And C ω (U, R) can be considered as closed subspace of the product of the fac-

tors C ω (R, R) indexed by all c ∈ C ω (R, U ) and the factors C ∞ (R, R) indexed by

all c ∈ C ∞ (R, U ). Since all factors are convenient so are the closed subspaces.

11.12. Theorem (General real analytic uniform boundedness principle).

Let E and F be convenient vector spaces and U ⊆ E be c∞ -open. Then C ω (U, F )

satis¬es the uniform S-boundedness principle, where S := {evx : x ∈ U }.

Proof. The convenient structure of C ω (U, F ) is induced by the cone of mappings

c— : C ω (U, F ) ’ C ω (R, F ) (c ∈ C ω (R, U )) together with the maps c— : C ω (U, F ) ’

C ∞ (R, F ) (c ∈ C ∞ (R, U )). Both spaces C ω (R, F ) and C ∞ (R, F ) satisfy the uni-

form T -boundedness principle, where T := {evt : t ∈ R}, by (11.6) and (5.26),

respectively. Hence, C ω (U, F ) satis¬es the uniform S-boundedness principle by

lemma (5.25), since evt —¦ c— = evc(t) .

11.12

110 Chapter II. Calculus of holomorphic and real analytic mappings 11.16

11.13. Remark. Let E and F be convenient vector spaces. Then L(E, F ), the

space of bounded linear mappings from E to F, are by (9.7) exactly the real analytic

ones.

11.14. Theorem. Let Ei for i = 1, . . . n and F be convenient vector spaces. Then

the bornology on L(E, . . . , En ; F ) (described in (5.1), see also (5.6)) is induced by

the embedding L(E1 , . . . , En ; F ) ’ C ω (E1 — . . . En , F ).

Thus, mapping f into L(E1 , . . . , En ; F ) is real analytic if and only if the composites

evx —¦ f are real analytic for all x ∈ E1 — . . . En , by (9.9).

Proof. Let S = {evx : x ∈ E1 — . . . — En }. Since C ω (E1 — . . . — En , F ) satis¬es

the uniform S-boundedness principle (11.12), the inclusion is bounded. On the

other hand L(E1 , . . . , En ; F ) also satis¬es the uniform S-boundedness principle by

(5.18), so the identity from L(E1 , . . . , En ; F ) with the bornology induced from

C ω (E1 — . . . — En , F ) into L(E1 , . . . , En ; F ) is bounded as well.

Since to be real analytic depends only on the bornology by (9.4) and since the conve-

nient vector space L(E1 , . . . , En ; F ) satis¬es the uniform S-boundedness principle,

the second assertion follows also.

The following two results will be generalized in (11.20). At the moment we will

make use of the following lemma only in case where E = C ∞ (R, R).

11.15. Lemma. For any convenient vector space E the ¬‚ip of variables induces

an isomorphism L(E, C ω (R, R)) ∼ C ω (R, E ) as vector spaces.

=

Proof. For c ∈ C ω (R, E ) consider c(x) := evx —¦c ∈ C ω (R, R) for x ∈ E. By the

˜

uniform S-boundedness principle (11.6) for S = {evt : t ∈ R} the linear mapping c

˜

is bounded, since evt —¦˜ = c(t) ∈ E .

c

If conversely ∈ L(E, C ω (R, R)), we consider ˜(t) = evt —¦ ∈ E = L(E, R) for

t ∈ R. Since the bornology of E is generated by S := {evx : x ∈ E}, ˜ : R ’ E is

real analytic, for evx —¦ ˜ = (x) ∈ C ω (R, R), by (11.14).

Corollary. We have C ∞ (R, C ω (R, R)) ∼ C ω (R, C ∞ (R, R)) as vector

11.16. =

spaces.

Proof. The dual C ∞ (R, R) is the free convenient vector space over R by (23.11),

and C ω (R, R) is convenient, so we have

C ∞ (R, C ω (R, R)) ∼ L(C ∞ (R, R) , C ω (R, R))

=

∼ C ω (R, C ∞ (R, R) ) by lemma (11.15)

=

∼ C ω (R, C ∞ (R, R)),

=

by re¬‚exivity of C ∞ (R, R), see (6.5.7).

11.16

11.18 11. The real analytic exponential law 111

11.17. Theorem. Let E be a convenient vector space, let U be c∞ -open in E,

let f : R — U ’ R be a real analytic mapping and let c ∈ C ∞ (R, U ). Then

c— —¦ f ∨ : R ’ C ω (U, R) ’ C ∞ (R, R) is real analytic.

This result on the mixing of C ∞ and C ω will become quite essential in the proof

of cartesian closedness. It will be generalized in (11.21), see also (42.15).

Proof. Let I ⊆ R be open and relatively compact, let t ∈ R and k ∈ N. Now

¯

choose an open and relatively compact J ⊆ R containing the closure I of I. There

is a bounded subset B ⊆ E such that c | J : J ’ EB is a Lipk -curve in the Banach

space EB generated by B, by (1.8). Let UB denote the open subset U © EB of the

Banach space EB . Since the inclusion EB ’ E is continuous, f is real analytic as

a function R — UB ’ R — U ’ R. Thus, by (10.1) there is a holomorphic extension

f : V — W ’ C of f to an open set V — W ⊆ C — (EB )C containing the compact

¯

set {t} — c(I). By cartesian closedness of the category of holomorphic mappings

f ∨ : V ’ H(W, C) is holomorphic. Now recall that the bornological structure of

H(W, C) is induced by that of C ∞ (W, C) := C ∞ (W, R2 ). And c— : C ∞ (W, C) ’

Lipk (I, C) is a bounded C-linear map, by the chain rule (12.8) for Lipk -mappings

and by the uniform boundedness principle for the point evaluations (12.9). Thus,

c— —¦ f ∨ : V ’ Lipk (I, C) is holomorphic, and hence its restriction to R © V , which

has values in Lipk (I, R), is (even topologically) real analytic by (9.5). Since t ∈ R

was arbitrary we conclude that c— —¦ f ∨ : R ’ Lipk (I, R) is real analytic. But

the bornology of C ∞ (R, R) is generated by the inclusions into Lipk (I, R), by the

uniform boundedness principles (5.26) for C ∞ (R, R) and (12.9) for Lipk (R, R), and

hence c— —¦ f ∨ : R ’ C ∞ (R, R) is real analytic.

11.18. Theorem. Cartesian closedness. The category of real analytic map-

pings between convenient vector spaces is cartesian closed. More precisely, for con-

venient vector spaces E, F and G and c∞ -open sets U ⊆ E and W ⊆ G a mapping

f : W — U ’ F is real analytic if and only if f ∨ : W ’ C ω (U, F ) is real analytic.

Proof. Step 1. The theorem is true for W = G = F = R.

(⇐) Let f ∨ : R ’ C ω (U, R) be C ω . We have to show that f : R — U ’ R is C ω .

We consider a curve c1 : R ’ R and a curve c2 : R ’ U .

If the ci are C ∞ , then c— —¦ f ∨ : R ’ C ω (U, R) ’ C ∞ (R, R) is C ω by assumption,

2

hence is C , so c2 —¦ f —¦ c1 : R ’ C ∞ (R, R) is C ∞ . By cartesian closedness of

∞ — ∨

smooth mappings, (c— —¦ f ∨ —¦ c1 )§ = f —¦ (c1 — c2 ) : R2 ’ R is C ∞ . By composing

2

with the diagonal mapping ∆ : R ’ R2 we obtain that f —¦ (c1 , c2 ) : R ’ R is C ∞ .

If the ci are C ω , then c— —¦ f ∨ : R ’ C ω (U, R) ’ C ω (R, R) is C ω by assumption,

2

— ∨

so c2 —¦ f —¦ c1 : R ’ C ω (R, R) is C ω . By theorem (11.7) the associated map

(c— —¦ f ∨ —¦ c1 )§ = f —¦ (c1 — c2 ) : R2 ’ R is C ω . So f —¦ (c1 , c2 ) : R ’ R is C ω .

2

(’) Let f : R — U ’ R be C ω . We have to show that f ∨ : R ’ C ω (U, R) is real

analytic. Obviously, f ∨ has values in this space. We consider a curve c : R ’ U .

If c is C ∞ , then by theorem (11.17) the associated mapping c— —¦f ∨ : R ’ C ∞ (R, R)

is C ω .

11.18

112 Chapter II. Calculus of holomorphic and real analytic mappings 11.19

If c is C ω , then f —¦ (Id —c) : R — R ’ R — U ’ R is C ω . By theorem (11.7) the

associated mapping (f —¦ (Id —c))∨ = c— —¦ f ∨ : R ’ C ω (R, R) is C ω .

Step 2. The theorem is true for F = R.

(⇐) Let f ∨ : W ’ C ω (U, R) be C ω . We have to show that f : W — U ’ R is C ω .

We consider a curve c1 : R ’ W and a curve c2 : R ’ U .

If the ci are C ∞ , then c— —¦ f ∨ : W ’ C ω (U, R) ’ C ∞ (R, R) is C ω by assumption,

2

hence is C , so c2 —¦ f —¦ c1 : R ’ C ∞ (R, R) is C ∞ . By cartesian closedness of

∞ — ∨

smooth mappings, the associated mapping (c— —¦ f ∨ —¦ c1 )§ = f —¦ (c1 — c2 ) : R2 ’ R

2

∞ ∞

is C . So f —¦ (c1 , c2 ) : R ’ R is C .

If the ci are C ω , then f ∨ —¦ c1 : R ’ W ’ C ω (U, R) is C ω by assumption, so

by step 1 the mapping (f ∨ —¦ c1 )§ = f —¦ (c1 — IdU ) : R — U ’ R is C ω . Hence,

f —¦ (c1 , c2 ) = f —¦ (c1 — IdU ) —¦ (Id, c2 ) : R ’ R is C ω .

(’) Let f : W — U ’ R be C ω . We have to show that f ∨ : W ’ C ω (U, R) is real

analytic. Obviously, f ∨ has values in this space. We consider a curve c1 : R ’ W .

If c1 is C ∞ , we consider a second curve c2 : R ’ U . If c2 is C ∞ , then f —¦ (c1 — c2 ) :

R — R ’ W — U ’ R is C ∞ . By cartesian closedness the associated mapping

(f —¦ (c1 — c2 ))∨ = c— —¦ f ∨ —¦ c1 : R ’ C ∞ (R, R) is C ∞ . If c2 is C ω , the mapping

2

f —¦ (IdW —c2 ) : W — R ’ R and also the ¬‚ipped one (f —¦ (IdW —c2 ))∼ : R — W ’ R

are C ω , hence by theorem (11.17) c— —¦ ((f —¦ (IdW —c2 ))∼ )∨ : R ’ C ∞ (R, R) is

1

C . By corollary (11.16) the associated mapping (c— —¦ ((f —¦ (IdW —c2 ))∼ )∨ )∼ =

ω

1

— ∨ ∞

ω

c2 —¦ f —¦ c1 : R ’ C (R, R) is C . So for both families describing the structure of

ˇ

C ω (U, R) we have shown that the composite with f —¦ c1 is C ∞ , so f ∨ —¦ c1 is C ∞ .

If c1 is C ω , then f —¦ (c1 — IdU ) : R — U ’ W — U ’ R is C ω . By step 1 the

associated mapping (f —¦ (c1 — IdU ))∨ = f ∨ —¦ c1 : R ’ C ω (U, R) is C ω .

Step 3. The general case.

f : W — U ’ F is C ω

» —¦ f : W — U ’ R is C ω for all » ∈ F

”

(» —¦ f )∨ = »— —¦ f ∨ : W ’ C ω (U, R) is C ω , by step 2 and (11.10)

”

f ∨ : W ’ C ω (U, F ) is C ω .

”

11.19. Corollary. Canonical mappings are real analytic. The following

mappings are C ω :

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

(1)

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

(2)

( )§ : C ω (U, C ω (V, G)) ’ C ω (U — V, G),

(3)

( )∨ : C ω (U — V, G) ’ C ω (U, C ω (V, G)),

(4)

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

(5)

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

(6)

’ C ω (C ω (E1 , F1 ), C ω (E2 , F2 )), (f, g) ’ (h ’ g —¦ h —¦ f ).

11.19

11.20 11. The real analytic exponential law 113

Proof. Just consider the canonically associated smooth mappings on multiple

products, as in (3.13).

11.20. Lemma. Canonical isomorphisms. One has the following natural iso-

morphisms:

(1) C ω (W1 , C ω (W2 , F )) ∼ C ω (W2 , C ω (W1 , F )),

=

(2) C ω (W1 , C ∞ (W2 , F )) ∼ C ∞ (W2 , C ω (W1 , F )).

=

(3) C ω (W1 , L(E, F )) ∼ L(E, C ω (W1 , F )).

=

(4) C (W1 , (X, F )) ∼ ∞ (X, C ω (W1 , F )).

∞

ω

=

(5) C ω (W1 , Lipk (X, F )) ∼ Lipk (X, C ω (W1 , F )).

=

In (4) the space X is a ∞ -space, i.e. a set together with a bornology induced by a

family of real valued functions on X, cf. [Fr¨licher, Kriegl, 1988, 1.2.4]. In (5) the

o

k

space X is a Lip -space, cf. [Fr¨licher, Kriegl, 1988, 1.4.1]. The spaces ∞ (X, F )

o

and Lipk (W, F ) are de¬ned in [Fr¨licher, Kriegl, 1988, 3.6.1 and 4.4.1].

o

Proof. All isomorphisms, as well as their inverse mappings, are given by the ¬‚ip of

˜ ˜

coordinates: f ’ f , where f (x)(y) := f (y)(x). Furthermore, all occurring function

spaces are convenient and satisfy the uniform S-boundedness theorem, where S is

the set of point evaluations, by (11.11), (11.14), (11.12), and by [Fr¨licher, Kriegl,

o

1988, 3.6.1, 4.4.2, 3.6.6, and 4.4.7].

˜ ˜

That f has values in the corresponding spaces follows from the equation f (x) =

˜

evx —¦ f . One only has to check that f itself is of the corresponding class, since it

˜

follows that f ’ f is bounded. This is a consequence of the uniform boundedness

principle, since

(evx —¦( ˜ ))(f ) = evx (f ) = f (x) = evx —¦f = (evx )— (f ).

˜ ˜

˜

That f is of the appropriate class in (1) and (2) follows by composing with c1 ∈

C β1 (R, W1 ) and C β2 (», c2 ) : C ±2 (W2 , F ) ’ C β2 (R, R) for all » ∈ F and c2 ∈

C β2 (R, W2 ), where βk and ±k are in {∞, ω} and βk ¤ ±k for k ∈ {1, 2}. Then

˜

C β2 (», c2 ) —¦ f —¦ c1 = (C β1 (», c1 ) —¦ f —¦ c2 )∼ : R ’ C β2 (R, R) is C β1 by (11.7) and

(11.16), since C β1 (», c1 ) —¦ f —¦ c2 : R ’ W2 ’ C ±1 (W1 , F ) ’ C β1 (R, R) is C β2 .

˜

That f is of the appropriate class in (3) follows, since L(E, F ) is the c∞ -closed

subspace of C ω (E, F ) formed by the linear C ω -mappings.

˜

That f is of the appropriate class in (4) or (5) follows from (3), using the free

convenient vector spaces 1 (X) or »k (X) over the ∞ -space X or the the Lipk -space

X, see [Fr¨licher, Kriegl, 1988, 5.1.24 or 5.2.3], satisfying ∞ (X, F ) ∼ L( 1 (X), F )

o =

or satisfying Lip (X, F ) ∼ L(»k (X), F ). Existence of these free convenient vector

k

=

spaces can be proved in a similar way as (23.6).

De¬nition. By a C ∞,ω -mapping f : U — V ’ F we mean a mapping f for which

f ∨ ∈ C ∞ (U, C ω (V, F )).

11.20

114 Chapter II. Calculus of holomorphic and real analytic mappings 11.23

11.21. Theorem. Composition of C ∞,ω -mappings. Let f : U — V ’ F and

g : U1 —V1 ’ V be C ∞,ω , and h : U1 ’ U be C ∞ . Then f —¦(h—¦pr1 , g) : U1 —V1 ’ F ,

(x, y) ’ f (h(x), g(x, y)) is C ∞,ω .

Proof. We have to show that the mapping x ’ (y ’ f (h(x), g(x, y))), U1 ’

C ω (V1 , F ) is C ∞ . It is well-de¬ned, since f and g are C ω in the second variable. In

order to show that it is C ∞ we compose with »— : C ω (V1 , F ) ’ C ω (V1 , R), where

» ∈ F is arbitrary. Thus, it is enough to consider the case F = R. Furthermore,

we compose with c— : C ω (V1 , R) ’ C ± (R, R), where c ∈ C ± (R, V1 ) is arbitrary for

± equal to ω and ∞.

In case ± = ∞ the composite with c— is C ∞ , since the associated mapping U1 —R ’

R is f —¦ (h —¦ pr1 , g —¦ (id — c)) which is C ∞ .

Now the case ± = ω. Let I ⊆ R be an arbitrary open bounded interval. Then

c— —¦ g ∨ : U1 ’ C ω (R, G) is C ∞ , where G is the convenient vector space containing

¯

V as an c∞ -open subset, and has values in {γ : γ(I) ⊆ V } ⊆ C ω (R, G). This set is

c∞ -open, since it is open for the topology of uniform convergence on compact sets

which is coarser than the bornological topology on C ∞ (R, E) and hence than the

c∞ -topology on C ω (R, G), see (11.10).

Thus, the composite with c— , comp —¦(f ∨ —¦ h, c— —¦ g ∨ ) is C ∞ , since f ∨ —¦ h : U1 ’

U ’ C ω (V, F ) is C ∞ , c— —¦ g ∨ : U1 ’ C ω (R, G) is C ∞ and comp : C ω (V, R) — {γ ∈

¯

C ω (R, G) : γ(I) ⊆ V } ’ C ω (I, R) is C ω , because it is associated to ev —¦(id — ev) :

¯ ¯

C ω (V, F ) — {γ ∈ C ω (R, G) : γ(I) ⊆ V } — I ’ V . That ev : {γ ∈ C ω (R, G) : γ(I) ⊆

V } — I ’ R is C ω follows, since the associated mapping is the restriction mapping

C ω (R, G) ’ C ω (I, G).

11.22. Corollary. Let w : W1 ’ W be C ω , let u : U ’ U1 be smooth, let v : V ’

V1 be C ω , and let f : U1 — V1 ’ W1 be C ∞,ω . Then w —¦ f —¦ (u — v) : U — V ’ W

is again C ∞,ω .

This is generalization of theorem (11.17).

Proof. Use (11.21) twice.

11.23. Corollary. Let f : E ⊇ U ’ F be C ω , let I ⊆ R be open and bounded,

¯

and ± be ω or ∞. Then f— : C ± (R, E) ⊇ {c : c(I) ⊆ U } ’ C ± (I, F ) is C ω .

Proof. Obviously, f— (c) := f —¦ c ∈ C ± (I, F ) is well-de¬ned for all c ∈ C ± (R, E)

¯

satisfying c(I) ⊆ U .

¯

Furthermore, the composite of f— with any C β -curve γ : R ’ {c : c(I) ⊆ U } ⊆

C ± (R, E) is a C β -curve in C ± (I, F ) for β equal to ω or ∞. For β = ± this follows

from cartesian closedness of the C ± -maps. For ± = β this follows from (11.22).

¯

Finally, {c : c(I) ⊆ U } ⊆ C ± (R, E) is c∞ -open, since it is open for the topology

of uniform convergence on compact sets which is coarser than the bornological and

hence than the c∞ -topology on C ± (R, E). Here is the only place where we make

use of the boundedness of I.

11.23

11.26 11. The real analytic exponential law 115

d

dt |t=0

11.24. Lemma. Derivatives. The derivative d, where df (x)(v) := f (x +

tv), is bounded and linear d : C ω (U, F ) ’ C ω (U, L(E, F )).

Proof. The di¬erential df (x)(v) makes sense and is linear in v, because every real

analytic mapping f is smooth. So it remains to show that (f, x, v) ’ df (x)(v) is

real analytic. So let f , x, and v depend real analytically (resp. smoothly) on a

real parameter s. Since (t, s) ’ x(s) + tv(s) is real analytic (resp. smooth) into

U ⊆ E, the mapping r ’ ((t, s) ’ f (r)(x(s)+tv(s)) is real analytic into C ω (R2 , F )

(resp. smooth into C ∞ (R2 , F ). Composing with ‚t |t=0 : C ω (R2 , F ) ’ C ω (R, F )

‚

(resp. : C ∞ (R2 , F ) ’ C ∞ (R, F )) shows that r ’ (s ’ d(f (r))(x(s))(v(s))), R ’

C ω (R, F ) is real analytic. Considering the associated mapping on R2 composed

with the diagonal map shows that (f, x, v) ’ df (x)(v) is real analytic.

The following examples as well as several others can be found in [Fr¨licher, Kriegl,

o

1988, 5.3.6].

11.25. Example. Let T : C ∞ (R, R) ’ C ∞ (R, R) be given by T (f ) = f . Then the

continuous linear di¬erential equation x (t) = T (x(t)) with initial value x(0) = x0

has a unique smooth solution x(t)(s) = x0 (t + s) which is however not real analytic.

Note the curious form x (t) = x(t) of this di¬erential equation. Beware of careless

notation!

Proof. A smooth curve x : R ’ C ∞ (R, R) is a solution of the di¬erential equation

‚ ‚ d

x (t) = T (x(t)) if and only if ‚t x(t, s) = ‚s x(t, s). Hence, we have dt x(t, r ’ t) = 0,

ˆ ˆ ˆ

i.e. x(t, r ’ t) is constant and hence equal to x(0, r) = x0 (r). Thus, x(t, s) =

ˆ ˆ ˆ

x0 (t + s).

Suppose x : R ’ C ∞ (R, R) were real analytic. Then the composite with ev0 :

C ∞ (R, R) ’ R were a real analytic function. But this composite is just x0 = ev0 —¦x,

which is not in general real analytic.

11.26. Example. Let E be either C ∞ (R, R) or C ω (R, R). Then the mapping

exp— : E ’ E is C ω , has invertible derivative at every point, but the image does

not contain an open neighborhood of exp— (0).

Proof. The mapping exp— is real analytic by (11.23). Its derivative is given by

(exp— ) (f )(g) : t ’ g(t)ef (t) and hence is invertible with g ’ (t ’ g(t)e’f (t) )

as inverse mapping. Now consider the real analytic curve c : R ’ E given by

c(t)(s) = 1 ’ (ts)2 . One has c(0) = 1 = exp— (0), but c(t) is not in the image of

exp— for any t = 0, since c(t)( 1 ) = 0 but exp— (g)(t) = eg(t) > 0 for all g and t.

t

11.26

116 Chapter II. Calculus of holomorphic and real analytic mappings

Historical Remarks on Holomorphic

and Real Analytic Calculus

The notion of holomorphic mappings used in section (15) was ¬rst de¬ned by the

Italian Luigi Fantappi´ in the papers [Fantappi´, 1930] and [Fantappi´, 1933]:

e e e

S.1: “Wenn jeder Funktion y(t) einer Funktionenmenge H eine bestimmte Zahl f entspricht,

d.h. die Zahl f von der Funktion y(t) (unabh¨ngige Ver¨nderliche in der Menge H) abh¨ngt,

a a a

werden wir sagen, daß ein Funktional von y(t):

f = F [y(t)]

ist; H heißt das De¬nitionsfeld des Funktionals F .

[ . . . ] gemischtes Funktional [ . . . ]

f = F [y1 (t1 , . . . ), . . . , yn (t1 , . . . ); z1 , . . . , zm ]”

He also considered the ˜functional transform™ and noticed the relation

f = F [y(t); z] corresponds to y ’ f (z)

S.4: “Sei jetzt F (y(t)) ein Funktional, das in einem Funktionenbereich H (von analytischen

Funktionen) de¬niert ist, und y0 (t) ein Funktion von H, die mit einer Umgebung (r) oder