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