(4) ’ (5) By the (1-dimensional) Cauchy integral formula we have

1 f (z + »v)

√

f (z)v = d».

»2

2π ’1 |»|=1

So f (z) is a linear functional which is bounded on compact sets K for which

{z + »v : |»| ¤ 1, v ∈ K} ⊆ U , thus it is bounded, by lemma (5.4).

(6) ’ (1) follows by composing the two locally uniformly converging power series,

see corollary (7.17).

Sublemma. Let E be a Fr´chet space and let U ⊆ E be open. Let f : U ’ C be

e

holomorphic along a¬ne lines which is also the pointwise limit on U of a power

series with bounded homogeneous composants. Then f is holomorphic on U .

Proof. By assumption, and the lemma in (7.18) the function f is almost con-

tinuous, since it is the pointwise limit of polynomials. For each z the derivative

f (z) : E ’ C as pointwise limit of di¬erence quotients is also almost continuous

on {v : z + »v ∈ U for |»| ¤ 1}, thus continuous on E since it is linear and by the

Baire property.

By (5) ’ (1) the function f is holomorphic on U .

(6) ’ (7) is obvious.

(7) ’ (1) [Zorn, 1945] We treat each connected component of U separately and

assume thus that U is connected. The set U0 := {z ∈ U : f is holomorphic near z}

is open. By (6) ’ (1) f is holomorphic near the point, where all derivatives are

bounded, so U0 is not empty. From the sublemma above we see that for any point

z in U0 the whole star {z + v : z + »v ∈ U for all |»| ¤ 1} is contained in U . Since

U is in particular polygonally connected, we have U0 = U .

(8) ’ (9) is trivial.

(9) ’ (3) Clearly, f is holomorphic along a¬ne lines and c∞ -continuous.

(1) ’ (8) All derivatives are again holomorphic by (7.10) and thus locally bounded.

So f is smooth by (5.20).

Now we treat the case where E is a general convenient vector space. Restricting to

suitable spaces EB transforms each of the statements into the weaker corresponding

one where E is a Banach space. These pairs of statements are equivalent: This is

obvious except the following two cases.

For (6) we argue as follows. The function f |(U © EB ) satis¬es condition (6) (so

all the others) for each bounded closed absolutely convex B ⊆ E. By (5.20) f is

smooth and it remains to show that the Taylor series at z converges pointwise on a

c∞ -open neighborhood of z. The star {z + v : z + »v ∈ U for all |»|le1} with center

z in U is again c∞ -open by (4.17) and on it the Taylor series of f at z converges

pointwise.

For (7) replace on both sides the condition ”at least one point” by the condition

”for all points”.

7.19

90 Chapter II. Calculus of holomorphic and real analytic mappings 7.24

7.20. Chain rule. The composition of holomorphic mappings is holomorphic and

the usual formula for the derivative of the composite holds.

Proof. Use (7.19.1) ” (7.19.8), and the real chain rule (3.18).

7.21. De¬nition. For convenient vector spaces E and F and for a c∞ -open subset

U ⊆ E we denote by H(U, F ) the space of all holomorphic mappings U ’ F . It

is a closed linear subspace of C ∞ (U, F ) by (7.19.8) and we give it the induced

convenient vector space structure.

7.22. Theorem. Cartesian closedness. For convenient vector spaces E1 , E2 ,

and F , and for c∞ -open subsets Uj ⊆ Ej a mapping f : U1 — U2 ’ F is holo-

morphic if and only if the canonically associated mapping f ∨ : U1 ’ H(U2 , F ) is

holomorphic.

Proof. Obviously, f ∨ has values in H(U2 , F ) and is smooth by smooth cartesian

closedness (3.12). Since its derivative is canonically associated to the ¬rst partial

derivative of f , it is complex linear. So f ∨ is holomorphic by (7.19.8).

If conversely f ∨ is holomorphic, then it is smooth into H(U2 , F ) by (7.19), thus

also smooth into C ∞ (U2 , F ). Thus, f : U1 — U2 ’ F is smooth by smooth carte-

sian closedness. The derivative df (x, y)(u, v) = (df ∨ (x)v)(y) + (d —¦ f ∨ )(x)(y)w is

obviously complex linear, so f is holomorphic.

7.23. Corollary. Let E etc. be convenient vector spaces and let U etc. be c∞ -open

subsets of such. Then the following canonical mappings are holomorphic.

ev : H(U, F ) — U ’ F, ev(f, x) = f (x)

ins : E ’ H(F, E — F ), ins(x)(y) = (x, y)

)§ : H(U, H(V, G)) ’ H(U — V, G)

(

)∨ : H(U — V, G) ’ H(U, H(V, G))

(

comp : H(F, G) — H(U, F ) ’ H(U, G)

H( ) : H(F, F ) — H(U , E) ’ H(H(E, F ), H(U , F ))

,

(f, g) ’ (h ’ f —¦ h —¦ g)

H(Ei , Fi ) ’ H(

: Ei , Fi )

Proof. Just consider the canonically associated holomorphic mappings on multiple

products.

7.24. Theorem (Holomorphic functions on Fr´chet spaces).

e

Let U ⊆ E be open in a complex Fr´chet space E. The following statements on

e

f : U ’ C are equivalent:

(1) f is holomorphic.

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

Taylor series.

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

7.24

8.2 8. Spaces of holomorphic mappings and germs 91

Proof. (1) ’ (2) follows from (7.14.1) ’ (7.14.2) and (7.19.1) ’ (7.19.6).

(2) ’ (3) is obvious.

(3) ’ (1) is the chain rule for converging power series (7.17).

8. Spaces of Holomorphic Mappings and Germs

8.1. Spaces of holomorphic functions. For a complex manifold N (always

assumed to be separable) let H(N, C) be the space of all holomorphic functions on

N with the topology of uniform convergence on compact subsets of N .

Let Hb (N, C) denote the Banach space of bounded holomorphic functions on N

equipped with the supremum norm.

For any open subset W of N let Hbc (N ⊇ W, C) be the closed subspace of Hb (W, C)

of all holomorphic functions on W which extend to continuous functions on the

closure W .

p

For a poly-radius r = (r1 , . . . , rn ) with ri > 0 and for 1 ¤ p ¤ ∞ let denote the

r

n

real Banach space x ∈ RN : (x± r± )±∈Nn p < ∞ .

8.2. Theorem (Structure of H(N, C) for complex manifolds N ).

The space H(N, C) of all holomorphic functions on N with the topology of uniform

convergence on compact subsets of N is a (strongly) nuclear Fr´chet space and

e

embeds bornologically as a closed subspace into C ∞ (N, R)2 .

Proof. By taking a countable covering of N with compact sets, one obtains a

countable neighborhood basis of 0 in H(N, C). Hence, H(N, C) is metrizable.

That H(N, C) is complete, and hence a Fr´chet space, follows since the limit of a

e

sequence of holomorphic functions with respect to the topology of uniform conver-

gence on compact sets is again holomorphic.

The vector space H(N, C) is a subspace of C ∞ (N, R2 ) = C ∞ (N, R)2 since a function

N ’ C is holomorphic if and only if it is smooth and the derivative at every point

is C-linear. It is a closed subspace, since it is described by the continuous linear

√ √

equations df (x)( ’1 · v) = ’1 · df (x)(v). Obviously, the identity from H(N, C)

with the subspace topology to H(N, C) is continuous, hence by the open mapping

theorem (52.11) for Fr´chet spaces it is an isomorphism.

e

That H(N, C) is nuclear and unlike C ∞ (N, R) even strongly nuclear can be shown

as follows. For N equal to the open polycylinder Dn ⊆ Cn this result can be found in

(52.36). For an arbitrary N the space H(N, C) carries the initial topology induced

by the linear mappings (u’1 )— : H(N, C) ’ H(u(U ), C) for all charts (u, U ) of

N , for which we may assume u(U ) = Dn , and hence by the stability properties of

strongly nuclear spaces, cf. (52.34), H(N, C) is strongly nuclear.

8.2

92 Chapter II. Calculus of holomorphic and real analytic mappings 8.4

8.3. Spaces of germs of holomorphic functions. For a subset A ⊆ N let

H(N ⊇ A, C) be the space of germs along A of holomorphic functions W ’ C for

open sets W in N containing A. We equip H(N ⊇ A, C) with the locally convex

topology induced by the inductive cone H(W, C) ’ H(N ⊇ A, C) for all W . This

is Hausdor¬, since iterated derivatives at points in A are continuous functionals

and separate points. In particular, H(N ⊇ W, C) = H(W, C) for W open in N .

For A1 ⊆ A2 ⊆ N the ”restriction” mappings H(N ⊇ A2 , C) ’ H(N ⊇ A1 , C) are

continuous.

The structure of H(S 2 ⊇ A, C), where A ⊆ S 2 is a subset of the Riemannian sphere,

has been studied by [Toeplitz, 1949], [Sebasti˜o e Silva, 1950b,] [Van Hove, 1952],

a

[K¨the, 1953], and [Grothendieck, 1953].

o

8.4. Theorem (Structure of H(N ⊇ K, C) for compact subsets K of com-

plex manifolds N ). The following inductive cones are co¬nal to each other.

H(N ⊇ K, C) ← {H(W, C), N ⊇ W ⊇ K}

H(N ⊇ K, C) ← {Hb (W, C), N ⊇ W ⊇ K}

H(N ⊇ K, C) ← {Hbc (N ⊇ W, C), N ⊇ W ⊇ K}

If K = {z} these inductive cones and the following ones for 1 ¤ p ¤ ∞ are co¬nal

to each other.

H(N ⊇ {z}, C) ← { p — C, r ∈ Rn }

r +

So all inductive limit topologies coincide. Furthermore, the space H(N ⊇ K, C) is a

Silva space, i.e. a countable inductive limit of Banach spaces, where the connecting

mappings between the steps are compact, i.e. mapping bounded sets to relatively

compact ones. The connecting mappings are even strongly nuclear. In particular,

the limit is regular, i.e. every bounded subset is contained and bounded in some

step, and H(N ⊇ K, C) is complete and (ultra-)bornological (hence a convenient

vector space), webbed, strongly nuclear and thus re¬‚exive, and its dual is a nuclear

Fr´chet space. The space H(N ⊇ K, C) is smoothly paracompact. It is however not

e

a Baire space.

Proof. Let K ⊆ V ⊆ V ⊆ W ⊆ N , where W and V are open and V is compact.

Then the obvious mappings

Hbc (N ⊇ W, C) ’ Hb (W, C) ’ H(W, C) ’ Hbc (N ⊇ V, C)

are continuous. This implies the ¬rst co¬nality assertion. For q ¤ p and multiradii

s < r the obvious maps q ’ p , ∞ ’ 1 , and 1 — C ’ Hb ({w ∈ Cn : |wi ’ zi | <

r r r s r

∞

ri }, C) ’ s —C are continuous, by the Cauchy inequalities from the proof of (7.7).

So the remaining co¬nality assertion follows.

Let us show next that the connecting mapping Hb (W, C) ’ Hb (V, C) is strongly

nuclear (hence nuclear and compact). Since the restriction mapping from E :=

H(W, C) to Hb (V, C) is continuous, it factors over E ’ E(U ) for some zero neigh-

borhood U in E, where E(U ) is the completed quotient of E with the Minkowski

8.4

8.6 8. Spaces of holomorphic mappings and germs 93

functional of U as norm, see (52.15). Since E is strongly nuclear by (8.2), there ex-

ists by de¬nition some larger 0-neighborhood U in E such that the natural mapping

E(U ) ’ E(U ) is strongly nuclear. So the claimed connecting mapping is strongly

nuclear, since it can be factorized as

Hb (W, C) ’ H(W, C) = E ’ E(U ) ’ E(U ) ’ Hb (V, C).

So H(N ⊇ K, C) is a Silva space. It is strongly nuclear by the permanence proper-

ties of strongly nuclear spaces (52.34). By (16.10) this also shows that H(N ⊇ K, C)

is smoothly paracompact. The remaining properties follow from (52.37).

Completeness of H(Cn ⊇ K, C) was shown in [Van Hove, 1952, th´or`me II], and

ee

for regularity of the inductive limit H(C ⊇ K, C) see e.g. [K¨the, 1953, Satz 12].

o

8.5. Lemma. For a closed subset A ⊆ C the spaces H(A ⊆ S 2 , C) and the space

H∞ (S 2 ⊇ S 2 \ A, C) of all germs vanishing at ∞ are strongly dual to each other.

Proof. This is due to [K¨the, 1953, Satz 12] and has been generalized by [Grothen-

o

dieck, 1953,] th´or`me 2 bis, to arbitrary subsets A ⊆ S 2 .

ee

Compare also the modern theory of hyperfunctions, cf. [Kashiwara, Kawai, Kimura,

1986].

8.6. Theorem (Structure of H(N ⊇ A, C) for closed subsets A of complex

manifolds N ). The inductive cone

H(N ⊇ A, C) ← { H(W, C) : A ⊆ W ⊆ N }

open

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

The projective cone

H(N ⊇ A, C) ’ { H(N ⊇ K, C) : K compact in A}

generates the bornology of H(N ⊇ A, C).

The space H(N ⊇ A, C) is Montel (hence quasi-complete and re¬‚exive), and ultra-

bornological (hence a convenient vector space). Furthermore, it is webbed and conu-

clear.

Proof. Compare also with the proof of the more general theorem (30.6).

We choose a continuous function f : N ’ R which is positive and proper. Then

(f ’1 ([n, n + 1]))n∈N0 is an exhaustion of N by compact subsets and (Kn := A ©

f ’1 ([n, n + 1])) is a compact exhaustion of A.

Let B ⊆ H(N ⊇ A, C) be bounded. Then B|K is also bounded in H(N ⊇ K, C) for

each compact subset K of A. Since the cone

{H(W, C) : K ⊆ W ⊆ N } ’ H(N ⊇ K, C)

open

8.6

94 Chapter II. Calculus of holomorphic and real analytic mappings 8.8

is regular by (8.4), there exist open subsets WK of N containing K such that B|K is

contained (so that the extension of each germ is unique) and bounded in H(WK , C).

In particular, we choose WKn ©Kn+1 ⊆ WKn © WKn+1 © f ’1 ((n, n + 2)). Then we

let W be the union of those connected components of

(WKn © f ’1 ((n, n + 1))) ∪

W := WKn ©Kn+1

n n

which meet A. Clearly, W is open and contains A. Each f ∈ B has an extension to

W : Extend f |Kn uniquely to fn on WKn . The function f |(Kn © Kn+1 ) has also

a unique extension fn,n+1 on WKn ©Kn+1 , so we have fn |WKn ©Kn+1 = fn,n+1 . This

extension of f ∈ B has a unique restriction to W . B is bounded in H(W, C) if it is

uniformly bounded on each compact subset K of W . Each K is covered by ¬nitely

many WKn and B|Kn is bounded in H(WKn , C), so B is bounded as required.

The space H(N ⊇ A, C) is ultra-bornological, Montel and in particular quasi-

complete, and conuclear, as regular inductive limit of the nuclear Fr´chet spaces

e

H(W, C).

And it is webbed because it is the (ultra-)bornologi¬cation of the countable pro-

jective limit of webbed spaces H(N ⊇ K, C), see (52.14) and (52.13).

8.7. Lemma. Let A be closed in C. Then the dual generated by the projective

cone

H(C ⊇ A, C) ’ { H(C ⊇ K, C), K compact in A }

is just the topological dual of H(C ⊇ A, C).

Proof. The induced topology is obviously coarser than the given one. So let »

be a continuous linear functional on H(C ⊇ A, C). Then we have » ∈ H∞ (S 2 ⊇

S 2 \ A, C) by (8.5). Hence, » ∈ H∞ (U, C) for some open neighborhood U of S 2 \ A,

so again by (8.5) » is a continuous functional on H(S 2 ⊇ K, C), where K = S 2 \ U

is compact in A. So » is continuous for the induced topology.

Problem. Does this cone generate even the topology of H(C ⊇ A, C)? This would

imply that the bornological topology on H(C ⊇ A, C) is complete and nuclear.

8.8. Lemma (Structure of H(N ⊇ A, C) for smooth closed submanifolds

A of complex manifolds N ). The projective cone

H(N ⊇ A, C) ’ { H(N ⊇ {z}, C) : z ∈ A}

generates the bornology.

Proof. Let B ⊆ H(N ⊇ A, C) be such that the set B is bounded in H(N ⊇ {z}, C)

for all z ∈ A. By the regularity of the inductive cone H(Cn ⊇ {0}, C) ← H(W, C)

we ¬nd arbitrary small open neighborhoods Wz such that the set Bz of the germs

at z of all germs in B is contained and bounded in H(Wz , C).

8.8

8.10 8. Spaces of holomorphic mappings and germs 95

Now choose a tubular neighborhood p : U ’ A of A in N . We may assume that Wz

is contained in U , has ¬bers which are star shaped with respect to the zero-section

and the intersection with A is connected. The union W of all the Wz , is therefore

an open subset of U containing A. And it remains to show that the germs in B

extend to W . For this it is enough to show that the extensions of the germs at

z1 and z2 agree on the intersection of Wz1 with Wz2 . So let w be a point in the

intersection. It can be radially connected with the base point p(w), which itself can

be connected by curves in A with z1 and z2 . Hence, the extensions of both germs

to p(w) coincide with the original germ, and hence their extensions to w are equal.

That B is bounded in H(W, C), follows immediately since every compact subset

K ⊆ W can be covered by ¬nitely many Wz .

8.9. The following example shows that (8.8) fails to be true for general closed

subsets A ⊆ N .

1

Example. Let A := { n : n ∈ N} ∪ {0}. Then A is compact in C but the projective

cone H(C ⊇ A, C) ’ {H(C ⊇ {z}, C) : z ∈ A} does not generate the bornology.

Proof. Let B ⊆ H(C ⊇ A, C) be the set of germs of the following locally constant

functions fn : {x + iy ∈ C : x = rn } ’ C, with fn (x + iy) equal to 0 for x < rn

2

and equal to 1 for x > rn , where rn := 2n+1 , for n ∈ N. Then B ⊆ H(C ⊇ A, C)

is not bounded, otherwise there would exist a neighborhood W of A such that the

germ of fn extends to a holomorphic mapping on W for all n. Since every fn is 0

on some neighborhood of 0, these extensions have to be zero on the component of

1

W containing 0, which is not possible, since fn ( n ) = 1.

But on the other hand the set Bz ⊆ H(C ⊇ {z}, C) of germs at z of all germs in B

is bounded, since it contains only the germs of the constant functions 0 and 1.

8.10. Theorem (Holomorphic uniform boundedness principle).

Let E and F be complex convenient vector spaces, and let U ⊆ E be a c∞ -open

subset. Then H(U, F ) satis¬es the uniform boundedness principle for the point

evaluations evx , x ∈ U .

For any closed subset A ⊆ N of a complex manifold N the locally convex space

H(N ⊇ A, C) satis¬es the uniform S-boundedness principle for every point sepa-

rating set S of bounded linear functionals.

Proof. By de¬nition (7.21) H(U, F ) carries the structure induced from the embed-

ding into C ∞ (U, F ) and hence satis¬es the uniform boundedness principle (5.26)

and (5.25).

The second part is an immediate consequence of (5.24) and (8.6), and (8.4).

Direct proof of a particular case of the second part. We prove the theorem

for a closed smooth submanifold A ⊆ C and the set S of all iterated derivatives at

points in A.

8.10

96 Chapter II. Calculus of holomorphic and real analytic mappings 8.10

Let us suppose ¬rst that A is the point 0. We will show that condition (5.22.3) is

satis¬ed. Let (bn ) be an unbounded sequence in H({0}, C) such that each Taylor

1 (k)

coe¬cient bn,k = k! bn (0) is bounded with respect to n:

sup{ |bn,k | : n ∈ N } < ∞.

(1)

1

We have to ¬nd (tn ) ∈ such that n tn b n is no longer the germ of a holomorphic

function at 0.

Each bn has positive radius of convergence, in particular there is an rn > 0 such

that

k

sup{ |bn,k rn | : k ∈ N } < ∞.

(2)

By theorem (8.4) the space H({0}, C) is a regular inductive limit of spaces ∞ . r

Hence, a subset B is bounded in H({0}, C) if and only if there exists an r > 0 such

1

that { k! b(k) (0) rk : b ∈ B, k ∈ N } is bounded. That the sequence (bn ) is unbounded

thus means that for all r > 0 there are n and k such that |bn,k | > ( 1 )k . We can

r

k

even choose k > 0 for otherwise the set { bn,k r : n, k ∈ N, k > 0 } is bounded, so

only { bn,0 : n ∈ N } can be unbounded. This contradicts (1).

Hence, for each m there are km > 0 such that Nm := { n ∈ N : |bn,km | > mkm }

is not empty. We can choose (km ) strictly increasing, for if they were bounded,