converging sequence of parameters tn the images xn := c(tn )

under a Lip-curve c are Mackey converging.

Let us show next that the cā -topology and cā -completeness are intimately related.

2.14. Theorem. Convenient vector spaces. Let E be a locally convex vector

space. E is said to be cā -complete or convenient if one of the following equivalent

(completeness) conditions is satisļ¬ed:

(1) Any Lipschitz curve in E is locally Riemann integrable.

(2) For any c1 ā C ā (R, E) there is c2 ā C ā (R, E) with c2 = c1 (existence of

an anti-derivative).

(3) E is cā -closed in any locally convex space.

(4) If c : R ā’ E is a curve such that ā—¦ c : R ā’ R is smooth for all ā E ā— ,

then c is smooth.

(5) Any Mackey-Cauchy sequence converges; i.e. E is Mackey complete, see

(2.2).

(6) If B is bounded closed absolutely convex, then EB is a Banach space. This

property is called locally complete in [Jarchow, 1981, p196].

(7) Any continuous linear mapping from a normed space into E has a continu-

ous extension to the completion of the normed space.

Condition (4) says that in a convenient vector space one can recognize smooth

curves by investigating compositions with continuous linear functionals. Condition

(5) says via (2.2.4) that cā -completeness is a bornological concept. In [FrĀØlicher,

o

Kriegl, 1988] a convenient vector space is always considered with its bornological

topology ā” an equivalent but not isomorphic category.

Proof. In (2.3) we showed (5) ā’ (4), in (2.7) we got (5) ā’ (1), and in (2.2) we

had (5) ā’ (6).

(1) ā’ (2) A smooth curve is Lipschitz, thus locally Riemann integrable. The

indeļ¬nite Riemann integral equals the āweakly deļ¬nedā integral of lemma (2.5),

hence is an anti-derivative.

(2) ā’ (3) Let E be a topological linear subspace of F . To show that E is cā -

closed we use (2.13). Let xn ā’ xā be fast falling, xn ā E but xā ā F . By

(2.8) the polygon c through (xn ) can be smoothly parameterized. Hence c is

smooth and has values in the vector space generated by {xn : n = ā}, which is

contained in E. Its anti-derivative c2 is up to a constant equal to c, and by (2)

x1 ā’ xā = c(1) ā’ c(0) = c2 (1) ā’ c2 (0) lies in E. So xā ā E.

(3) ā’ (5) Let F be the completion E of E. Any Mackey Cauchy sequence in E

has a limit in F , and since E is by assumption cā -closed in F the limit lies in E.

Hence, the sequence converges in E.

(6) ā’ (7) Let f : F ā’ E be a continuous mapping on a normed space F . Since the

image of the unit ball is bounded, it is a bounded mapping into EB for some closed

absolutely convex B. But into EB it can be extended to the completion, since EB

is complete.

2.14

2.15 2. Completeness 21

(7) ā’ (1) Let c : R ā’ E be a Lipschitz curve. Then c is locally a continuous curve

into EB for some absolutely convex bounded set B. The inclusion of EB into E

has a continuous extension to the completion of EB , and c is Riemann integrable

in this Banach space, so also in E.

(4) ā’ (3) Let E be embedded in some space F . We use again (2.13) in order to

show that E is cā -closed in F . So let xn ā’ xā fast falling, xn ā E for n = 0,

but xā ā F . By (2.8) the polygon c through (xn ) can be smoothly symmetrically

parameterized in F , and c(t) ā E for t = 0. We consider c(t) := tc(t). This is a

Ė

curve in E which is smooth in F , so it is scalarwise smooth in E, thus smooth in

E by (4). Then xā = c (0) ā E.

Ė

2.15. Theorem. Inheritance of cā -completeness. The following construc-

tions preserve cā -completeness: limits, direct sums, strict inductive limits of se-

quences of closed embeddings, as well as formation of ā (X, ), where X is a set

together with a family B of subsets of X containing the ļ¬nite ones, which are called

bounded and ā (X, F ) denotes the space of all functions f : X ā’ F , which are

bounded on all B ā B, supplied with the topology of uniform convergence on the

sets in B.

Note that the deļ¬nition of the topology of uniform convergence as initial topology

shows, that adding all subsets of ļ¬nite unions of elements in B to B does not change

this topology. Hence, we may always assume that B has this stability property; this

is the concept of a bornology on a set.

Proof. The projective limit (52.8) of F is the cā -closed linear subspace

(xĪ± ) ā F(Ī±) : F(f )xĪ± = xĪ² for all f : Ī± ā’ Ī² ,

hence is cā -complete, since the product of cā -complete factors is obviously cā -

complete.

Since the coproduct (52.7) of spaces XĪ± is the topological direct sum, and has as

bounded sets those which are contained and bounded in some ļ¬nite subproduct, it

is cā -complete if all factors are.

For colimits this is in general not true. For strict inductive limits of sequences of

closed embeddings it is true, since bounded sets are contained and bounded in some

step, see (52.8).

For the result on ā (X, F ) we consider ļ¬rst the case, where X itself is bounded.

Then cā -completeness can be proved as in (52.4) or reduced to this result. In fact

let B be bounded in ā (X, F ). Then B(X) is bounded in F and hence contained

in some absolutely convex bounded set B, for which FB is a Banach space. So we

may assume that B := {f ā ā (X, F ) : f (X) ā B}. The space ā (X, F )B is just

the space ā (X, FB ) with the supremum norm, which is a Banach space by (52.4).

ā

(X, F ) ā’ ā (B, F )

Let now X and B be arbitrary. Then the restriction maps

give an embedding Ī¹ of ā (X, F ) into the product BāB ā

(B, F ). Since this

2.15

22 Chapter I. Calculus of smooth mappings 3.2

product is complete, by what we have shown above, it is enough to show that this

embedding has a closed image. So let fĪ± |B converge to some fB in ā (B, F ).

Deļ¬ne f (x) := f{x} (x). For any B ā B containing x we have that fB (x) =

(limĪ± fĪ± |B )(x) = limĪ± (fĪ± (x)) = limĪ± fĪ± |{x} = f{x} (x) = f (x), and f (B) is boun-

ded for all B ā B, since f |B = fB ā ā (B, F ).

Example. In general, a quotient and an inductive limit of cā -complete spaces

need not be cā -complete. In fact, let ED := {x ā RN : supp x ā D} for any

subset D ā N of density dens D := lim sup{ |Dā©[1,n]| } = 0. It can be shown that

n

E := dens D=0 ED ā‚ R is the inductive limit of the FrĀ“chet subspaces ED ā¼ RD .

N

e =

ā

It cannot be c -complete, since ļ¬nite sequences are contained in E and are dense

in RN ā E.

3. Smooth Mappings and the Exponential Law

Now let us start proving the exponential law C ā (U Ć— V, F ) ā¼ C ā (U, C ā (V, F ))

=

ļ¬rst for U = V = F = R.

3.1. Proposition. For a continuous map f : R Ć— [0, 1] ā’ R the partial derivative

ā‚1 f exists and is continuous if and only if f āØ : R ā’ C([0, 1], R) is continuously

1 1ā‚

diļ¬erentiable. And in this situation I((f āØ ) (t)) = dt 0 f (t, s) ds = 0 ā‚t f (t, s) ds,

d

where I : C([0, 1], R) ā’ R is integration.

Proof. We assume that ā‚1 f exists and is continuous. Hence, (ā‚1 f )āØ : R ā’

C([0, 1], R) is continuous. We want to show that f āØ : R ā’ C([0, 1], R) is dif-

ferentiable (say at 0) with this function (at 0) as derivative. So we have to show

āØ āØ

that the mapping t ā’ f (t)ā’f (0) is continuously extendable to R by deļ¬ning its

t

āØ

value at 0 as (ā‚1 f ) (0). Or equivalently, by what is obvious for continuous maps,

that the map

f (t,s)ā’f (0,s)

for t = 0

t

(t, s) ā’

ā‚1 f (0, s) otherwise

is continuous. This follows immediately from the continuity of ā‚1 f and of integra-

1

tion since it can be written as 0 ā‚1 f (r t, s) dr by the fundamental theorem.

1

So we arrive under this assumption at the conclusion, that f (t, s) ds is diļ¬eren-

0

tiable with derivative

1 1

d ā‚

āØ

f (t, s) ds = I((f ) (t)) = f (t, s) ds.

dt ā‚t

0 0

The converse implication is obvious.

3.2. Theorem. Simplest case of exponential law. Let f : R2 ā’ R be an

arbitrary mapping. Then all iterated partial derivatives exist and are locally bounded

if and only if the associated mapping f āØ : R ā’ C ā (R, R) exists as a smooth curve,

3.2

3.2 3. Smooth mappings and the exponential law 23

where C ā (R, R) is considered as the FrĀ“chet space with the topology of uniform

e

convergence of each derivative on compact sets. Furthermore, we have (ā‚1 f )āØ =

d(f āØ ) and (ā‚2 f )āØ = d ā—¦ f āØ = dā— (f āØ ).

Proof. We have several possibilities to prove this result. Either we show Mackey

convergence of the diļ¬erence quotients, via the boundedness of 1 c(t)ā’c(0) ā’ c (0) ,

t t

ā¼ ā (X, ā (Y, R)); or we use

ā

(X Ć—Y, R) =

and then use the trivial exponential law

the induction step proved in (3.1), namely that f āØ : R ā’ C(R, R) is diļ¬erentiable

if and only if ā‚1 f exists and is continuous R2 ā’ R, together with the exponential

law C(R2 , R) ā¼ C(R, C(R, R)). We choose the latter method.

=

For this we have to note ļ¬rst that if for a function g the partial derivatives ā‚1 g and

ā‚2 g exist and are locally bounded then g is continuous:

g(x, y) ā’ g(0, 0) = g(x, y) ā’ g(x, 0) + g(x, 0) ā’ g(0, 0)

= yā‚2 g(x, r2 y) + xā‚1 g(r1 x, 0)

for suitable r1 , r2 ā [0, 1], which goes to 0 with (x, y).

Proof of (ā’) By what we just said, all iterated partial derivatives of f are contin-

uous. First observe that f āØ : R ā’ C ā (R, R) makes sense and that for all t ā R we

have

q

dq (f āØ (t)) = (ā‚2 f )āØ (t).

(1)

Next we claim that f āØ : R ā’ C ā (R, R) is diļ¬erentiable, with derivative d(f āØ ) =

(ā‚1 f )āØ , or equivalently that for all a the curve

f āØ (t+a)ā’f āØ (a)

for t = 0

t

c:tā’ āØ

(ā‚1 f ) (a) otherwise

is continuous as curve R ā’ C ā (R, R). Without loss of generality we may assume

that a = 0. Since C ā (R, R) carries the initial structure with respect to the linear

mappings dp : C ā (R, R) ā’ C(R, R) we have to show that dp ā—¦ c : R ā’ C(R, R)

is continuous, or equivalently by the exponential law for continuous maps, that

(dp ā—¦ c)ā§ : R2 ā’ R is continuous. For t = 0 and s ā R we have

f āØ (t) ā’ f āØ (0)

ā§

p p p

(d ā—¦ c) (t, s) = d (c(t))(s) = d (s)

t

p p

ā‚2 f (t, s) ā’ ā‚2 f (0, s)

= by (1)

t

1

p

= ā‚1 ā‚2 f (t Ļ„, s) dĻ„ by the fundamental theorem.

0

For t = 0 we have

(dp ā—¦ c)ā§ (0, s) = dp (c(0))(s) = dp ((ā‚1 f )āØ (0))(s)

p

= (ā‚2 (ā‚1 f ))āØ (0)(s) by (1)

p

= ā‚2 ā‚1 f (0, s)

p

= ā‚1 ā‚2 f (0, s) by the theorem of Schwarz.

3.2

24 Chapter I. Calculus of smooth mappings 3.3

1 p

So we see that (dp ā—¦ c)ā§ (t, s) = 0 ā‚1 ā‚2 f (t Ļ„, s) dĻ„ for all (t, s). This function is con-

p p

tinuous in (t, s), since ā‚1 ā‚2 f : R2 ā’ R is continuous, hence (t, s, Ļ„ ) ā’ ā‚1 ā‚2 f (t Ļ„, s)

p

is continuous, and therefore also (t, s) ā’ (Ļ„ ā’ ā‚1 ā‚2 f (t Ļ„, s)) from R2 ā’ C([0, 1], R).

1

Composition with the continuous linear mapping 0 : C([0, 1], R) ā’ R gives the

continuity of (dp ā—¦ c)ā§ .

Now we proceed by induction. By the induction hypothesis applied to ā‚1 f , we

obtain that d(f āØ ) = (ā‚1 f )āØ and (ā‚1 f )āØ : R ā’ C ā (R, R) is n times diļ¬erentiable,

so f āØ is (n + 1)-times diļ¬erentiable.

Proof of (ā) First remark that for a smooth map f : R ā’ C ā (R, R) the asso-

ciated map f ā§ : R2 ā’ R is locally bounded: Since f is smooth f (I1 ) is com-

pact, hence bounded in C ā (R, R) for all compact intervals I1 . In particular,

f (I1 )(I2 ) = f ā§ (I1 Ć— I2 ) has to be bounded in R for all compact intervals I1 and I2 .

Since f is smooth both curves df and d ā—¦ f = dā— f are smooth (use (1.3) and that

d is continuous and linear). An easy calculation shows that the partial derivatives

of f ā§ exist and are given by ā‚1 f ā§ = (df )ā§ and ā‚2 f ā§ = (d ā—¦ f )ā§ . So one obtains

inductively that all iterated derivatives of f ā§ exist and are locally bounded, since

they are associated to smooth curves R ā’ C ā (R, R).

In order to proceed to more general cases of the exponential law we need a deļ¬nition

of C ā -maps deļ¬ned on inļ¬nite dimensional spaces. This deļ¬nition should at least

guarantee the chain rule, and so one could take the weakest notion that satisļ¬es

the chain rule. However, consider the following

3.3. Example. We consider the following 3-fold āsingular coveringā f : R2 ā’ R2

given in polar coordinates by (r, Ļ•) ā’ (r, 3Ļ•). In cartesian coordinates we obtain

the following formula for the values of f :

(r cos(3Ļ•), r sin(3Ļ•)) = r (cos Ļ•)3 ā’ 3 cos Ļ•(sin Ļ•)2 , 3 sin Ļ•(cos Ļ•)2 ā’ (sin Ļ•)3

x3 ā’ 3xy 2 3x2 y ā’ y 3

= ,2 .

x2 + y 2 x + y2

Note that the composite from the left with any orthonormal projection is just the

composite of the ļ¬rst component of f with a rotation from the right (Use that f

intertwines the rotation with angle Ī“ and the rotation with angle 3Ī“).

Obviously, the map f is smooth on R2 \ {0}. It is homogeneous of degree 1, and

ā‚

hence the directional derivative is f (0)(v) = ā‚t |t=0 f (tv) = f (v). However, both

components are nonlinear with respect to v and thus are not diļ¬erentiable at (0, 0).

Obviously, f : R2 ā’ R2 is continuous.

We claim that f is diļ¬erentiable along diļ¬erentiable curves, i.e. (f ā—¦ c) (0) exists,

provided c (0) exists.

Only the case c(0) = 0 is not trivial. Since c is diļ¬erentiable at 0 the curve c1

deļ¬ned by c1 (t) := c(t) for t = 0 and c (0) for t = 0 is continuous at 0. Hence

t

f (c(t))ā’f (c(0)) f (t c1 (t))ā’0

= = f (c1 (t)). This converges to f (c1 (0)), since f is con-

t t

tinuous.

3.3

3.3 3. Smooth mappings and the exponential law 25

Furthermore, if f (x)(v) denotes the directional derivative, which exists everywhere,

then (f ā—¦ c) (t) = f (c(t))(c (t)). Indeed for c(t) = 0 this is clear and for c(t) = 0 it

follows from f (0)(v) = f (v).

The directional derivative of the 1-homogeneous mapping f is 0-homogeneous: In

fact, for s = 0 we have

ā‚ ā‚ t 1

f (sx)(v) = f (s x + tv) = s f (x + v) = s f (x)( v) = f (x)(v).

ā‚t ā‚t s s

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