9.7. Theorem. Linear real analytic mappings. Let E and F be convenient

vector spaces. For any linear mapping » : E ’ F the following assertions are

equivalent.

(1) » is bounded.

(2) » —¦ c : R ’ F is real analytic for all real analytic c : R ’ E.

(3) »—¦c : R ’ F is bornologically real analytic for all bornologically real analytic

curves c : R ’ E

(4) » —¦ c : R ’ F is real analytic for all bornologically real analytic curves

c:R’E

This will be generalized in (10.4) to non-linear mappings.

9.7

10.1 10. Real analytic mappings 101

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

(4) ’ (1) Let » satisfy (4) and suppose that » is unbounded. By composing with

an ∈ E we may assume that » : E ’ R and there is a bounded sequence

(xk ) such that »(xk ) is unbounded. By passing to a subsequence we may suppose

that |»(xk )| > k 2k . Let ak := k ’k xk , then (rk ak ) is bounded and (rk »(ak )) is

∞

unbounded for all r > 0. Hence, the curve c(t) := k=0 tk ak is given by a Mackey

convergent power series. So » —¦ c is real analytic and near 0 we have »(c(t)) =

∞ N

k k N k’N

k=0 bk t for some bk ∈ R. But »(c(t)) = k=0 »(ak )t + t »( k>N ak t )

and t ’ k>N ak tk’N is still a Mackey converging power series in E. Comparing

coe¬cients we see that bk = »(ak ) and consequently »(ak )rk is bounded for some

r > 0, a contradiction.

Proof of (1) ’ (2) Let c : R ’ E be real analytic. By theorem (9.3) the set

1

{ k! c(k) (a) rk : a ∈ K, k ∈ N} is bounded for all compact sets K ‚ R and for all

sequences (rk ) with rk tk ’ 0 for all t > 0. Since c is smooth and bounded linear

mappings are smooth by (2.11), the function » —¦ c is smooth and (» —¦ c)(k) (a) =

»(c(k) (a)). By applying (9.3) we obtain that » —¦ c is real analytic.

9.8. Corollary. For two convenient vector space structures on a vector space E

the following statements are equivalent:

(1) They have the same bounded sets.

(2) They have the same smooth curves.

(3) They have the same real analytic curves.

Proof. (1) ” (2) was shown in (2.11). The implication (1) ’ (3) follows from (9.3),

which shows that real analyticity is a bornological concept, whereas the implication

(1) ⇐ (3) follows from (9.7).

9.9. Corollary. If a cone of linear maps T± : E ’ E± between convenient vector

spaces generates the bornology on E, then a curve c : R ’ E is C ω resp. C ∞

provided all the composites T± —¦ c : R ’ E± are.

Proof. The statement on the smooth curves is shown in (3.8). That on the real

analytic curves follows again from the bornological condition of (9.3).

10. Real Analytic Mappings

10.1. Theorem (Real analytic functions on Fr´chet spaces). Let U ⊆ E

e

be open in a real Fr´chet space E. The following statements on f : U ’ R are

e

equivalent:

(1) f is smooth and is real analytic along topologically real analytic curves.

(2) f is smooth and is real analytic along a¬ne lines.

(3) f is smooth and is locally given by its pointwise converging Taylor series.

(4) f is smooth and is locally given by its uniformly and absolutely converging

Taylor series.

10.1

102 Chapter II. Calculus of holomorphic and real analytic mappings 10.4

(5) f is locally given by a uniformly and absolutely converging power series.

˜˜ ˜

(6) f extends to a holomorphic mapping f : U ’ C for an open subset U in the

˜

complexi¬cation EC with U © E = U .

Proof. (1) ’ (2) is obvious. The implication (2) ’ (3) follows from (7.14), (1)

’ (2), whereas (3) ’ (4) follows from (7.14),(2) ’ (3), and (4) ’ (5) is obvious.

Proof of (5) ’ (6) Locally we can extend converging power series into the complex-

˜

i¬cation by (7.14). Then we take the union U of their domains of de¬nition and

˜

use uniqueness to glue f which is holomorphic by (7.24).

Proof of (6) ’ (1) Obviously, f is smooth. Any topologically real analytic curve c

in E can locally be extended to a holomorphic curve in EC by (9.5). So f —¦ c is real

analytic.

10.2. The assumptions ˜f is smooth™ cannot be dropped in (10.1.1) even in ¬nite

dimensions, as shown by the following example, due to [Boman, 1967].

n+2

xy

Example. The mapping f : R2 ’ R, de¬ned by f (x, y) := x2 +y2 is real analytic

along real analytic curves, is n-times continuous di¬erentiable but is not smooth

and hence not real analytic.

Proof. Take a real analytic curve t ’ (x(t), y(t)) into R2 . The components can be

factored as x(t) = tk u(t), y(t) = tk v(t) for some k and real analytic curves u, v with

uv n+2

u(0)2 +v(0)2 = 0. The composite f —¦(x, y) is then the function t ’ tk(n+1) u2 +v2 (t),

which is obviously real analytic near 0. The mapping f is n-times continuous

di¬erentiable, since it is real analytic on R2 \{0} and the directional derivatives of

order i are (n + 1 ’ i)-homogeneous, hence continuously extendable to R2 . But f

cannot be (n + 1)-times continuous di¬erentiable, otherwise the derivative of order

n + 1 would be constant, and hence f would be a polynomial.

10.3. De¬nition (Real analytic mappings). Let E be a convenient vector

space. Let us denote by C ω (R, E) the space of all real analytic curves.

Let U ⊆ E be c∞ -open, and let F be a second convenient vector space. A mapping

f : U ’ F will be called real analytic or C ω for short, if f is real analytic along

real analytic curves and is smooth (i.e. is smooth along smooth curves); so f —¦

c ∈ C ω (R, F ) for all c ∈ C ω (R, E) with c(R) ⊆ U and f —¦ c ∈ C ∞ (R, F ) for all

c ∈ C ∞ (R, E) with c(R) ⊆ U . Let us denote by C ω (U, F ) the space of all real

analytic mappings from U to F .

10.4. Analogue of Hartogs™ Theorem for real analytic mappings. Let E

and F be convenient vector spaces, let U ⊆ E be c∞ -open, and let f : U ’ F .

Then f is real analytic if and only if f is smooth and » —¦ f is real analytic along

each a¬ne line in E, for all » ∈ F .

Proof. One direction is clear, and by de¬nition (10.3) we may assume that F = R.

Let c : R ’ U be real analytic. We show that f —¦ c is real analytic by using theorem

(9.3). So let (rk ) be a sequence such that rk r ≥ rk+ and rk tk ’ 0 for all t > 0

10.4

10.4 10. Real analytic mappings 103

and let K ‚ R be compact. We have to show, that there is an µ > 0 such that the

set { 1! (f —¦ c)( ) (a) rl ( 2 ) : a ∈ K, ∈ N} is bounded.

µ

1

By theorem (9.3) the set { n! c(n) (a) rn : n ≥ 1, a ∈ K} is contained in some bounded

absolutely convex subset B ⊆ E, such that EB is a Banach space. Clearly, for the

inclusion iB : EB ’ E the function f —¦ iB is smooth and real analytic along a¬ne

lines. Since EB is a Banach space, by (10.1.2) ’ (10.1.4) f —¦ iB is locally given

by its uniformly and absolutely converging Taylor series. Then for each a ∈ K by

1

(7.14.2) ’ (7.14.4) there is an µ > 0 such that the set { k! dk f (c(a))(x1 , . . . , xk ) :

k ∈ N, xj ∈ µB} is bounded. For each y ∈ 1 µB termwise di¬erentiation gives

2

1

p k

d f (c(a) + y)(x1 , . . . , xp ) = k≥p (k’p)! d f (c(a))(x1 , . . . , xp , y, . . . , y), so we may

assume that {dk f (c(a))(x1 , . . . , xk )/k! : k ∈ N, xj ∈ µB, a ∈ K} is contained in

[’C, C] for some C > 0 and some uniform µ > 0.

The Taylor series of f —¦ c at a is given by

k!

dk f (c(a)) ( n! c(n) (a))mn t ,

1 1

(f —¦ c)(a + t) = k!

n mn ! n

≥0 k≥0 (mn )∈NN0

mn =k

n

n mn n=

xmn := (x1 , . . . , x1 , . . . , xn , . . . , xn , . . . ).

where n

n

m1 mn

This follows easily from composing the Taylor series of f and c and ordering by

powers of t. Furthermore, we have

k! ’1

= k’1

mn !

n

(mn )∈NN0

mn =k

n

n mn n=

by the following argument: It is the -th Taylor coe¬cient at 0 of the function

∞

( n≥0 tn ’ 1)k = ( 1’t )k = tk j=0 ’k (’t)j , which turns out to be the binomial

t

j

coe¬cient in question.

By the foregoing considerations we may estimate as follows.

—¦ c)( ) (a)| rl ( 2 ) ¤

1 µ

! |(f

k!

dk f (c(a)) ( n! c(n) (a))mn

1 1 µ

¤ r (2)

k!

n mn ! n

k≥0 N

(mn )∈N0

n mn =k

n mn n=

k!

dk f (c(a)) ( n! c(n) (a) rn µn )mn

1 1 1

¤ k! 2

n mn ! n

k≥0 (mn )∈NN0

mn =k

n

n mn n=

’1

C 2 = 1 C,

1

¤ k’1 2

k≥0

10.4

104 Chapter II. Calculus of holomorphic and real analytic mappings 10.8

because

k! ’1

( n! c(n) (a) µn rn )mn ∈

1

(µB)k ⊆ (EB )k .

k’1

n mn ! n

(mn )∈NN0

mn =k

n

n mn n=

10.5. Corollary. Let E and F be convenient vector spaces, let U ⊆ E be c∞ -open,

and let f : U ’ F . Then f is real analytic if and only if f is smooth and » —¦ f —¦ c

is real analytic for every periodic (topologically) real analytic curve c : R ’ U ⊆ E

and all » ∈ F .

Proof. By (10.4) f is real analytic if and only if f is smooth and » —¦ f is real

analytic along topologically real analytic curves c : R ’ E. Let h : R ’ R be

de¬ned by h(t) = t0 + µ · sin t. Then c —¦ h : R ’ R ’ U is a (topologically)

real analytic, periodic function with period 2π, provided c is (topologically) real

analytic. If c(t0 ) ∈ U we can choose µ > 0 such that h(R) ⊆ c’1 (U ). Since sin is

locally around 0 invertible, real analyticity of » —¦ f —¦ c —¦ h implies that » —¦ f —¦ c is

real analytic near t0 . Hence, the proof is completed.

10.6. Corollary. Reduction to Banach spaces. Let E be a convenient vector

space, let U ⊆ E be c∞ -open, and let f : U ’ R be a mapping. Then f is real

analytic if and only if the restriction f : EB ⊇ U © EB ’ R is real analytic for all

bounded absolutely convex subsets B of E.

So any result valid on Banach spaces can be translated into a result valid on con-

venient vector spaces.

Proof. By theorem (10.4) it su¬ces to check f along bornologically real analytic

curves. These factor by de¬nition (9.4) locally to real analytic curves into some

EB .

10.7. Corollary. Let U be a c∞ -open subset in a convenient vector space E and

let f : U ’ R be real analytic. Then for every bounded B there is some rB > 0

1k k

such that the Taylor series y ’ k! d f (x)(y ) converges to f (x + y) uniformly

and absolutely on rB B.

Proof. Use (10.6) and (10.1.4).

10.8. Scalar analytic functions on convenient vector spaces E are in general not

germs of holomorphic functions from EC to C:

Example. Let fk : R ’ R be real analytic functions with radius of convergence at

zero converging to 0 for k ’ ∞. Let f : R(N) ’ R be the mapping de¬ned on the

∞

countable sum R(N) of the reals by f (x0 , x1 , . . . ) := k=1 xk fk (x0 ). Then f is real

10.8

11.1 11. The real analytic exponential law 105

˜

analytic, but there is no complex valued holomorphic mapping f on some neigh-

borhood of 0 in C(N) which extends f , and the Taylor series of f is not pointwise

convergent on any c∞ -open neighborhood of 0.

Proof. Claim. f is real analytic.

Since the limit R(N) = ’ n Rn is regular, every smooth curve (and hence every real

lim

’

(N)

is locally smooth (resp. real analytic) into Rn for some n.

analytic curve) in R

Hence, f —¦ c is locally just a ¬nite sum of smooth (resp. real analytic) functions

and is therefore smooth (resp. real analytic).

Claim. f has no holomorphic extension.

˜

Suppose there exists some holomorphic extension f : U ’ C, where U ⊆ C(N) is c∞ -

open neighborhood of 0, and is therefore open in the locally convex Silva topology by

(4.11.2). Then U is even open in the box-topology (52.7), i.e., there exist µk > 0 for

all k, such that {(zk ) ∈ C(N) : |zk | ¤ µk for all k} ⊆ U . Let U0 be the open disk in C

˜ ˜ ˜

with radius µ0 and let fk : U0 ’ C be de¬ned by fk (z) := f (z, 0, . . . , 0, µk , 0, . . . ) µ1 ,

k

˜ is an extension of

where µk is inserted instead of the variable xk . Obviously, fk

fk , which is impossible, since the radius of convergence of fk is less than µ0 for k

su¬ciently large.

Claim. The Taylor series does not converge.

If the Taylor series would be pointwise convergent on some U , then the previous

arguments for R(N) instead of C(N) would show that the radii of convergence of the

fk were bounded from below.

11. The Real Analytic Exponential Law

11.1. Spaces of germs of real-analytic functions. Let M be a real analytic

¬nite dimensional manifold. If f : M ’ M is a mapping between two such

manifolds, then f is real analytic if and only if f maps smooth curves into smooth

ones and real analytic curves into real analytic ones, by (10.1).

For each real analytic manifold M of real dimension m there is a complex manifold

MC of complex dimension m containing M as a real analytic closed submanifold,

whose germ along M is unique ([Whitney, Bruhat, 1959, Prop. 1]), and which can

be chosen even to be a Stein manifold, see [Grauert, 1958, section 3]. The complex

charts are just extensions of the real analytic charts of an atlas of M into the

complexi¬cation of the modeling real vector space.

Real analytic mappings f : M ’ M are the germs along M of holomorphic

mappings W ’ MC for open neighborhoods W of M in MC .

Let C ω (M, F ) be the space of real analytic functions f : M ’ F , for any convenient

vector space F , and let H(MC ⊇ M, C) be the space of germs along M of holomor-

phic functions as in (8.3). Furthermore, for a subset A ⊆ M let C ω (M ⊇ A, R)

denotes the space of germs of real analytic functions along A, de¬ned on some

neighborhood of A.

11.1

106 Chapter II. Calculus of holomorphic and real analytic mappings 11.4

11.2. Lemma. For any subset A of M the complexi¬cation of the real vector space

C ω (M ⊇ A, R) is the complex vector space H(MC ⊇ A, C).

De¬nition. For any A ⊆ M of a real analytic manifold M we will topologize the

space sections C ω (M ⊇ A, R) as subspace of H(MC ⊇ A, C), in fact as the real part

of it.

Proof. Let f, g ∈ C ω (M ⊇ A, R). These are germs of real analytic mappings

de¬ned on some open neighborhood of A in M . Inserting complex numbers into

the locally convergent Taylor series in local coordinates shows, that f and g can be

considered as holomorphic mappings from some neighborhood W of A in MC , which

√

have real values if restricted to W © M . The mapping h := f + ’1g : W ’ C

gives then an element of H(MC ⊇ A, C).

Conversely, let h ∈ H(MC ⊇ A, C). Then h is the germ of a holomorphic function

h : W ’ C for some open neighborhood W of A in MC . The decomposition of h

¯ ¯

into real and imaginary part f = 1 (h + h) and g = 2√’1 (h ’ h), which are real

1

2

analytic functions if restricted to W © M , gives elements of C ω (M ⊇ A, R).

These correspondences are inverse to each other since a holomorphic germ is deter-

mined by its restriction to a germ of mappings M ⊇ A ’ C.

11.3. Lemma. For a ¬nite dimensional real analytic manifold M the inclusion

C ω (M, R) ’ C ∞ (M, R) is continuous.

Proof. Consider the following diagram, where W is an open neighborhood of M

in its complexi¬cation MC .

y wC ∞

inclusion

C ω (M, R) (M, R)

u u

direct summand (11.2) direct summand

y w C (M, R )

u u

∞

inclusion 2

H(MC ⊇ M, C)

restriction (8.4) restriction

y wC ∞

inclusion

(W, R2 )

H(W, C) (8.2)

11.4. Theorem (Structure of C ω (M ⊇ A, R) for closed subsets A of real

analytic manifolds M ). The inductive cone

C ω (M ⊇ A, R) ← { C ω (W, R) : A ⊆ W ⊆ M }

open

is regular, i.e. every bounded set is contained and bounded in some step.

The projective cone

C ω (M ⊇ A, R) ’ { C ω (M ⊇ K, R) : K compact in A}

generates the bornology of C ω (M ⊇ A, R).

11.4

11.6 11. The real analytic exponential law 107

If A is even a smooth submanifold, then the following projective cone also generates

the bornology.

C ω (M ⊇ A, R) ’ { C ω (M ⊇ {x}, R) : x ∈ A}

The space C ω (Rm ⊇ {0}, R) is also the regular inductive limit of the spaces p

∈

r (r

Rm ) for all 1 ¤ p ¤ ∞, see (8.1).

+

For general closed A ⊆ N the space C ω (M ⊇ A, R) is Montel (hence quasi-complete

and re¬‚exive), and ultra-bornological (hence a convenient vector space). It is also

webbed and conuclear. If A is compact then it is even a strongly nuclear Silva space

and its dual is a nuclear Fr´chet space and it is smoothly paracompact. It is however

e

not a Baire space.

Proof. This follows using (11.2) from (8.4), (8.6), and (8.8) by passing to the real

parts and from the fact that all properties are inherited by complemented subspaces

as C ω (M ⊇ A, R) of H(MC ⊇ A, C).

11.5. Corollary. A subset B ⊆ C ω (Rm ⊇ {0}, R) is bounded if and only if there

(±)

exists an r > 0 such that { f (0) |±|

: f ∈ B, ± ∈ Nm } is bounded in R.

±! r 0

Proof. The space C ω (Rm ⊇ {0}, R) is the regular inductive limit of the spaces

∞ m

r for r ∈ R+ by (11.4). Hence, B is bounded if and only if it is contained and

bounded in ∞ for some r ∈ Rm , which is the looked for condition.

r +

11.6. Theorem (Special real analytic uniform boundedness principle).

For any closed subset A ⊆ M of a real analytic manifold M , the space C ω (M ⊇

A, R) satis¬es the uniform S-boundedness principle for any point separating set S

of bounded linear functionals.

If A has no isolated points and M is 1-dimensional this applies to the set of all

point evaluations evt , t ∈ A.

Proof. Again this follows from (5.24) using now (11.4). If A has no isolated points

and M is 1-dimensional the point evaluations are separating, by the uniqueness

theorem for holomorphic functions.

Direct proof of a particular case. We show that C ω (R, R) satis¬es the uniform

S-boundedness principle for the set S of all point evaluations.

We check property (5.22.2). Let B ⊆ C ω (R, R) be absolutely convex such that

evt (B) is bounded for all t and such that C ω (R, R)B is complete. We have to show

that B is complete.

By lemma (11.3) the set B satis¬es the conditions of (5.22.2) in the space C ∞ (R, R).

Since C ∞ (R, R) satis¬es the uniform S-boundedness principle, cf. [Fr¨licher, Kriegl,

o

1988], the set B is bounded in C ∞ (R, R). Hence, all iterated derivatives at points

are bounded on B, and a fortiori the conditions of (5.22.2) are satis¬ed for B in

H(R, C). By the particular case of theorem (8.10) the set B is bounded in H(R, C)

and hence also in the direct summand C ω (R, R).

11.6

108 Chapter II. Calculus of holomorphic and real analytic mappings 11.9

11.7. Theorem. The real analytic curves in C ω (R, R) correspond exactly to the

real analytic functions R2 ’ R.

Proof. (’) Let f : R ’ C ω (R, R) be a real analytic curve. Then f : R ’

C ω (R ⊇ {t}, R) is also real analytic. We use theorems (11.4) and (9.6) to conclude

that f is even a topologically real analytic curve in C ω (R ⊇ {t}, R). By lemma

(9.5) for every s ∈ R the curve f can be extended to a holomorphic mapping from

an open neighborhood of s in C to the complexi¬cation (11.2) H(C ⊇ {t}, C) of

C ω (R ⊇ {t}, R).

From (8.4) it follows that H(C ⊇ {t}, C) is the regular inductive limit of all spaces

H(U, C), where U runs through some neighborhood basis of t in C. Lemma (7.7)

shows that f is a holomorphic mapping V ’ H(U, C) for some open neighborhoods

U of t and V of s in C.

By the exponential law for holomorphic mappings (see (7.22)) the canonically asso-

ciated mapping f § : V — U ’ C is holomorphic. So its restriction is a real analytic

function R — R ’ R near (s, t) which coincides with f § for the original f .

(⇐) Let f : R2 ’ R be a real analytic mapping. Then f (t, ) is real analytic, so

the associated mapping f ∨ : R ’ C ω (R, R) makes sense. It remains to show that

it is real analytic. Since the mappings C ω (R, R) ’ C ω (R ⊇ K, R) generate the

bornology, by (11.4), it is by (9.9) enough to show that f ∨ : R ’ C ω (R ⊇ K, R)

is real analytic for each compact K ⊆ R, which may be checked locally near each

s ∈ R.

f : R2 ’ R extends to a holomorphic function on an open neighborhood V — U of

{s} — K in C2 . By cartesian closedness for the holomorphic setting the associated

mapping f ∨ : V ’ H(U, C) is holomorphic, so its restriction V © R ’ C ω (U ©

R, R) ’ C ω (K, R) is real analytic as required.

11.8. Remark. From (11.7) it follows that the curve c : R ’ C ω (R, R) de¬ned in

(9.1) is real analytic, but it is not topologically real analytic. In particular, it does

not factor locally to a real analytic curve into some Banach space C ω (R, R)B for a

bounded subset B and it has no holomorphic extension to a mapping de¬ned on a

neighborhood of R in C with values in the complexi¬cation H(R, C) of C ω (R, R),

cf. (9.5).

11.9. Lemma. For a real analytic manifold M , the bornology on C ω (M, R) is

induced by the following cone:

c—

C (M, R) ’ C ± (R, R) for all C ± -curves c : R ’ M , where ± equals ∞ and ω.

ω

’

Proof. The maps c— are bornological since C ω (M, R) is convenient by (11.4), and

by the uniform S-boundedness principle (11.6) for C ω (R, R) and by (5.26) for

C ∞ (R, R) it su¬ces to check that evt —¦c— = evc(t) is bornological, which is obvious.

Conversely, we consider the identity mapping i from the space E into C ω (M, R),

where E is the vector space C ω (M, R), but with the locally convex structure in-

duced by the cone.

11.9

11.12 11. The real analytic exponential law 109

Claim. The bornology of E is complete.

The spaces C ω (R, R) and C ∞ (R, R) are convenient by (11.4) and (2.15), respec-

tively. So their product

C ∞ (R, R)

C ω (R, R) —

c∈C ∞ (R,M )

c∈C ω (R,M )

is also convenient. By theorem (10.1.1) ” (10.1.5) the embedding of E into this

product has closed image, hence the bornology of E is complete.

Now we may apply the uniform S-boundedness principle for C ω (M, R) (11.6), since

obviously evp —¦i = ev0 —¦c— is bounded, where cp is the constant curve with value p,

p

for all p ∈ M .

11.10. Structure on C ω (U, F ). Let E be a real convenient vector space and let

U be c∞ -open in E. We equip the space C ω (U, R) of all real analytic functions (cf.

(10.3)) with the locally convex topology induced by the families of mappings

c—

C (U, R) ’ C ω (R, R), for all c ∈ C ω (R, U )

ω

’

c—

C ω (U, R) ’ C ∞ (R, R), for all c ∈ C ∞ (R, U ).

’

For a ¬nite dimensional vector spaces E this de¬nition gives the same bornology

as the one de¬ned in (11.1), by lemma (11.9).

If F is another convenient vector space, we equip the space C ω (U, F ) of all real

analytic mappings (cf. (10.3)) with the locally convex topology induced by the

family of mappings

»

C ω (U, F ) ’ — C ω (U, R), for all » ∈ F .

’’

Obviously, the injection C ω (U, F ) ’ C ∞ (U, F ) is bounded and linear.

11.11. Lemma. Let E and F be convenient vector spaces and let U ⊆ E be

c∞ -open. Then C ω (U, F ) is also convenient.

Proof. This follows immediately from the fact that C ω (U, F ) can be considered

as closed subspace of the product of factors C ω (U, R) indexed by all » ∈ F .

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

(r, σ) zu H angeh¨rt. Wenn f¨r jede analytische Mannigfaltigkeit y(t; ±1 , . . . , ±m ), die in

o u

diese Umgebung eindringt (d.h. eine solche, die f¨r alle Wertesysteme ±1 , . . . , ±m ) eines

u

Bereichs “ eine Funktion von t der Umgebung liefert), der Wert des Funktionals

Ft [y(t; ±1 , . . . , ±m )] = f (±1 , . . . , ±m )

immer eine Funktion der Parameter ±1 , . . . , ±m ist, die nicht nur in “ de¬niert, sondern

dort noch eine analytische Funktion ist, werden wir sagen, daß das Funktional F regul¨r ist

a

in der betrachteten Umgebung y0 (t). Wenn ein Funktional F regul¨r ist in einer Umgebung

a

jeder Funktion seines De¬nitionsbereiches, so heißt F analytisch.”

The development in the complex case was much faster than in the smooth case

since one did not have to explain the concept of higher derivatives.

The Portuguese Jos´ Sebasti˜o e Silva showed that analyticity in the sense of

e a

Fantappi´ coincides with other concepts, in his dissertation [Sebasti˜o e Silva,

e a

1948], published as [Sebasti˜o e Silva, 1950a], and in [Sebasti˜o e Silva, 1953].

a a

An overview over various notions of holomorphicity was given by the Brasilian

Domingos Pisanelli in [Pisanelli, 1972a] and [Pisanelli, 1972b].

117

Chapter III

Partitions of Unity

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

The main aim of this chapter is to discuss the abundance or scarcity of smooth

functions on a convenient vector space: E.g. existence of bump functions and parti-

tions of unity. This question is intimately related to di¬erentiability of seminorms

and norms, and in many examples these are, if at all, only ¬nitely often di¬eren-

tiable. So we start this chapter with a short (but complete) account of ¬nite order

di¬erentiability, based on Lipschitz conditions on higher derivatives, since with this

notion we can get as close as possible to exponential laws. A more comprehensive

exposition of ¬nite order Lipschitz di¬erentiability can be found in the monograph

[Fr¨licher, Kriegl, 1988].

o

Then we treat di¬erentiability of seminorms and convex functions, and we have

tried to collect all relevant information from the literature. We give full proofs of

all what will be needed later on or is of central interest. We also collect related

results, mainly on ˜generic di¬erentiability™, i.e. di¬erentiability on a dense Gδ -set.

If enough smooth bump functions exist on a convenient vector space, we call it

˜smoothly regular™. Although the smooth (i.e. bounded) linear functionals separate

points on any convenient vector space, stronger separation properties depend very

much on the geometry. In particular, we show that 1 and C[0, 1] are not even

C 1 -regular. We also treat more general ˜smooth spaces™ here since most results do

not depend on a linear structure, and since we will later apply them to manifolds.

In many problems like E. Borel™s theorem (15.4) that any power series appears

as Taylor series of a smooth function, or the existence of smooth functions with

given carrier (15.3), one uses in ¬nite dimensions the existence of smooth functions

with globally bounded derivatives. These do not exist in in¬nite dimensions in

general; even for bump functions this need not be true globally. Extreme cases

are Hilbert spaces where there are smooth bump functions with globally bounded

derivatives, and c0 which does not even admit C 2 -bump functions with globally

bounded derivatives.

In the ¬nal section of this chapter a space which admits smooth partitions of unity

subordinated to any open cover is called smoothly paracompact. Fortunately, a

118 Chapter III. Partitions of unity 12.2

wide class of convenient vector spaces has this property, among them all spaces of

smooth sections of ¬nite dimensional vector bundles which we shall need later as

modeling spaces for manifolds of mappings. The theorem (16.15) of [Toru´czyk,

n

1973] characterizes smoothly paracompact metrizable spaces, and we will give a

full proof. It is the only tool for investigating whether non-separable spaces are

smoothly paracompact and we give its main applications.

12. Di¬erentiability of Finite Order

12.1. De¬nition. A mapping f : E ⊇ U ’ F , where E and F are convenient

vector spaces, and U ⊆ E is c∞ -open, is called Lipk if f —¦ c is a Lipk -curve (see

(1.2)) for each c ∈ C ∞ (R, U ).

This is equivalent to the property that f —¦c is Lipk on c’1 (U ) for each c ∈ C ∞ (R, E).

This can be seen by reparameterization.

12.2. General curve lemma. Let E be a convenient vector space, and let cn ∈

C ∞ (R, E) be a sequence of curves which converges fast to 0, i.e., for each k ∈ N

the sequence nk cn is bounded. Let sn ≥ 0 be reals with n sn < ∞.

Then there exists a smooth curve c ∈ C ∞ (R, E) and a converging sequence of reals

tn such that c(t + tn ) = cn (t) for |t| ¤ sn , for all n.

rn +rn+1

2

. Let h : R ’ [0, 1] be

Proof. Let rn := k<n ( k2 + 2sk ) and tn := 2

smooth with h(t) = 1 for t ≥ 0 and h(t) = 0 for t ¤ ’1, and put hn (t) := h(n2 (sn +

1

t)).h(n2 (sn ’t)). Then we have hn (t) = 0 for |t| ≥ n2 +sn and hn (t) = 1 for |t| ¤ sn ,

(j)

and for the derivatives we have |hn (t)| ¤ n2j .Hj , where Hj := max{|h(j) | : t ∈ R}.

Thus, in the sum

hn (t ’ tn ).cn (t ’ tn )

c(t) :=

n

at most one summand is non-zero for each t ∈ R, and c is a smooth curve since for

each ∈ E we have

( —¦ c)(t) = fn (t), where fn (t + tn ) := hn (t). (cn (t)),

n

n2 . sup |fn (t)| = n2 . sup |fn (s + tn )| : |s| ¤

(k) (k) 1

+ sn

n2

t

k

k

2

n2j Hj . sup |( —¦ cn )(k’j) (s)| : |s| ¤ 1

¤n + sn

n2

j

j=0

k

k

n2j+2 Hj . sup |( —¦ cn )(i) (s)| : |s| ¤ max( n2 + sn ) and i ¤ k ,

1

¤ j n

j=0

which is uniformly bounded with respect to n, since cn converges to 0 fast.

12.2

12.4 12. Di¬erentiability of ¬nite order 119

12.3. Corollary. Let cn : R ’ E be polynomials of bounded degree with values in

a convenient vector space E. If for each ∈ E the sequence n ’ sup{|( —¦ cn )(t) :

|t| ¤ 1} converges to 0 fast, then the sequence cn converges to 0 fast in C ∞ (R, E),

so the conclusion of (12.2) holds.

Proof. The structure on C ∞ (R, E) is the initial one with respect to the cone

∞ ∞

— : C (R, E) ’ C (R, R) for all ∈ E , by (3.9). So we only have to show the

result for E = R. On the ¬nite dimensional space of all polynomials of degree at

most d the expression in the assumption is a norm, and the inclusion into C ∞ (R, R)

is bounded.

12.4. Di¬erence quotients. For a curve c : R ’ E with values in a vector space

E the di¬erence quotient δ k c of order k is given recursively by

δ 0 c := c,

δ k’1 c(t0 , . . . , tk’1 ) ’ δ k’1 c(t1 , . . . , tk )

k

δ c(t0 , . . . , tk ) := k ,

t 0 ’ tk

for pairwise di¬erent ti . The constant factor k in the de¬nition of δ k is chosen in

such a way that δ k approximates the k-th derivative. By induction, one can easily

see that

k

δ k c(t0 , . . . , tk ) = k! 1

c(ti ) ti ’tj .

i=0 0¤j¤k

j=i

k

We shall mainly need the equidistant di¬erence quotient δeq c of order k, which is

given by

k

k!

k k 1

δeq c(t; v) := δ c(t, t + v, . . . , t + kv) = k c(t + iv) i’j .

v i=0 0¤j¤k

j=i

Lemma. For a convenient vector space E and a curve c : R ’ E the following

conditions are equivalent:

(1) c is Lipk’1 .

(2) The di¬erence quotient δ k c of order k is bounded on bounded sets.

(3) —¦ c is continuous for each ∈ E , and the equidistant di¬erence quotient

k

δeq c of order k is bounded on bounded sets in R — (R \ {0}).

Proof. All statements can be tested by composing with bounded linear functionals

∈ E , so we may assume that E = R.

(3) ’ (2) Let I ‚ R be a bounded interval. Then there is some K > 0 such that

k

|δeq c(x; v)| ¤ K for all x ∈ I and kv ∈ I. Let ti ∈ I be pairwise di¬erent points.

We claim that |δ k c(t0 , . . . , tk )| ¤ K. Since δ k c is symmetric we may assume that

t0 < t1 < · · · < tk , and since it is continuous (c is continuous) we may assume that

’t

all tk’t0 are of the form ni for ni , N ∈ N. Put v := tkN 0 , then δ k c(t0 , . . . , tk ) =

ti

’t0 N

12.4

120 Chapter III. Partitions of unity 12.4

δ k c(t0 , t0 + n1 v, . . . , t0 + nk v) is a convex combination of δeq c(t0 + rv; v) for 0 ¤ r ¤

k

maxi ni ’ k. This follows by recursively inserting intermediate points of the form

t0 + mv, and using

δ k (t0 + m0 v, . . . , t0 + mi v, . . . , t0 + mk+1 v) =

mi ’ m0 k

= δ (t0 + m0 v, . . . , t0 + mk v)

mk+1 ’ m0

mk+1 ’ mi k

+ δ (t1 + m1 v, . . . , t0 + mk+1 v)

mk+1 ’ m0

which itself may be proved by induction on k.

(2) ’ (1) We have to show that c is k times di¬erentiable and that δ 1 c(k) is bounded

on bounded sets. We use induction, k = 0 is clear.

Let T = S be two subsets of R of cardinality j + 1. Then there exist enumerations

T = {t0 , . . . , tj } and S = {s0 , . . . , sj } such that ti = sj for i ¤ j; then we have

j

δ j c(t0 , . . . , tj ) ’ δ j c(s0 , . . . , sj ) = (ti ’ si )δ j+1 c(t0 , . . . , ti , si , . . . , sj ).

1

j+1

i=0

For the enumerations we put the elements of T © S at the end in T and at the

beginning in S. Using the recursive de¬nition of δ j+1 c and symmetry the right

hand side becomes a telescoping sum.

Since δ k c is bounded we see from the last equation that all δ j c are also bounded,

in particular this is true for δ 2 c. Then

c(t + s) ’ c(t) c(t + s ) ’ c(t)

δ 2 c(t, t + s, t + s )

s’s

’ = 2

s s

shows that the di¬erence quotient of c forms a Mackey Cauchy net, and hence the

limit c (t) exists.

Using the easily checked formula

j i’1

(tj ’ tl ) δ j c(t0 , . . . , tj ),

1

c(tj ) = i!

i=0 l=0

induction on j and di¬erentiability of c one shows that

j

δ j c (t0 , . . . , tj ) = δ j+1 c(t0 , . . . , tj , ti ),

1

(4) j+1

i=0

where δ j+1 c(t0 , . . . , tj , ti ) := limt’ti δ j+1 c(t0 , . . . , tj , t). The right hand side of (4)

is bounded, so c is Lipk’2 by induction on k.

(1) ’ (2) For a di¬erentiable function f : R ’ R and t0 < · · · < tj there exist si

with ti < si < ti+1 such that

δ j f (t0 , . . . , tj ) = δ j’1 f (s0 , . . . , sj’1 ).

(5)

12.4

12.5 12. Di¬erentiability of ¬nite order 121

Let p be the interpolation polynomial

j i’1

(t ’ tl ) δ j f (t0 , . . . , tj ).

1

(6) p(t) := i!

i=0 l=0

Since f and p agree on all tj , by Rolle™s theorem the ¬rst derivatives of f and p

agree on some intermediate points si . So p is the interpolation polynomial for

f at these points si . Comparing the coe¬cient of highest order of p and of the

interpolation polynomial (6) for f at the points si (5) follows.

Applying (5) recursively for f = c(k’2) , c(k’3) , . . . , c shows that δ k c is bounded on

bounded sets, and (2) follows.

(2) ’ (3) is obvious.

12.5. Let r0 , . . . , rk be the unique rational solution of the linear equation

k

1 for j = 1

i j ri =

0 for j = 0, 2, 3, . . . , k.

i=0

Lemma. If f : R2 ’ R is Lipk for k ≥ 1 and I is a compact interval then there

exists M such that for all t, v ∈ I we have

k

ri f (t, iv) ¤ M |v|k+1 .

‚

‚s |0 f (t, s).v ’

i=0

Proof. We consider ¬rst the case 0 ∈ I so that v stays away from 0. For this it

/

‚

su¬ces to show that the derivative ‚s |0 f (t, s) is locally bounded. If it is unbounded

near some point x∞ , there are xn with |xn ’x∞ | ¤ 21 such that ‚s |0 f (xn , s) ≥ n.2n .

‚

n

We apply the general curve lemma (12.2) to the curves cn : R ’ R2 given by

cn (t) := (xn , 2tn ) and to sn := 21 in order to obtain a smooth curve c : R ’ R2

n

and scalars tn ’ 0 with c(t + tn ) = cn (t) for |t| ¤ sn . Then (f —¦ c) (tn ) =

1

1‚

2n ‚s |0 f (xn , s) ≥ n, which contradicts that f is Lip .

Now we treat the case 0 ∈ I. If the assertion does not hold there are xn , vn ∈

k

‚

I, such that ‚s |0 f (xn , s).vn ’ i=0 ri f (xn , ivn ) ≥ n.2n(k+1) |vn |k+1 . We may

assume xn ’ x∞ , and by the case 0 ∈ I we may assume that vn ’ 0, even with

/

|xn ’ x∞ | ¤ 21 and |vn | ¤ 21 . We apply the general curve lemma (12.2) to the

n n

curves cn : R ’ R2 given by cn (t) := (xn , 2tn ) and to sn := 21 to obtain a smooth

n

2

curve c : R ’ R and scalars tn ’ 0 with c(t + tn ) = cn (t) for |t| ¤ sn . Then we

have

k

(f —¦ c) (tn )2n vn ’ ri (f —¦ c)(tn + i2n vn ) =

i=0

k

n

ri (f —¦ cn )(i2n vn )

= (f —¦ cn ) (0)2 vn ’

i=0

k

n

ri f (xn , ivn ) ≥ n(2n |vn |)k+1 .

1‚

2n ‚s |0 f (xn , s)2 vn ’

=

i=0

12.5

122 Chapter III. Partitions of unity 12.6

This contradicts the next claim for g = f —¦ c.

Claim. If g : R ’ R is Lipk for k ≥ 1 and I is a compact interval then there is

k

M > 0 such that for t, v ∈ I we have g (t).v ’ i=0 ri g(t + iv) ¤ M |v|k+1 .

k

Consider gt (v) := g (t).v ’ i=0 ri g(t + iv). Then the derivatives up to order k at

v = 0 of gt vanish by the choice of the ri . Since g (k) is locally Lipschitzian there

(k)

exists an M such that |gt (v)| ¤ M |v| for all t, v ∈ I, which we may integrate in

|v|k+1

turn to obtain |gt (v)| ¤ M (k+1)! .

12.6. Lemma. Let f : R2 ’ R be Lipk+1 . Then t ’ is Lipk .

‚

‚s |0 f (t, s)

Proof. Suppose that g : t ’ ‚s |0 f (t, s) is not Lipk . Then by lemma (12.4) the

‚

k+1

equidistant di¬erence quotient δeq g is not locally bounded at some point which we

may assume to be 0. Then there are xn and vn with |xn | ¤ 1/4n and 0 < vn < 1/4n

such that

|δeq g(xn ; vn )| > n.2n(k+2) .

k+1

(1)

We apply the general curve lemma (12.2) to the curves cn : R ’ R2 given by

cn (t) := en ( 2tn + xn ) := ( 2tn + xn ’ vn , 2tn ) and to sn := k+2 in order to obtain a

2n

2

smooth curve c : R ’ R and scalars tn ’ 0 with c(t + tn ) = cn (t) for 0 ¤ t ¤ sn .

k

Put f0 (t, s) := i=0 ri f (t, is) for ri as in (12.5), put f1 (t, s) := g(t)s, ¬nally put

f2 := f1 ’f0 . Then f0 in Lipk+1 , so f0 —¦c is Lipk+1 , hence the equidistant di¬erence

quotient δeq (f0 —¦ c)(xn ; 2n vn ) is bounded.

k+2

By lemma (12.5) there exists M > 0 such that |f2 (t, s)| ¤ M |s|k+2 for all t, s ∈

[’(k + 1), k + 1], so we get

|δeq (f2 —¦ c)(xn ; 2n vn )| = |δeq (f2 —¦ cn )(0; 2n vn )|

k+2 k+2

|δ k+2 (f2 —¦ en )(xn ; vn )|

1

= 2n(k+2) eq

k+2

i(k+2)

|f2 ((i ’ 1)vn + xn , ivn )|

(k+2)!

¤ 2n(k+2) |ivn |(k+2) j=i |i ’ j|

i=1

k+2

i(k+2)

(k+2)!

¤ M .

2n(k+2) |i ’ j|

j=i

i=1

This is bounded, and so for f1 = f0 + f2 the expression |δeq (f1 —¦ c)(xn ; 2n vn )| is

k+2

also bounded, with respect to n. However, on the other hand we get

δeq (f1 —¦ c)(xn ; 2n vn ) = δeq (f1 —¦ cn )(0; 2n vn )

k+2 k+2

δ k+2 (f1 —¦ en )(xn ; vn )

1

= 2n(k+2) eq

k+2

f1 ((i ’ 1)vn + xn , ivn )

(k+2)! 1

= 2n(k+2) i’j

(k+2)

vn

i=0 0¤j¤k+2

j=i

12.6

12.8 12. Di¬erentiability of ¬nite order 123

k+2

g((i ’ 1)vn + xn )ivn

(k+2)! 1

= 2n(k+2) i’j

(k+2)

vn

i=0 0¤j¤k+2

j=i

k+1

g(lvn + xn )

(k+2)! 1

= 2n(k+2) l’j

(k+1)

vn

l=0 0¤j¤k+1

j=l

δ k+1 g(xn ; vn ),

k+2

= 2n(k+2) eq

which in absolute value is larger than (k + 2)n by (1), a contradiction.

12.7. Lemma. Let E be a normed space and F be a convenient vector space, U

open in E. Then, a mapping f : U ’ F is Lip0 if and only if f is locally Lipschitz,

i.e., f (x)’f (y) is locally bounded.

x’y

Proof. (’) If f is Lip0 but not locally Lipschitz near z ∈ U , there are ∈ F

and points xn = yn in U with xn ’ z ¤ 1/2n and yn ’ z ¤ 1/2n , such that

(f (yn ) ’ f (xn )) ≥ n. yn ’ xn . Now we apply the general curve lemma (12.2)

with sn := yn ’ xn and cn (t) := xn ’ z + t(yn ’ xn ) to get a smooth curve c with

c(t + tn ) = cn (t) for 0 ¤ t ¤ sn . Then s1 (( —¦ f —¦ c)(tn + sn ) ’ ( —¦ f —¦ c)(tn )) =

n

1

yn ’xn (f (yn ) ’ f (xn )) ≥ n.

(⇐) This is obvious, since the composition of locally Lipschitzian mappings is again

locally Lipschitzian.

12.8. Theorem. Let f : E ⊇ U ’ F be a mapping, where E and F are convenient

vector spaces, and U ⊆ E is c∞ -open. Then the following assertions are equivalent

for each k ≥ 0:

(1) f is Lipk+1 .

(2) The directional derivative

‚

‚t |t=0 (f (x

(dv f )(x) := + tv))

exists for x ∈ U and v ∈ E and de¬nes a Lipk -mapping U — E ’ F .

Note that this result gives a di¬erent (more algebraic) proof of Boman™s theorem

(3.4) and (3.14).

Proof. (1) ’ (2) Clearly, t ’ f (x+tv) is Lipk+1 , and so the directional derivative

exists and is the Mackey-limit of the di¬erence quotients, by lemma (1.7). In order

to show that df : (x, v) ’ dv f (x) is Lipk we take a smooth curve (x, v) : R ’

U — E and ∈ F , and we consider g(t, s) := x(t) + s.v(t), g : R2 ’ E. Then

—¦ f —¦ g : R2 ’ R is Lipk+1 , so by lemma (12.6) the curve

‚ ‚

t ’ (df (x(t), v(t))) = ‚s |0 f (g(t, s)) ‚s |0

= (f (g(t, s)))

is of class Lipk .

12.8

124 Chapter III. Partitions of unity 12.10

(2) ’ (1) If c ∈ C ∞ (R, U ) then

f (c(t)) ’ f (c(0))

’ df (c(0), c (0)) =

t

1

df (c(0) + s(c(t) ’ c(0)), c(t)’c(0) ) ’ df (c(0), c (0)) ds

= t

0

converges to 0 for t ’ 0 since g : (t, s) ’ df (c(0) + s(c(t) ’ c(0)), c(t)’c(0) ) ’

t

k

df (c(0), c (0)) is Lip , thus by lemma (12.7) g is locally Lipschitz, so the set of all

1

g(t1 ,s)’g(t2 ,s)

is locally bounded, and ¬nally t ’ 0 g(t, s)ds is locally Lipschitz.

t1 ’t2

Thus, f —¦ c is di¬erentiable with derivative (f —¦ c) (0) = df (c(0), c (0)).

Since df is Lipk and (c, c ) is smooth we get that (f —¦ c) is Lipk , hence f —¦ c is

Lipk+1 .