2 µ

f (x ’ y) •µ (y) dy = f (x ’ µy)•(y)dy.

fµ (x) := (f •µ )(x) =

R2 R2

Since the convolution fµ := f •µ of a continuous function f with a smooth function

•µ with compact support is di¬erentiable with directional derivative dv (f •µ )(x) =

(f dv •µ )(x), we obtain that fµ is smooth. And since f •µ ’ f in C(R2 , R) for

µ ’ 0 and any continuous function f , we conclude dp fµ = dp f •µ ’ dp f uniformly

v v v

on compact sets.

We remark now that for a smooth map g : R2 ’ R we have by the chain rule

d

g(x + tv) = ‚1 g(x + tv) · v1 + ‚2 g(x + tv) · v2

dv g(x + tv) =

dt

and by induction that

p

p i p’i i p’i

dp g(x) = v v ‚1 ‚2 g(x).

v

i 12

j=0

i p’i

Hence, we can calculate the iterated derivatives ‚1 ‚2 g(x) for 0 ¤ i ¤ p from

p + 1 many derivatives dpj g(x) provided the v j are chosen in such a way, that the

v

j j

Vandermonde™s determinant det((v1 )i (v2 )p’i )ij = 0. For this choose v2 = 1 and all

j j j

the v1 pairwise distinct, then det((v1 )i (v2 )p’i )ij = j>k (v1 ’ v1 ) = 0.

k

Hence, the iterated derivatives of fµ are linear combinations of the derivatives dp fµ

v

for p + 1 many vectors v, where the coe¬cients depend only on the v™s. So we

conclude that the iterated derivatives of fµ form a Cauchy sequence in C(R2 , R),

and hence converge to continuous functions f ± . Thus, all iterated derivatives ‚ ± f

of f exist and are equal to these continuous functions f ± , by the following lemma

(3.5).

3.5. Lemma. Let fµ ’ f in C(R2 , R) and dv fµ ’ fv in C(R2 , R). Then dv f

exists and equals fv .

Proof. We have to show that for ¬xed x, v ∈ R2 the curve

f (x+tv)’f (x)

for t = 0

t

c:t’

fv (x) otherwise

3.5

28 Chapter I. Calculus of smooth mappings 3.9

is continuous from R ’ R. The corresponding curve cµ for fµ can be rewritten

1

as cµ (t) = 0 dv fµ (x + „ t v) d„ , which converges by assumption uniformly for t in

1

compact sets to the continuous curve t ’ 0 fv (x + „ t v) d„ . Pointwise it converges

to c(t), hence c is continuous.

For the vector valued case of the exponential law we need a locally convex structure

on C ∞ (R, E).

3.6. De¬nition. Space of curves. Let C ∞ (R, E) be the locally convex vector

space of all smooth curves in E, with the pointwise vector operations, and with

the topology of uniform convergence on compact sets of each derivative separately.

dk

This is the initial topology with respect to the linear mappings C ∞ (R, E) ’ ’

C ∞ (R, E) ’ ∞ (K, E), where k runs through N, where K runs through all compact

subsets of R, and where ∞ (K, E) carries the topology of uniform convergence, see

also (2.15).

Note that the derivatives dk : C ∞ (R, E) ’ C ∞ (R, E), the point evaluations evt :

C ∞ (R, E) ’ E and the pull backs g — : C ∞ (R, E) ’ C ∞ (R, E) for all g ∈ C ∞ (R, R)

are continuous and linear.

3.7. Lemma. A space E is c∞ -complete if and only if C ∞ (R, E) is.

Proof. (’) The mapping c ’ (c(n) )n∈N is by de¬nition an embedding of C ∞ (R, E)

into the c∞ -complete product n∈N ∞ (R, E). Its image is a closed subspace, since

the previous lemma can be easily generalized to curves c : R ’ E.

(⇐) Consider the continuous linear mapping const : E ’ C ∞ (R, E) given by

x ’ (t ’ x). It has as continuous left inverse the evaluation at any point (e.g. ev0 :

C ∞ (R, E) ’ E, c ’ c(0)). Hence, E can be identi¬ed with the closed subspace of

C ∞ (R, E) given by the constant curves, and is thereby itself c∞ -complete.

3.8. Lemma. Curves into limits. A curve into a c∞ -closed subspace of a space

is smooth if and only if it is smooth into the total space. In particular, a curve is

smooth into a projective limit if and only if all its components are smooth.

Proof. Since the derivative of a smooth curve is the Mackey limit of the di¬erence

quotient, the c∞ -closedness implies that this limit belongs to the subspace. Thus,

we deduce inductively that all derivatives belong to the subspace, and hence the

curve is smooth into the subspace.

The result on projective limits now follows, since obviously a curve is smooth into

a product, if all its components are smooth.

We show now that the bornology, but obviously not the topology, on function spaces

can be tested with the linear functionals on the range space.

3.9. Lemma. Bornology of C ∞ (R, E). The family

: C ∞ (R, E) ’ C ∞ (R, R) : ∈ E — }

{ —

3.9

3.10 3. Smooth mappings and the exponential law 29

generates the bornology of C ∞ (R, E). This also holds for E — replaced by E .

A set in C ∞ (R, E) is bounded if and only if each derivative is uniformly bounded

on compact subsets.

Proof. A set B ⊆ C ∞ (R, E) is bounded if and only if the sets {dn c(t) : t ∈ I, c ∈ B}

are bounded in E for all n ∈ N and compact subsets I ‚ R.

This is furthermore equivalent to the condition that the set { (dn c(t)) = dn ( —¦c)(t) :

t ∈ I, c ∈ B} is bounded in R for all ∈ E — , n ∈ N, and compact subsets I ‚ R

and in turn equivalent to: { —¦ c : c ∈ B} is bounded in C ∞ (R, R).

For E — replaced by E ⊇ E — the statement holds, since ∈E

is bounded for all

—

by the explicit description of the bounded sets.

3.10. Proposition. Vector valued simplest exponential law. For a map-

ping f : R2 ’ E into a locally convex space (which need not be c∞ -complete) the

following assertions are equivalent:

(1) f is smooth along smooth curves.

All iterated directional derivatives dp f exist and are locally bounded.

(2) v

±

(3) All iterated partial derivatives ‚ f exist and are locally bounded.

f ∨ : R ’ C ∞ (R, E) exists as a smooth curve.

(4)

Proof. We prove this result ¬rst for c∞ -complete spaces E. Then each of the

statements (1-4) are valid if and only if the corresponding statement for —¦ f is

valid for all ∈ E — . Only (4) needs some arguments: In fact, f ∨ (t) ∈ C ∞ (R, E)

if and only if — (f ∨ (t)) = ( —¦ f )∨ (t) ∈ C ∞ (R, R) for all ∈ E — by (2.14). Since

C ∞ (R, E) is c∞ -complete, its bornologically isomorphic image in ∞

∈E — C (R, R)

is c∞ -closed. So f ∨ : R ’ C ∞ (R, E) is smooth if and only if — —¦ f ∨ = ( —¦ f )∨ :

R ’ C ∞ (R, R) is smooth for all ∈ E — . So the proof is reduced to the scalar valid

case, which was proved in (3.2) and (3.4).

Now the general case. For the existence of certain derivatives we know by (1.9) that

it is enough that we have some candidate in the space, which is the corresponding

derivative of the map considered as map into the c∞ -completion (or even some

larger space). Since the derivatives required in (1-4) depend linearly on each other,

this is true. In more detail this means:

(1) ’ (2) is obvious.

|±|

(2) ’ (3) is the fact that ‚ ± is a universal linear combination of dv f .

(3) ’ (1) follows from the chain rule which says that (f —¦c)(p) (t) is a universal linear

(p )

(p )

combination of ‚i1 . . . ‚iq f (c(t))ci1 1 (t) . . . ciq q (t) for ij ∈ {1, 2} and pj = p, see

also (10.4).

(3) ” (4) holds by (1.9) since (‚1 f )∨ = d(f ∨ ) and (‚2 f )∨ = d —¦ f ∨ = d— (f ∨ ).

For the general case of the exponential law we need a notion of smooth mappings

and a locally convex topology on the corresponding function spaces. Of course, it

3.10

30 Chapter I. Calculus of smooth mappings 3.12

would be also handy to have a notion of smoothness for locally de¬ned mappings.

Since the idea is to test smoothness with smooth curves, such curves should have

locally values in the domains of de¬nition, and hence these domains should be

c∞ -open.

3.11. De¬nition. Smooth mappings and spaces of them. A mapping f :

E ⊇ U ’ F de¬ned on a c∞ -open subset U is called smooth (or C ∞ ) if it maps

smooth curves in U to smooth curves in F .

Let C ∞ (U, F ) denote the locally convex space of all smooth mappings U ’ F with

pointwise linear structure and the initial topology with respect to all mappings

c— : C ∞ (U, F ) ’ C ∞ (R, F ) for c ∈ C ∞ (R, U ).

For U = E = R this coincides with our old de¬nition. Obviously, any composition

of smooth mappings is also smooth.

Lemma. The space C ∞ (U, F ) is the (inverse) limit of spaces C ∞ (R, F ), one for

each c ∈ C ∞ (R, U ), where the connecting mappings are pull backs g — along repa-

rameterizations g ∈ C ∞ (R, R).

Note that this limit is the closed linear subspace in the product

C ∞ (R, F )

c∈C ∞ (R,U )

consisting of all (fc ) with fc—¦g = fc —¦ g for all c and all g ∈ C ∞ (R, R).

Proof. The mappings c— : C ∞ (U, F ) ’ C ∞ (R, F ) de¬ne a continuous linear em-

g—

bedding C (U, F ) ’ limc {C (R, F ) ’ C ∞ (R, F )}, since c— (f ) —¦ g = f —¦ c —¦ g =

∞ ∞

’

(c —¦ g) (f ). It is surjective since for any (fc ) ∈ limc C ∞ (R, F ) one has fc = f —¦ c

—

where f is de¬ned as x ’ fconstx (0).

3.12. Theorem. Cartesian closedness. Let Ui ⊆ Ei be c∞ -open subsets in

locally convex spaces, which need not be c∞ -complete. Then a mapping f : U1 —

U2 ’ F is smooth if and only if the canonically associated mapping f ∨ : U1 ’

C ∞ (U2 , F ) exists and is smooth.

Proof. We have the following implications:

f ∨ : U1 ’ C ∞ (U2 , F ) is smooth.

f ∨ —¦c1 : R ’ C ∞ (U2 , F ) is smooth for all smooth curves c1 in U1 , by (3.11).

”

c— —¦ f ∨ —¦ c1 : R ’ C ∞ (R, F ) is smooth for all smooth curves ci in Ui , by

” 2

(3.11) and (3.8).

f —¦ (c1 — c2 ) = (c— —¦ f ∨ —¦ c1 )§ : R2 ’ F is smooth for all smooth curves ci

” 2

in Ui , by (3.10).

” f : U1 — U2 ’ F is smooth.

Here the last equivalence is seen as follows: Each curve into U1 — U2 is of the form

(c1 , c2 ) = (c1 — c2 ) —¦ ∆, where ∆ is the diagonal mapping. Conversely, f —¦ (c1 —

c2 ) : R2 ’ F is smooth for all smooth curves ci in Ui , since the product and the

composite of smooth mappings is smooth by (3.11) (and by (3.4)).

3.12

3.15 3. Smooth mappings and the exponential law 31

3.13. Corollary. Consequences of cartesian closedness. Let E, F , G, etc. be

locally convex spaces, and let U , V be c∞ -open subsets of such. Then the following

canonical mappings are smooth.

ev : C ∞ (U, F ) — U ’ F , (f, x) ’ f (x);

(1)

ins : E ’ C ∞ (F, E — F ), x ’ (y ’ (x, y));

(2)

( )§ : C ∞ (U, C ∞ (V, G)) ’ C ∞ (U — V, G);

(3)

( )∨ : C ∞ (U — V, G) ’ C ∞ (U, C ∞ (V, G));

(4)

comp : C ∞ (F, G) — C ∞ (U, F ) ’ C ∞ (U, G), (f, g) ’ f —¦ g;

(5)

C ∞ ( , ) : C ∞ (E2 , E1 ) — C ∞ (F1 , F2 ) ’

(6)

’ C ∞ (C ∞ (E1 , F1 ), C ∞ (E2 , F2 )), (f, g) ’ (h ’ g —¦ h —¦ f );

: C ∞ (Ei , Fi ) ’ C ∞ ( Ei , Fi ), for any index set.

(7)

Proof. (1) The mapping associated to ev via cartesian closedness is the identity

on C ∞ (U, F ), which is C ∞ , thus ev is also C ∞ .

(2) The mapping associated to ins via cartesian closedness is the identity on E — F ,

hence ins is C ∞ .

(3) The mapping associated to ( )§ via cartesian closedness is the smooth com-

position of evaluations ev —¦(ev — Id) : (f ; x, y) ’ f (x)(y).

(4) We apply cartesian closedness twice to get the associated mapping (f ; x; y) ’

f (x, y), which is just a smooth evaluation mapping.

(5) The mapping associated to comp via cartesian closedness is (f, g; x) ’ f (g(x)),

which is the smooth mapping ev —¦(Id — ev).

(6) The mapping associated to the one in question by applying cartesian closed twice

is (f, g; h, x) ’ g(h(f (x))), which is the C ∞ -mapping ev —¦(Id — ev) —¦ (Id — Id — ev).

(7) Up to a ¬‚ip of factors the mapping associated via cartesian closedness is the

product of the evaluation mappings C ∞ (Ei , Fi ) — Ei ’ Fi .

Next we generalize (3.4) to ¬nite dimensions.

3.14. Corollary. [Boman, 1967]. The smooth mappings on open subsets of Rn in

the sense of de¬nition (3.11) are exactly the usual smooth mappings.

Proof. Both conditions are of local nature, so we may assume that the open subset

of Rn is an open box and in turn even Rn itself.

(’) If f : Rn ’ F is smooth then by cartesian closedness (3.12), for each coordinate

the respective associated mapping f ∨i : Rn’1 ’ C ∞ (R, F ) is smooth, so again by

(3.12) we have ‚i f = (d— f ∨i )§ , so all ¬rst partial derivatives exist and are smooth.

Inductively, all iterated partial derivatives exist and are smooth, thus continuous,

so f is smooth in the usual sense.

(⇐) Obviously, f is smooth along smooth curves by the usual chain rule.

3.15. Di¬erentiation of an integral. We return to the question of di¬erentiat-

ing an integral. So let f : E — R ’ F be smooth, and let F be the completion of

the locally convex space F . Then we may form the function f0 : E ’ F de¬ned by

3.15

32 Chapter I. Calculus of smooth mappings 3.16

1

x ’ 0 f (x, t) dt. We claim that it is smooth, and that the directional derivative is

1

given by dv f0 (x) = 0 dv (f ( , t))(x) dt. By cartesian closedness (3.12) the associ-

1

ated mapping f ∨ : E ’ C ∞ (R, F ) is smooth, so the mapping 0 —¦f ∨ : E ’ F is

smooth since integration is a bounded linear operator, and

1

‚ ‚

dv f0 (x) = f0 (x + sv) = f (x + sv, t)dt

‚s s=0 ‚s s=0

0

1 1

‚

= f (x + sv, t)dt = dv (f ( , t))(x) dt.

‚s s=0

0 0

But we want to generalize this to functions f de¬ned only on some c∞ -open subset

U ⊆ E—R, so we have to show that the natural domain U0 := {x ∈ E : {x}—[0, 1] ⊆

U } of f0 is c∞ -open in E. We will do this in lemma (4.15). From then on the proof

runs exactly the same way as for globally de¬ned functions. So we obtain the

Proposition. Let f : E — R ⊇ U ’ F be smooth with c∞ -open U ⊆ E — R. Then

1

x ’ 0 f (x, t) dt is smooth on the c∞ -open set U0 := {x ∈ E : {x} — [0, 1] ⊆ U }

1

with values in the completion F and dv f0 (x) = 0 dv (f ( , t))(x) dt for all x ∈ U0

and v ∈ E.

Now we want to de¬ne the derivative of a general smooth map and prove the chain

rule for them.

3.16. Corollary. Smoothness of the di¬erence quotient. For a smooth curve

c : R ’ E the di¬erence quotient

c(t) ’ c(s)

±

for t = s

t’s

(t, s) ’

c (t) for t = s

is a smooth mapping R2 ’ E.

1

c(t)’c(s)

Proof. By (2.5) we have f : (t, s) ’ c (s + r(t ’ s))dr, and by (3.15)

=

t’s 0

it is smooth R2 ’ E. The left hand side has values in E, and for t = s this is also

true for all iterated directional derivatives. It remains to consider the derivatives

for t = s. The iterated directional derivatives are given by (3.15) as

1

dp f (t, s) dp c (s + r(t ’ s)) dr

=

v v

0

1

dp c (s + r(t ’ s)) dr,

= v

0

where dv acts on the (t, s)-variable. The later integrand is for t = s just a linear

combination of derivatives of c which are independent of r, hence dp f (t, s) ∈ E. By

v

(3.10) the mapping f is smooth into E.

3.16

3.18 3. Smooth mappings and the exponential law 33

3.17. De¬nition. Spaces of linear mappings. Let L(E, F ) denote the space

of all bounded (equivalently smooth by (2.11)) linear mappings from E to F . It is

a closed linear subspace of C ∞ (E, F ) since f is linear if and only if for all x, y ∈ E

and » ∈ R we have (evx +» evy ’ evx+»y )f = 0. We equip it with this topology

and linear structure.

So a mapping f : U ’ L(E, F ) is smooth if and only if the composite mapping

f

U ’ L(E, F ) ’ C ∞ (E, F ) is smooth.

’

3.18. Theorem. Chain rule. Let E and F be locally convex spaces, and let

U ⊆ E be c∞ -open. Then the di¬erentiation operator

d : C ∞ (U, F ) ’ C ∞ (U, L(E, F )),

f (x + tv) ’ f (x)

df (x)v := lim ,

t

t’0

exists, is linear and bounded (smooth). Also the chain rule holds:

d(f —¦ g)(x)v = df (g(x))dg(x)v.

Proof. Since t ’ x+tv is a smooth curve we know that d§§ : C ∞ (U, F )—U —E ’

F exists. We want to show that it is smooth, so let (f, x, v) : R ’ C ∞ (U, F )—U —E

be a smooth curve. Then

f (t)(x(t) + sv(t)) ’ f (t)(x(t))

d§§ (f (t), x(t), v(t)) = lim = ‚2 h(t, 0),

s

s’0

which is smooth in t, where the smooth mapping h : R2 ’ F is given by (t, s) ’

f § (t, x(t)+sv(t)). By cartesian closedness (3.12) the mapping d§ : C ∞ (U, F )—U ’

C ∞ (E, F ) is smooth.

Now we show that this mapping has values in the subspace L(E, F ): d§ (f, x)

is obviously homogeneous. It is additive, because we may consider the smooth

mapping (t, s) ’ f (x + tv + sw) and compute as follows, using (3.14).

‚

df (x)(v + w) = f (x + t(v + w))

‚t 0

‚ ‚

= f (x + tv + 0w) + f (x + 0v + tw) = df (x)v + df (x)w.

‚t 0 ‚t 0

So we see that d§ : C ∞ (U, F ) — U ’ L(E, F ) is smooth, and the mapping d :

C ∞ (U, F ) ’ C ∞ (U, L(E, F )) is smooth by (3.12) and obviously linear.

We ¬rst prove the chain rule for a smooth curve c instead of g. We have to show

that the curve

f (c(t))’f (c(0))

for t = 0

t

t’

df (c(0)).c (0) for t = 0