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Mathematical
Surveys
and
Monographs
Volume 53




The Convenient
Setting of
Global Analysis


Andreas Kriegl
Peter W. Michor




HEMATIC
AT A
M
L




¤Ρ—¤ΟΣ Μ—
AMERICAN




•™Σ™¤„¦

SOCIETY
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American Mathematical Society
FO
88
8
UN
DED 1
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



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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
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 III“V treat special material
which can be looked up later whenever properties like smooth partitions of unity
in in¬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 ˜Fonds zur F¨rderung der wissenschaftlichen
o
Forschung, Projekt P 10037 PHY™.
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 c∞ -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 in¬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 ∞ (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 di¬erence quotients converge to the derivative much better (2.1)
than arbitrary converging nets or ¬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 ˜exponential
law™ (3.12) and (3.13): We have

C ∞ (E — F, G) ∼ C ∞ (E, C ∞ (F, G)),
=

without any restriction, for a natural structure on C ∞ (F, G).
Smooth curves have integrals in E if and only if a weak completeness condition
is satis¬ed: it appeared as bornological completeness, Mackey completeness, or
local completeness in the literature, we call it c∞ -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 c∞ -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 —β F, G) ∼ L(E, F ; G) ∼ 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 ¬nal topology with respect to all smooth
curves, which we call the c∞ -topology, since all smooth mappings are continuous
for it: the vector space E, equipped with this topology is denoted by c∞ E, with
lower case c in analogy to kE for the Kelley-¬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 c∞ -topology coincides with the usual Mackey closure topology.
The space c∞ E is not a topological vector space in general. This is related to the
fact that the evaluation E — E ’ R is jointly continuous only for normable E, but
it is always smooth and hence continuous on c∞ (E — E ). The c∞ -open subsets are
the natural domains of de¬nitions of locally de¬ned functions. For nice spaces (e.g.
Fr´chet and strong duals of Fr´chet-Schwartz spaces, see (4.11)) the c∞ -topology
e e
coincides with the given locally convex topology. In general, the c∞ -topology is
¬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 ¬nite dimensional manifolds and, more generally, of smooth sections
of ¬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 re¬‚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 Hausdor¬. The family of all bounded sets in E plays an im-
portant rˆle. 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 — 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 — , because of di¬erenti-
ation.
See also the appendix (52) for some background on functional analysis.

1.2. Di¬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 ’ E is called di¬erentiable if the derivative

1.2
1.3 1. Smooth curves 9

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

smooth or C if all iterated derivatives exist. It is called C n for some ¬nite n if its
iterated derivatives up to order n exist and are continuous.
Likewise, a mapping f : Rn ’ E is called smooth if all iterated partial derivatives
‚ ‚
‚i1 ,...,ip f := ‚xi1 . . . ‚xip f exist for all i1 , . . . , ip ∈ {1, . . . , n}.
A curve c : R ’ E is called locally Lipschitzian if every point r ∈ R has a neigh-
borhood U such that the Lipschitz condition is satis¬ed on U , i.e., the set

1
c(t) ’ c(s) : t = s; t, s ∈ U
t’s


is bounded. Note that this implies that the curve satis¬es the Lipschitz condition
on each bounded interval, since for (ti ) increasing

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

is in the absolutely convex hull of a ¬nite union of bounded sets.
A curve c : R ’ E is called Lipk or C (k+1)’ if all derivatives up to order k exist and
are locally Lipschitzian. For these properties we have the following implications:

C n+1 =’ Lipn =’ C n ,
di¬erentiable =’ 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 di¬erence quotient.
See also the lemma (1.3) below.

1.3. Lemma. Continuous linear mappings are smooth. A continuous linear
mapping : E ’ F between locally convex vector spaces maps Lipk -curves in E to
Lipk -curves in F , for all 0 ¤ k ¤ ∞, and for k > 0 one has ( —¦ c) (t) = (c (t)).

Proof. As a linear map commutes with di¬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 ( —¦ c) (t) = (c (t)).
Note that a di¬erentiable curve is continuous, and that a continuously di¬erentiable
curve is locally Lipschitz: For ∈ E — we have

1
c(t) ’ c(s) ( —¦ c)(t) ’ ( —¦ c)(s)
( —¦ c) (s + (t ’ s)r)dr,
= =
t’s t’s 0


which is bounded, since ( —¦ 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 in¬nite dimensions. For this let c : R ’ E be a
di¬erentiable curve. If E = R the classical mean value theorem states, that the
di¬erence quotient (c(a)’c(b))/(a’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) ’ c(b))/(a ’ 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 di¬erentiable in at
most countably many points. Recall however, that there exist strictly monotone
functions f : R ’ 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 di¬erentiable function h with non-vanishing deriv-
ative the quotient (c(a)’c(b))/(h(a)’h(b)) equals some intermediate value of c /h .
A version for vector valued c (for real valued h) is that (c(a) ’ c(b))/(h(a) ’ 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) ∈ h (t) · A, where A is some closed convex set, and we should be
able to conclude that c(b) ’ c(a) ∈ (h(b) ’ h(a)) · 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 di¬erentiable in the interior.

Proposition. Mean value theorem. Let c : [a, b] =: I ’ E be a continuous
curve, which is di¬erentiable except at points in a countable subset D ⊆ I. Let h
be a continuous monotone function h : I ’ R, which is di¬erentiable on I \ D. Let
A be a convex closed subset of E, such that c (t) ∈ h (t) · A for all t ∈ D.
/
Then c(b) ’ c(a) ∈ (h(b) ’ h(a)) · 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)’c(a)) ∈ ((h(b) ’ h(a)) · A).
/
But then —¦ 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 [±, β]. We may
furthermore assume that h is monotonely increasing. Then h (t) ≥ 0, and h(b) ’
h(a) ≥ 0. Thus the assumption says that ±h (t) ¤ c (t) ¤ βh (t), and we want to
conclude that ±(h(b) ’ h(a)) ¤ c(b) ’ c(a) ¤ β(h(b) ’ h(a)). If we replace c by
c ’ βh or by ±h ’ c it is enough to show that c (t) ¤ 0 implies that c(b) ’ c(a) ¤ 0.
For given µ > 0 we will show that c(b) ’ c(a) ¤ µ(b ’ a + 1). For this let J be
the set {t ∈ [a, b] : c(s) ’ c(a) ¤ µ ((s ’ a) + tn <s 2’n ) for a ¤ s < t}, where
D =: {tn : n ∈ N}. Obviously, J is a closed interval containing a, say [a, b ]. By
continuity of c we obtain that c(b ) ’ c(a) ¤ µ ((b ’ a) + tn <b 2’n ). Suppose
b < b. If b ∈ D, then there exists a subinterval [b , b + δ] of [a, b] such that for
/
b ¤ s < b + δ we have c(s) ’ c(b ) ’ c (b )(s ’ b ) ¤ µ(s ’ b ). Hence we get
c(s) ’ c(b ) ¤ c (b )(s ’ b ) + µ(s ’ b ) ¤ µ(s ’ b ),

1.4
1.5 1. Smooth curves 11

and consequently

c(s) ’ c(a) ¤ c(s) ’ c(b ) + c(b ) ’ c(a)
2’n ¤ µ s ’ a + 2’n .
¤ µ(s ’ b ) + µ b ’ a +
tn <s
tn <b


On the other hand if b ∈ D, i.e., b = tm for some m, then by continuity of c we
can ¬nd an interval [b , b + δ] contained in [a, b] such that for all b ¤ s < b + δ we
have
c(s) ’ c(b ) ¤ µ2’m .
Again we deduce that

c(s) ’ c(a) ¤ µ2’m + µ b ’ a + 2’n ¤ µ s ’ a + 2’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] = [’1, 1]. Then c (t) ∈ h (t)[’2, 2], but c(1) ’ c(’1) = 2 ∈
/
{0} = (h(1) ’ h(’1))[’2, 2].

1.5. Testing with functionals. Recall that in classical analysis vector valued
curves c : R ’ Rn are often treated by considering their components ck := prk —¦c,
where prk : Rn ’ 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 ’ E is scalarly true, if —¦ c : R ’ E ’ R has this property for all
continuous linear functionals on E.
We want to compare scalar di¬erentiability with di¬erentiability. For ¬nite dimen-
sional spaces we know the trivial fact that these to notions coincide. For in¬nite
dimensions we ¬rst consider Lip-curves c : R ’ E. Since by (52.19) boundedness
can be tested by the continuous linear functionals we see, that c is Lip if and only if
—¦c : R ’ R is Lip for all ∈ E — . Moreover, if for a bounded interval J ‚ R we take
B as the absolutely convex hull of the bounded set c(J)∪{ c(t)’c(s) : t = s; t, s ∈ J},
t’s
then we see that c|J : J ’ EB is a well de¬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{» > 0 : v ∈ ».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) —¦ c is Lip for all ∈ E — .
For continuous instead of Lipschitz curves we obviously have the analogous impli-
cations (1 ’ 2 ’ 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 in¬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 —¦ c is continuous for all
∈ E—.
Furthermore, the “worst” continuous curve - i.e. c : R ’ C(R,R) R =: E given
by (c(t))f := f (t) for all t ∈ R and f ∈ 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 µn ’ ∞ with {µn (c(tn ) ’ c(0)) : n ∈
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 ∈ C(R, R) the sequence µn (f (tn ) ’ f (0))
has to be bounded. But we can choose a continuous function f with f (0) = 0 and

f (tn ) = √1 n and conclude that µn (f (tn ) ’ f (0)) = µn is unbounded.
µ
Similarly, one shows that the reverse implications do not hold for di¬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 (xγ )γ∈“ be a net in EB . Then the following two conditions are
equivalent:
(1) xγ converges to 0 in the normed space EB ;
(2) There exists a net µγ ’ 0 in R, such that xγ ∈ µγ · B.
In (2) we may assume that µγ ≥ 0 and is decreasing with respect to γ, at least for
large γ. In the particular case of a sequence (or where we have a co¬nal countable
subset of “) we can choose all µn > 0 and hence we may divide.

A net (xγ ) for which a bounded absolutely convex B ⊆ E exists, such that xγ
converges to x in EB is called Mackey convergent to x or short M -convergent.

Proof. (‘) Let xγ = µγ ·bγ with bγ ∈ B and µγ ’ 0. Then pB (xγ ) = |µγ | pB (bγ ) ¤
|µγ | ’ 0, i.e. xγ ’ x in EB .
x
(“) Let δ > 1, and set µγ := δ pB (xγ ). By assumption, µγ ’ 0 and xγ = µγ µγ , γ
xγ xγ xγ
1
where µγ := 0 if µγ = 0. Since pB ( µγ ) = δ < 1 or is 0, we conclude that µγ ∈ B.
For the ¬nal assertions, choose γ1 such that |µγ | ¤ 1 for γ ≥ γ1 , and for those γ we
replace µγ by sup{|µγ | : γ ≥ γ}. Thus we may choose µ ≥ 0 and decreasing with
respect to γ.
If we have a sequence (γn )n∈N which is co¬nal in “, i.e. for every γ ∈ “ there exists
an n ∈ N with γ ¤ γn , then we may replace µγ by
1
max({µγ } ∪ {µγm : γm ≥ γ} ∪ { : γm ≥ γ})
m
to conclude that µγ = 0 for all γ.

If “ is the ordered set of all countable ordinals, then it is not possible to ¬nd a net
(µγ )γ∈“ , which is positive everywhere and converges to 0, since a converging net is
¬nally constant.

1.6
1.8 1. Smooth curves 13

1.7. The di¬erence quotient converges Mackey. Now we show how to de-
scribe the quality of convergence of the di¬erence quotient.

Corollary. Let c : R ’ E be a Lip1 -curve. Then the curve t ’ 1 ( 1 (c(t) ’ c(0)) ’
tt
c (0)) is bounded on bounded subsets of R \ {0}.

Proof. We apply (1.4) to c and obtain:
c(t) ’ c(0)
’ c (0) ∈ c (r) : 0 < |r| < |t| ’ c (0)
t closed, convex

= c (r) ’ c (0) : 0 < |r| < |t|
closed, convex
c (r) ’ c (0)
: 0 < |r| < |t|
=r
r closed, convex

Let a > 0. Since { c (r)’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) ’ c(0) r c (r) ’ c (0)
1
’ c (0) ∈ : 0 < |r| < |t| ⊆ B.
t t t r closed, convex


1.8. Corollary. Smoothness of curves is a bornological concept. For
0 ¤ k < ∞ a curve c in a locally convex vector space E is Lipk if and only if for
each bounded open interval J ‚ R there exists an absolutely convex bounded set
B ⊆ 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
¬nite k only, in general. Take the “worst” smooth curve c : R ’ C ∞ (R,R) R,
analogously to (1.5), and, using Borel™s theorem, deduce from c(k) (0) ∈ EB for all
k ∈ N a contradiction.

Proof. For k = 0 this was shown before. For k ≥ 1 take a closed absolutely convex
bounded set B ⊆ E containing all derivatives c(i) on J up to order k as well as their
di¬erence quotients on {(t, s) : t = s, t, s ∈ J}. We show ¬rst that c is di¬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)’c(0) ’ c (0)) lies in B. So c(t)’c(0) ’ 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 doesn™t change if we pass from a given locally convex topology
to its bornologi¬cation, see (4.2), which by de¬nition is the ¬nest locally convex
topology having the same bounded sets.
Let us now return to scalar di¬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 ’ E for k < n + 1 be
curves such that —¦ c0 is Lipn and ( —¦ c0 )(k) = —¦ ck for all k < n + 1 and all
∈ E — . Then c0 is Lipn and (c0 )(k) = ck .

Proof. For n = 0 this was shown in (1.5). For n ≥ 1, by (1.7) applied to —¦ c we
have that
1 c0 (t) ’ c0 (0)
’ c1 (0)
t t
is locally bounded, and hence by (52.19) the set

c0 (t) ’ c0 (0)
1
’ c1 (0) :t∈I
t t
0 0
is bounded. Thus c (t)’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
¬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 (xγ )γ∈“ in E is called Mackey-Cauchy provided that there exist a bounded
(absolutely convex) set B and a net (µγ,γ )(γ,γ )∈“—“ in R converging to 0, such
that xγ ’ xγ ∈ µγ,γ B. As in (1.6) one shows that for a net xγ in EB this is
equivalent to the condition that xγ is Cauchy in the normed space EB .

2.1. Lemma. The di¬erence quotient is Mackey-Cauchy. Let c : R ’ E be
scalarly a Lip1 -curve. Then t ’ c(t)’c(0) is a Mackey-Cauchy net for t ’ 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 t’s c(t)’c(0) ’ c(s)’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) ’ c(0) c(s) ’ c(0) c (tr) ’ c (sr)
1
’ = dr
t’s t’s
t s 0
1
c (tr) ’ c (sr)
= r dr.
tr ’ sr
0


Since c (tr)’c (sr) is locally bounded by assumption, the same is true for the integral,
tr’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 ⊇
(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) ’ (2), and (3) ’ (4) are trivial.
(2) ’ (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 ’ xm ) ’ 0 there exists for every
µ > 0 an N ∈ N such that for all n, m ≥ N we have pB (xn ’ xm ) < µ, and hence
xn ’ xm ∈ µB. Taking the limit for m ’ ∞, and using closedness of B we conclude
that xn ’ x ∈ µB for all n > N . In particular x ∈ EB and xn ’ x in EB .
(4) ’ (1) Let (xγ )γ∈“ be a Mackey-Cauchy net in E. So there is some net µγ,γ ’ 0,
such that xγ ’ xγ ∈ µγ,γ B for some bounded set B. Let γ0 be arbitrary. By (4)
we may assume that B is absolutely convex and contains xγ0 , and that EB is
complete. For γ ∈ “ we have that xγ = xγ0 + xγ ’ xγ0 ∈ xγ0 + µγ,γ0 B ∈ EB , and
pB (xγ ’ xγ ) ¤ µγ,γ ’ 0. So (xγ ) is a Cauchy net in EB , hence converges in EB ,
and thus also in E.

2.3. Corollary. Scalar testing of di¬erentiable curves. Let E be Mackey
complete and c : R ’ E be a curve for which —¦ c is Lipn for all ∈ E — . Then c
is Lipn .

Proof. For n = 0 this was shown in (1.5) without using any completeness, so let
n ≥ 1. Since we have shown in (2.1) that the di¬erence quotient is a Mackey-Cauchy
net we conclude that the derivative c exists, and hence ( —¦ c) = —¦ c . So we may
apply the induction hypothesis to conclude that c is Lipn’1 , and consequently c is
Lipn .

Next we turn to integration. For continuous curves c : [0, 1] ’ E one can show
completely analogously to 1-dimensional analysis that the Riemann sums R(c, Z, ξ),
de¬ned by k (tk ’ tk’1 )c(ξk ), where 0 = t0 < t1 < · · · < tn = 1 is a partition
Z of [0, 1] and ξk ∈ [tk’1 , tk ], form a Cauchy net with respect to the partial strict
ordering given by the size of the mesh max{|tk ’ tk’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 — which are


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

δ : E ’ L(Eequi , R) given by δ(x)( ) := (x).

Proof. Let U be a basis of absolutely convex closed 0-neighborhoods in E. Then
the family of polars U o := { ∈ E — : | (x)| ¤ 1 for all x ∈ U }, with U ∈ U form a
basis for the equicontinuous sets, and hence the bipolars U oo := { — ∈ L(Eequi , R) :


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


bipolar theorem (52.18) we have U = δ ’1 (U oo ) for all U ∈ U. This shows that δ is
a homeomorphism onto its image.

2.5. Lemma. Integral of continuous curves. Let c : R ’ E be a continuous
curve in a locally convex vector space. Then there is a unique di¬erentiable curve
c : R ’ E in the completion E of E such that ( c)(0) = 0 and ( c) = c.

Proof. We show uniqueness ¬rst. Let c1 : R ’ E be a curve with derivative c and
c1 (0) = 0. For every ∈ E — the composite —¦ c1 is an anti-derivative of —¦ c with
initial value 0, so it is uniquely determined, and since E — separates points c1 is also
uniquely determined.
Now we show the existence. By the previous lemma we have that E is (isomorphic

to) the closure of E in the obviously complete space L(Eequi , R). We de¬ne ( c)(t) :
t
E — ’ R by ’ 0 ( —¦ c)(s)ds. It is a bounded linear functional on Eequi since for


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

So c : R ’ L(Eequi , R).
c is di¬erentiable with derivative δ —¦ c.
Now we show that

( c)(t + r) ’ ( c)(r)
’ (δ —¦ c)(r) ( ) =
t
t+r r
1
( —¦ c)(s)ds ’ ( —¦ c)(s)ds ’ t( —¦ c)(r)
= =
t 0 0
r+t 1
1
( —¦ c)(s) ’ ( —¦ c)(r) ds = c(r + ts) ’ c(r) ds.
=
t r 0


Let E ⊆ E — be equicontinuous, and let µ > 0. Then there exists a neighborhood U
of 0 such that | (U )| < µ for all ∈ E. For su¬ciently small t, all s ∈ [0, 1] and ¬xed
1
r we have c(r + ts) ’ c(r) ∈ U . So | 0 (c(r + ts) ’ c(r))ds| < µ. This shows that
the di¬erence quotient of c at r converges to δ(c(r)) uniformly on equicontinuous
subsets.
It remains to show that ( c)(t) ∈ E. By the mean value theorem (1.4) the di¬erence

quotient 1 (( c)(t) ’ ( 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.

De¬nition of the integral. For continuous curves c : R ’ E the de¬nite integral
b b
c ∈ E is given by a c = ( c)(b) ’ ( c)(a).
a



2.5
2.8 2. Completeness 17

2.6. Corollary. Basics on the integral. For a continuous curve c : R ’ E we
have:
b b
∈ E—.
(1) ( a c) = a ( —¦ c) for all
b d d
(2) a c + b c = a c.
b •(b)
c for • ∈ C 1 (R, R).
(c —¦ •)•
(3) =
a •(a)
b
(4) c lies in
the closed convex hull in E of the set
a
{(b ’ a)c(t) : a < t < b} in E.
b
(5) a : C(R, E) ’ E is linear.
(6) (Fundamental theorem of calculus.) For each C 1 -curve c : R ’ E we have
s
c(s) ’ 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] ’ E
be a smooth or just a Lip-curve, and take a partition Z with mesh µ(Z) at most
δ. If we have a second partition, then we can take the common re¬nement. Let
[a, b] be one interval of the original partition with intermediate point t, and let
a = t0 < t1 < · · · < tn = b be the re¬nement. Note that |b ’ a| ¤ δ and hence
|t ’ tk | ¤ δ. Then we can estimate as follows.

(b ’ a) c(t) ’ (tk ’ tk’1 )c(tk ) = (tk ’ tk’1 ) (c(t) ’ c(tk )) = µk bk ,
k k k

c(t)’c(tk )
where bk := is contained in the absolutely convex Lipschitz bound
δ

c(t) ’ c(s)
: t, s ∈ [0, 1]
B :=
t’s abs.conv

of c and µk := (tk ’tk’1 )δ ≥ 0 and satis¬es k µk = (b’a)δ. Hence we have for the
Riemann sums with respect to the original partition Z1 and the re¬nement Z that
R(c, Z1 ) ’ R(c, Z ) lies in δ · B. So R(c, Z1 ) ’ R(c, Z2 ) ∈ 2δB for any two partitions
Z1 and Z2 of mesh at most δ, i.e. the Riemann sums form a Mackey-Cauchy net
with coe¬cients µZ1 ,Z2 := 2 max{µ(Z1 ), µ(Z2 )} and we have proved:

2.7. Proposition. Integral of Lipschitz curves. Let c : [0, 1] ’ 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 di¬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 in¬nite polygon)
and tn 0 with c(tn ) = xn . The corresponding result for smooth curves uses the
following notion.

De¬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 ∈ N the sequence nk (xn ’ 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 in¬nite polygon through the xn can be parameterized as a smooth curve
1
c : R ’ E such that c( n ) = xn and c(0) = x.

Proof. Let • : R ’ [0, 1] be a smooth map, which is 0 on {t : t ¤ 0} and 1 on
{t : t ≥ 1}. The parameterization c is de¬ned as follows:
for t ¤ 0,
x
±
1

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


for t ≥ 1
x1
1 1
Obviously, c is smooth on R \ {0}, and the p-th derivative of c for ¤t¤ is
n+1 n
given by
1
t ’ n+1
c(p) (t) = •(p) 1 (n(n + 1))p (xn ’ xn+1 ).
1
n ’ n+1

Since xn converges fast to x, we have that c(p) (t) ’ 0 for t ’ 0, because the ¬rst
factor is bounded and the second goes to zero. Hence c is smooth on R, by the
following lemma.

2.9. Lemma. Di¬erentiable extension to an isolated point. Let c : R ’ E
be continuous and di¬erentiable on R \ {0}, and assume that the derivative c :
R \ {0} ’ E has a continuous extension to R. Then c is di¬erentiable at 0 and
c (0) = limt’0 c (t).

Proof. Let a := limt’0 c (t). By the mean value theorem (1.4) we have c(t)’c(0) ∈ t
c (s) : 0 = |s| ¤ |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 δ > 0 such
that c (t) ∈ a + U for all 0 < |t| ¤ δ. Hence c(t)’c(0) ’ a ∈ U , i.e. c (0) = a.
t

The next result shows that we can pass through certain sequences xn ’ x even
with given velocities vn ’ 0.

2.10. Corollary. If xn ’ x fast and vn ’ 0 fast in E, then there are smoothly
parameterized polygon c : R ’ E and tn ’ 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 de¬ned by
1 1
y2n+1 := xn ’
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 • which satis¬es •(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 ¬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 ’ 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 ’ 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 » ∈ F such
that |(» —¦ )(bn )| ≥ nn+1 . The sequence (n’n bn ) converges fast to 0, hence lies on
some compact part of a smooth curve by (2.8). Consequently, (» —¦ )(n’n bn ) =
n’n (» —¦ )(bn ) is bounded, a contradiction.

2.12. De¬nition. The c∞ -topology on a locally convex space E is the ¬nal topol-
ogy with respect to all smooth curves R ’ E. Its open sets will be called c∞ -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 ¬nest locally convex topology coarser than the
c∞ -topology is the bornologi¬cation of the locally convex topology.
Let (µn ) be a sequence of real numbers converging to ∞. Then a sequence (xn ) in
E is called µ-converging to x if the sequence (µn (xn ’ x)) is bounded in E.

2.13. Theorem. c∞ -open subsets. Let µn ’ ∞ be a real valued sequence.
Then a subset U ⊆ E is open for the c∞ -topology if it satis¬es any of the following
equivalent conditions:
(1) All inverse images under Lipk -curves are open in R (for ¬xed k ∈ N∞ ).
(2) All inverse images under µ-converging sequences are open in N∞ .
(3) The traces to EB are open in EB for all absolutely convex bounded subsets
B ⊆ E.
Note that for closed subsets an equivalent statement reads as follows: A set A is c∞ -
closed if and only if for every sequence xn ∈ A, which is µ-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) ’ (2) Suppose (xn ) is µ-converging to x ∈ U , but xn ∈ U for in¬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 ∈ U but c(0) = x ∈ U . This is a contradiction.
/
(2) ’ (3) A sequence (xn ), which converges in EB to x with respect to pB , is Mackey
convergent, hence has a µ-converging subsequence. Note that EB is normed, and
hence it is enough to consider sequences.
(3) ’ (2) Suppose (xn ) is µ-converging to x. Then the absolutely convex hull B of
{µn (xn ’ x) : n ∈ N} ∪ {x} is bounded, and xn ’ x in (EB , pB ), since µn (xn ’ x)
is bounded.

2.13
20 Chapter I. Calculus of smooth mappings 2.14

(2) ’ (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 c∞ -topology and c∞ -completeness are intimately related.

2.14. Theorem. Convenient vector spaces. Let E be a locally convex vector
space. E is said to be c∞ -complete or convenient if one of the following equivalent
(completeness) conditions is satis¬ed:
(1) Any Lipschitz curve in E is locally Riemann integrable.
(2) For any c1 ∈ C ∞ (R, E) there is c2 ∈ C ∞ (R, E) with c2 = c1 (existence of
an anti-derivative).
(3) E is c∞ -closed in any locally convex space.
(4) If c : R ’ E is a curve such that —¦ c : R ’ R is smooth for all ∈ E — ,
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 c∞ -completeness is a bornological concept. In [Fr¨licher,
o
Kriegl, 1988] a convenient vector space is always considered with its bornological
topology ” an equivalent but not isomorphic category.

Proof. In (2.3) we showed (5) ’ (4), in (2.7) we got (5) ’ (1), and in (2.2) we
had (5) ’ (6).
(1) ’ (2) A smooth curve is Lipschitz, thus locally Riemann integrable. The
inde¬nite Riemann integral equals the “weakly de¬ned” integral of lemma (2.5),
hence is an anti-derivative.
(2) ’ (3) Let E be a topological linear subspace of F . To show that E is c∞ -
closed we use (2.13). Let xn ’ x∞ be fast falling, xn ∈ E but x∞ ∈ 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 = ∞}, which is
contained in E. Its anti-derivative c2 is up to a constant equal to c, and by (2)
x1 ’ x∞ = c(1) ’ c(0) = c2 (1) ’ c2 (0) lies in E. So x∞ ∈ E.
(3) ’ (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 c∞ -closed in F the limit lies in E.
Hence, the sequence converges in E.
(6) ’ (7) Let f : F ’ 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) ’ (1) Let c : R ’ 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) ’ (3) Let E be embedded in some space F . We use again (2.13) in order to
show that E is c∞ -closed in F . So let xn ’ x∞ fast falling, xn ∈ E for n = 0,
but x∞ ∈ F . By (2.8) the polygon c through (xn ) can be smoothly symmetrically
parameterized in F , and c(t) ∈ E for t = 0. We consider c(t) := tc(t). This is a
˜
curve in E which is smooth in F , so it is scalarwise smooth in E, thus smooth in
E by (4). Then x∞ = c (0) ∈ E.
˜

2.15. Theorem. Inheritance of c∞ -completeness. The following construc-
tions preserve c∞ -completeness: limits, direct sums, strict inductive limits of se-
quences of closed embeddings, as well as formation of ∞ (X, ), where X is a set
together with a family B of subsets of X containing the ¬nite ones, which are called
bounded and ∞ (X, F ) denotes the space of all functions f : X ’ F , which are
bounded on all B ∈ B, supplied with the topology of uniform convergence on the
sets in B.

Note that the de¬nition of the topology of uniform convergence as initial topology
shows, that adding all subsets of ¬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 c∞ -closed linear subspace

(x± ) ∈ F(±) : F(f )x± = xβ for all f : ± ’ β ,

hence is c∞ -complete, since the product of c∞ -complete factors is obviously c∞ -
complete.
Since the coproduct (52.7) of spaces X± is the topological direct sum, and has as
bounded sets those which are contained and bounded in some ¬nite subproduct, it
is c∞ -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 ∞ (X, F ) we consider ¬rst the case, where X itself is bounded.
Then c∞ -completeness can be proved as in (52.4) or reduced to this result. In fact
let B be bounded in ∞ (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 ∈ ∞ (X, F ) : f (X) ⊆ B}. The space ∞ (X, F )B is just
the space ∞ (X, FB ) with the supremum norm, which is a Banach space by (52.4).

(X, F ) ’ ∞ (B, F )
Let now X and B be arbitrary. Then the restriction maps
give an embedding ι of ∞ (X, F ) into the product B∈B ∞
(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 f± |B converge to some fB in ∞ (B, F ).
De¬ne f (x) := f{x} (x). For any B ∈ B containing x we have that fB (x) =
(lim± f± |B )(x) = lim± (f± (x)) = lim± f± |{x} = f{x} (x) = f (x), and f (B) is boun-
ded for all B ∈ B, since f |B = fB ∈ ∞ (B, F ).

Example. In general, a quotient and an inductive limit of c∞ -complete spaces
need not be c∞ -complete. In fact, let ED := {x ∈ RN : supp x ⊆ D} for any
subset D ⊆ N of density dens D := lim sup{ |D©[1,n]| } = 0. It can be shown that
n
E := dens D=0 ED ‚ R is the inductive limit of the Fr´chet subspaces ED ∼ RD .
N
e =

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



3. Smooth Mappings and the Exponential Law

Now let us start proving the exponential law C ∞ (U — V, F ) ∼ C ∞ (U, C ∞ (V, F ))
=
¬rst for U = V = F = R.

3.1. Proposition. For a continuous map f : R — [0, 1] ’ R the partial derivative
‚1 f exists and is continuous if and only if f ∨ : R ’ C([0, 1], R) is continuously
1 1‚
di¬erentiable. And in this situation I((f ∨ ) (t)) = dt 0 f (t, s) ds = 0 ‚t f (t, s) ds,
d

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

Proof. We assume that ‚1 f exists and is continuous. Hence, (‚1 f )∨ : R ’
C([0, 1], R) is continuous. We want to show that f ∨ : R ’ C([0, 1], R) is dif-
ferentiable (say at 0) with this function (at 0) as derivative. So we have to show
∨ ∨
that the mapping t ’ f (t)’f (0) is continuously extendable to R by de¬ning its
t

value at 0 as (‚1 f ) (0). Or equivalently, by what is obvious for continuous maps,
that the map
f (t,s)’f (0,s)
for t = 0
t
(t, s) ’
‚1 f (0, s) otherwise
is continuous. This follows immediately from the continuity of ‚1 f and of integra-
1
tion since it can be written as 0 ‚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 di¬eren-
0
tiable with derivative
1 1
d ‚

f (t, s) ds = I((f ) (t)) = f (t, s) ds.
dt ‚t
0 0

The converse implication is obvious.

3.2. Theorem. Simplest case of exponential law. Let f : R2 ’ R be an
arbitrary mapping. Then all iterated partial derivatives exist and are locally bounded
if and only if the associated mapping f ∨ : R ’ C ∞ (R, R) exists as a smooth curve,

3.2
3.2 3. Smooth mappings and the exponential law 23

where C ∞ (R, R) is considered as the Fr´chet space with the topology of uniform
e
convergence of each derivative on compact sets. Furthermore, we have (‚1 f )∨ =
d(f ∨ ) and (‚2 f )∨ = d —¦ f ∨ = d— (f ∨ ).

Proof. We have several possibilities to prove this result. Either we show Mackey
convergence of the di¬erence quotients, via the boundedness of 1 c(t)’c(0) ’ c (0) ,
t t
∼ ∞ (X, ∞ (Y, R)); or we use

(X —Y, R) =
and then use the trivial exponential law
the induction step proved in (3.1), namely that f ∨ : R ’ C(R, R) is di¬erentiable
if and only if ‚1 f exists and is continuous R2 ’ R, together with the exponential
law C(R2 , R) ∼ C(R, C(R, R)). We choose the latter method.
=
For this we have to note ¬rst that if for a function g the partial derivatives ‚1 g and
‚2 g exist and are locally bounded then g is continuous:
g(x, y) ’ g(0, 0) = g(x, y) ’ g(x, 0) + g(x, 0) ’ g(0, 0)
= y‚2 g(x, r2 y) + x‚1 g(r1 x, 0)
for suitable r1 , r2 ∈ [0, 1], which goes to 0 with (x, y).
Proof of (’) By what we just said, all iterated partial derivatives of f are contin-
uous. First observe that f ∨ : R ’ C ∞ (R, R) makes sense and that for all t ∈ R we
have
q
dq (f ∨ (t)) = (‚2 f )∨ (t).
(1)
Next we claim that f ∨ : R ’ C ∞ (R, R) is di¬erentiable, with derivative d(f ∨ ) =
(‚1 f )∨ , or equivalently that for all a the curve
f ∨ (t+a)’f ∨ (a)
for t = 0
t
c:t’ ∨
(‚1 f ) (a) otherwise
is continuous as curve R ’ C ∞ (R, R). Without loss of generality we may assume
that a = 0. Since C ∞ (R, R) carries the initial structure with respect to the linear
mappings dp : C ∞ (R, R) ’ C(R, R) we have to show that dp —¦ c : R ’ C(R, R)
is continuous, or equivalently by the exponential law for continuous maps, that
(dp —¦ c)§ : R2 ’ R is continuous. For t = 0 and s ∈ R we have
f ∨ (t) ’ f ∨ (0)
§
p p p
(d —¦ c) (t, s) = d (c(t))(s) = d (s)
t
p p
‚2 f (t, s) ’ ‚2 f (0, s)
= by (1)
t
1
p
= ‚1 ‚2 f (t „, s) d„ by the fundamental theorem.
0

For t = 0 we have
(dp —¦ c)§ (0, s) = dp (c(0))(s) = dp ((‚1 f )∨ (0))(s)
p
= (‚2 (‚1 f ))∨ (0)(s) by (1)
p
= ‚2 ‚1 f (0, s)
p
= ‚1 ‚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 —¦ c)§ (t, s) = 0 ‚1 ‚2 f (t „, s) d„ for all (t, s). This function is con-
p p
tinuous in (t, s), since ‚1 ‚2 f : R2 ’ R is continuous, hence (t, s, „ ) ’ ‚1 ‚2 f (t „, s)
p
is continuous, and therefore also (t, s) ’ („ ’ ‚1 ‚2 f (t „, s)) from R2 ’ C([0, 1], R).
1
Composition with the continuous linear mapping 0 : C([0, 1], R) ’ R gives the
continuity of (dp —¦ c)§ .
Now we proceed by induction. By the induction hypothesis applied to ‚1 f , we
obtain that d(f ∨ ) = (‚1 f )∨ and (‚1 f )∨ : R ’ C ∞ (R, R) is n times di¬erentiable,
so f ∨ is (n + 1)-times di¬erentiable.
Proof of (⇐) First remark that for a smooth map f : R ’ C ∞ (R, R) the asso-
ciated map f § : R2 ’ R is locally bounded: Since f is smooth f (I1 ) is com-
pact, hence bounded in C ∞ (R, R) for all compact intervals I1 . In particular,
f (I1 )(I2 ) = f § (I1 — I2 ) has to be bounded in R for all compact intervals I1 and I2 .
Since f is smooth both curves df and d —¦ f = d— f are smooth (use (1.3) and that
d is continuous and linear). An easy calculation shows that the partial derivatives
of f § exist and are given by ‚1 f § = (df )§ and ‚2 f § = (d —¦ f )§ . So one obtains
inductively that all iterated derivatives of f § exist and are locally bounded, since
they are associated to smooth curves R ’ C ∞ (R, R).

In order to proceed to more general cases of the exponential law we need a de¬nition
of C ∞ -maps de¬ned on in¬nite dimensional spaces. This de¬nition should at least
guarantee the chain rule, and so one could take the weakest notion that satis¬es
the chain rule. However, consider the following

3.3. Example. We consider the following 3-fold “singular covering” f : R2 ’ R2
given in polar coordinates by (r, •) ’ (r, 3•). In cartesian coordinates we obtain
the following formula for the values of f :

(r cos(3•), r sin(3•)) = r (cos •)3 ’ 3 cos •(sin •)2 , 3 sin •(cos •)2 ’ (sin •)3
x3 ’ 3xy 2 3x2 y ’ 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 ¬rst component of f with a rotation from the right (Use that f
intertwines the rotation with angle δ and the rotation with angle 3δ).
Obviously, the map f is smooth on R2 \ {0}. It is homogeneous of degree 1, and

hence the directional derivative is f (0)(v) = ‚t |t=0 f (tv) = f (v). However, both
components are nonlinear with respect to v and thus are not di¬erentiable at (0, 0).
Obviously, f : R2 ’ R2 is continuous.
We claim that f is di¬erentiable along di¬erentiable curves, i.e. (f —¦ c) (0) exists,
provided c (0) exists.
Only the case c(0) = 0 is not trivial. Since c is di¬erentiable at 0 the curve c1
de¬ned by c1 (t) := c(t) for t = 0 and c (0) for t = 0 is continuous at 0. Hence
t
f (c(t))’f (c(0)) f (t c1 (t))’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 —¦ 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
‚ ‚ t 1
f (sx)(v) = f (s x + tv) = s f (x + v) = s f (x)( v) = f (x)(v).
‚t ‚t s s

. 1
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