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