ńņš. 4 |

exposer la thĀ“orie dans un autre volume.ā

e

GĖteaux derivative. Another student of Hadamard deļ¬ned the derivative in

a

[GĖteaux, 1913] with proofs in [GĖteaux, 1922] as follows, see also [GĖteaux, 1922]:

a a a

āConsidĀ“rons U (z + Ī»t1 ) (t1 fonction analogue ` z). Supposons que

e a

d

U (z + Ī»t1 )

dĪ» Ī»=0

existe quel que soit t1 . On lā™appelle la variation premi`re de U au point z: Ī“U (z, t1 ). Cā™est

e

une fonctionnelle de z et de t1 , quā™on suppose habituellement linĀ“aire, en chaque point z,

e

par rapport ` t1 .ā

a

Several mathematicians gave conditions implying the linearity of the GĖteaux-de-

a

rivative. In [Daniell, 1919] is was shown that this holds for a Lipschitz function

whose GĖteaux-derivative exists locally. Another student of Hadamard assumed

a

linearity in [LĀ“vy, 1922], see again [FrĀ“chet, 1937]:

e e

S.51: āUne fonction abstraite X = F (x) sera dite diļ¬Ā“rentiable au sens de M. Paul Levy

e

pour x = x0 , sā™il existe une transformation vectorielle linĀ“aire ĪØ(āx) de lā™accroissement āx

e

telle que, pour chaque vecteur āx,

ā’ā’ ā’ ā’ ā’ ā’ ā’ā’

ā’ā’ā’ā’ā’ā’ā’

F (x0 )F (x0 + Ī»āx)

lim existe et = ĪØ(āx).ā

Ī»

Ī»ā’0

Hadamard diļ¬erentiability. In [Hadamard, 1923] a function f : R2 ā’ R was

called diļ¬erentiable if all compositions with diļ¬erentiable curves are again diļ¬er-

entiable and satisfy the chain rule. He refers to a lecture of PoincarĀ“ in 1904.

e

In [FrĀ“chet, 1937] it was shown that Hadamardā™s notion is equivalent to that of

e

Stolz-Pierpoint-Young:

S.244: āUne fonctionelle U [f ] sera dite diļ¬Ā“rentiable pour f ā” f0 au sens de M. Hadamard

e

gĀ“nĀ“ralisĀ“, sā™il existe une fonctionnelle W [df, f0 ], linĀ“aire par rapport ` df , telle que si lā™on

ee e e a

consid`re une fonction f (t, Ī») dĀ“rivable par rapport ` Ī» pour Ī» = 0, avec f (t, 0) = f0 (t), la

e e a

fonction de Ī», U [f (t, Ī»)] soit dĀ“rivable en Ī» pour Ī» = 0 et quā™on ait pour Ī» = 0

e

d df

U f (t, Ī») = W , f0

dĪ» dĪ»

ou avec les notations des āvariationsā

Ī“U [f ] = W [Ī“f, f0 ].ā

Historical remarks on the development of smooth calculus 75

S.245: āla diļ¬Ā“rentielle au sens de M. Hadamard gĀ“nĀ“ralisĀ“ qui est Ā“quivalente ` la nĖtre

e ee e e a o

dans lā™Analyse classique est plus gĀ“nĀ“rale dans lā™Analyse fonctionnelle.ā

ee

He also realized the importance of Hadamardā™s deļ¬nition:

S.249: āLā™intĀ“rĖt de la dĀ“ļ¬nition de M. Hadamard nā™est pas Ā“puisĀ“ par son utilization en

ee e e e

Analyse fonctionnelle. Il est peut-Ėtre plus encore dans la possibilitĀ“ de son extension en

e e

Analyse gĀ“nĀ“rale.

ee

Dans ce domaine, on peut gĀ“nĀ“raliser la notion de fonctionnelle et considĀ“rer des transfor-

ee e

mations X = F [x] dā˜un Ā“lĀ“ment abstrait x en un Ā“lĀ“ment abstrait X. Nous avons pu en 1925

ee ee

[FrĀ“chet, 1925b] Ā“tendre notre dĀ“ļ¬nition (rappelĀ“e plus haut p.241 et 242) de la diļ¬Ā“rentielle

e e e e e

dā˜une fonctionnelle, dĀ“ļ¬nir la diļ¬Ā“rentielle de F [x] quand X et x appartiennent ` des espaces

e e a

āvectoriels abstraits distanciĀ“sā et en etablir les propriĀ“tĀ“s les plus importantes.

e ee

La dĀ“ļ¬nition au sens de M. Hadamard gĀ“nĀ“ralisĀ“ prĀ“sente sur notre dĀ“ļ¬nition lā™avantage de

e ee ee e

garder un sens pour des espaces abstraits vectoriels non distanciĀ“s o` notre dĀ“ļ¬nition ne

eu e

sā˜applique pas. [ . . . ]

Il reste ` voir si elle conserve les propriĀ“tes les plus importantes de la diļ¬Ā“rentielle classique

a e e

en dehors de la propriĀ“tĀ“ (gĀ“nĀ“ralisant le thĀ“or`me des fonctions composĀ“es) qui lui sert de

ee e e ee e

dĀ“ļ¬nition. Cā™est un point sur lequel nous reviendrons ultĀ“rieurement.ā

e e

Hadamardā™s notion of diļ¬erentiability was later extended to inļ¬nite dimensions by

[Michal, 1938] who deļ¬ned a mapping f : E ā’ F between topological vector spaces

to be diļ¬erentiable at x if there exists a continuous linear mapping : E ā’ F

such that f ā—¦ c : R ā’ F is diļ¬erentiable at 0 with derivative ( ā—¦ c )(0) for each

everywhere diļ¬erentiable curve c : R ā’ E with c(0) = x.

Independently, a student of FrĀ“chet extended in [Ky Fan, 1942] diļ¬erentiability in

e

the sense of Hadamard to normed spaces, and proved the basic properties like the

chain rule:

S.307: āM. FrĀ“chet a eu lā™obligeance de me conseiller dā™Ā“tudier cette question quā™il avait

e e

dā™abord lā™intention de traiter lui-mĖme.ā

e

Hadamard diļ¬erentiability was further generalized to metrizable vector spaces in

[Balanzat, 1949] and to vector spaces with a sequential limit structure in [Long de

Foglio, 1960]. Finally, in [Balanzat, 1960] the theory was developed for topological

vector spaces. There he proved the chain rule and made the observation that the

implication ādiļ¬erentiable implies continuousā is equivalent to the property that

the closure of a set coincides with the sequential adherence.

Diļ¬erentiability via bornology. Here the basic observation is that convergence

which appears in questions of diļ¬erentiability is much better than just topological,

cf. (1.7). The relevant notion of convergence was introduced by [Mackey, 1945].

Diļ¬erentiability based on the von Neumann bornology was ļ¬rst considered in [Se-

bastiĖo e Silva, 1956a, 1956b, 1957]. In [SebastiĖo e Silva, 1961] he extended this

a a

to bornological vector spaces and referred to Waelbroeck and FantappiĀ“ for these

e

spaces:

ā . . . de gĀ“nĀ“raliser aux espaces localement convexes, rĀ“els ou complexes, la notion de fonc-

ee e

tion diļ¬Ā“rentiable, ainsi que les thĀ“or`mes fondamentaux du calcul diļ¬Ā“rentiel et intĀ“gral,

e ee e e

et de la thĀ“orie des fonctions analytiques de plusieurs variables complexes.

e

Je me suis persuadĀ“ que, pour cette gĀ“nĀ“ralisation, cā™est la notion dā™ensemble bornĀ“, plutĖt

e ee e o

que celle de voisinage, qui doit jouer un rĖle essentiel.ā

o

76 Chapter I. Calculus of smooth mappings

In [Waelbroeck, 1967a, 1967b] the notion of ā˜b-spaceā™ was introduced, and diļ¬er-

entiability in them was discussed. He showed that for Mackey complete spaces a

scalar-wise smooth mapping is already smooth, see (2.14.5) ā’ (2.14.4). He refers to

[Mikusinski, 1960], [Waelbroeck, 1960], [Marinescu, 1963], and [Buchwalter, 1965].

Bornological vector spaces were developed in full detail in [Hogbe-Nlend, 1970, 1971,

1977], and diļ¬erential calculus in them was further developed by [Lazet, 1971], and

[Colombeau, 1973], see also [Colombeau, 1982]. The importance of diļ¬erentiability

with respect to the bornology generated by the compact subsets was realized in

[Sova, 1966b].

An overview on diļ¬erentiability of ļ¬rst order can be found in [Averbukh, Smol-

yanov, 1968]. One ļ¬nds there 25 inequivalent deļ¬nitions of the ļ¬rst derivative in

a single point, and one sees how complicated ļ¬nite order diļ¬erentiability really is

beyond Banach spaces.

Higher derivatives. In [Maissen, 1963] it was shown that only for normed spaces

there exists a topology on L(E, E) such that the evaluation mapping L(E, E)Ć—E ā’

E is jointly continuous, and [Keller, 1965] generalized this. We have given the

archetypical argument in the introduction.

Thus, a ā˜satisfactoryā™ calculus seemed to stop at the level of Banach spaces, where

an elaborated theory including existence theorems was presented already in the

very inļ¬‚uential text book [DieudonnĀ“, 1960].

e

Beyond Banach spaces one had to use convergence structures in order to force

the continuity of the composition of linear mappings and the general chain rule.

Respective theories based on convergence were presented by [Marinescu, 1963],

[Bastiani, 1964], [FrĀØlicher, Bucher, 1966], and by [Binz, 1966]. A review is [Keller,

o

1974], where the following was shown: Continuity of the derivative implied stronger

remainder convergence conditions. So for continuously diļ¬erentiable mappings the

many possible notions collapse to 9 inequivalent ones (fewer for FrĀ“chet spaces).

e

And if one looks for inļ¬nitely often diļ¬erentiable mappings, then one ends up with

6 inequivalent notions (only 3 for FrĀ“chet spaces). Further work in this direction

e

culminated in the two huge volumes [GĀØhler, 1977, 1978], and in the historically

a

very detailed study [Ver Eecke, 1983] and [Ver Eecke, 1985].

Exponential law. The notion of homotopy makes more sense if it is viewed as a

curve I ā’ C(X, Y ). The ā˜exponential lawā™

Z XĆ—Y ā¼ (Z Y )X , or C(X Ć— Y, Z) ā¼ C(X, C(Y, Z)),

= =

however, is not true in general. It holds only for compactly generated spaces, as was

shown by [Brown, 1961], see also [Gabriel, Zisman, 1963/64], or for compactly con-

tinuous mappings between arbitrary topological spaces, due to [Brown, 1963] and

[Brown, 1964]. Without referring to Brown in the text, [Steenrod, 1967] made this

result really popular under the title ā˜a convenient category of topological spacesā™,

which is the source of the widespread use of ā˜convenientā™, also in this book. See

also [Vogt, 1971].

Historical remarks on the development of smooth calculus 77

Following the advise of A. FrĀØlicher, [Seip, 1972] used compactly generated vector

o

spaces for calculus. In [Seip, 1976] he obtained a cartesian closed category of

smooth mappings between compactly generated vector spaces, and in [Seip, 1979]

he modiļ¬ed his calculus by assuming both smoothness along curves and compact

continuity, for all derivatives. Based on this, he obtained a cartesian closed category

of ā˜smooth manifoldsā™ in [Seip, 1981] by replacing atlas of charts by the set of smooth

curves and assuming a kind of (Riemannian) exponential mapping which he called

local addition.

Motivated by Seipā™s work in the thesis [Kriegl, 1980], supervised by Peter Mi-

chor, smooth mappings between arbitrary subsets ā˜Vektormengenā™ of locally convex

spaces were supposed to respect smooth curves and to induce ā˜tangent mappingsā™

which again should respect smooth curves, and so on. On open subsets of E map-

pings turned out to be smooth if they were smooth along smooth mappings Rn ā’ E

for all n. This gave a cartesian closed setting of calculus without any assumptions

on compact continuity of derivatives. A combination of this with the result of [Bo-

man, 1967] then quickly lead to [Kriegl, 1982] and [Kriegl, 1983], one of the sources

of this book.

Independently, [FrĀØlicher, 1980] considered categories generated by monoids of real

o

valued functions and characterized cartesian closedness in terms of the monoid.

[FrĀØlicher, 1981] used the result of [Boman, 1967] to show that on FrĀ“chet spaces

o e

usual smoothness is equivalent to smoothness in the sense of the category generated

by the monoid C ā (R, R). That this category is cartesian closed was shown in the

unpublished paper [Lawvere, Schanuel, Zame, 1981].

Already [Boman, 1967] used Lipschitz conditions for his result on ļ¬nite order dif-

ferentiability, since it fails to be true for C n -functions. Motivated by this, ļ¬nite

diļ¬erentiability based on Lipschitz conditions has then been developed by [FrĀØlicher,

o

Gisin, Kriegl, 1983]. A careful presentation can be found in the monograph [FrĀØli-o

cher, Kriegl, 1988]. Finite diļ¬erentiability based on HĀØlder conditions were studied

o

by [Faure, 1989] and [Faure, 1991].

78

79

Chapter II

Calculus of Holomorphic

and Real Analytic Mappings

7. Calculus of Holomorphic Mappings . . . . . . . .... . . . . . . 80

8. Spaces of Holomorphic Mappings and Germs . . .... . . . . . . 91

9. Real Analytic Curves . . . . . . . . . . . . . .... . . . . . . 97

10. Real Analytic Mappings . . . . . . . . . . . .... . . . . . . 101

11. The Real Analytic Exponential Law . . . . . . .... . . . . . . 105

Historical Remarks on Holomorphic and Real Analytic Calculus . . . . . 116

This chapter starts with an investigation of holomorphic mappings between inļ¬nite

dimensional vector spaces along the same lines as we investigated smooth mappings

in chapter I. This theory is rather easy if we restrict to convenient vector spaces.

The basic tool is the set of all holomorphic mappings from the unit disk D ā‚ C

into a complex convenient vector space E, where all possible deļ¬nitions of being

holomorphic coincide, see (7.4). This replaces the set of all smooth curves in the

smooth theory. A mapping between cā -open sets of complex convenient vector

spaces is then said to be holomorphic if it maps holomorphic curves to holomorphic

curves. This can be tested by many equivalent descriptions (see (7.19)), the most

important are that f is smooth and df (x) is complex linear for each x (i.e. f satisļ¬es

the Cauchy-Riemann diļ¬erential equation); or that f is holomorphic along each

aļ¬ne complex line and is cā -continuous (generalized Hartogā™s theorem). Again

(multi-) linear mappings are holomorphic if and only if they are bounded (7.12).

The space H(U, F ) of all holomorphic mappings from a cā -open set U ā E into

a convenient vector space F carries a natural structure of a complex convenient

vector space (7.21), and satisļ¬es the holomorphic uniform boundedness principle

(8.10). Of course our general aim of cartesian closedness (7.22), (7.23) is valid also

in this setting: H(U, H(V, F )) ā¼ H(U Ć— V, F ).

=

As in the smooth case we have to pay a price for cartesian closedness: holomorphic

mappings can be expanded into power series, but these converge only on a cā -open

subset in general, and not on open subsets.

The second part of this chapter is devoted to real analytic mappings in inļ¬nite di-

mensions. The ideas are similar as in the case of smooth and holomorphic mappings,

but our wish to obtain cartesian closedness forces us to some modiļ¬cations: In (9.1)

we shall see that for the real analytic mapping f : R2 (s, t) ā’ (st)1 +1 ā R there is

2

no reasonable topology on C Ļ (R, R), such that the mapping f āØ : R ā’ C Ļ (R, R) is

80 Chapter II. Calculus of holomorphic and real analytic mappings 7.1

locally given by its convergent Taylor series, which looks like a counterexample to

cartesian closedness. Recall that smoothness (holomorphy) of curves can be tested

by applying bounded linear functionals (see (2.14), (7.4)). The example above

shows at the same time that this is not true in the real analytic case in general; if

E carries a Baire topology then it is true (9.6).

So we are forced to take as basic tool the space C Ļ (R, E) of all curves c such that

ā—¦ c : R ā’ R is real analytic for each bounded linear functional, and we call these

the real analytic curves. In order to proceed we have to show that real analyticity of

a curve can be tested with any set of bounded linear functionals which generates the

bornology. This is done in (9.4) with the help of an unusual bornological description

of real analytic functions R ā’ R (9.3).

Now a mapping f : U ā’ F is called real analytic if f ā—¦ c is smooth for smooth c

and is real analytic for real analytic c : R ā’ U . The second condition alone is not

suļ¬cient, even for f : R2 ā’ R. Then a version of Hartogā™s theorem is true: f is real

analytic if and only if it is smooth and real analytic along each aļ¬ne line (10.4).

In order to get to the aim of cartesian closedness we need a natural structure of a

convenient vector space on C Ļ (U, F ). We start with C Ļ (R, R) which we consider as

real part of the space of germs along R of holomorphic functions. The latter spaces

of holomorphic germs are investigated in detail in section (8). At this stage of

the theory we can prove the real analytic uniform boundedness theorem (11.6) and

(11.12), but unlike in the smooth and holomorphic case for the general exponential

law (11.18) we still have to investigate mixing of smooth and real analytic variables

in (11.17). The rest of the development of section (11) then follows more or less

standard (categorical) arguments.

7. Calculus of Holomorphic Mappings

7.1. Basic notions in the complex setting. In this section all locally convex

spaces E will be complex ones, which we can view as real ones ER together with

continuous linear mapping J with J 2 = ā’ Id (the complex structure). So all con-

cepts for real locally convex spaces from sections (1) to (5) make sense also for

complex locally convex spaces.

A set which is absolutely convex in the real sense need not be absolutely convex

in the complex sense. However, the C-absolutely convex hull of a bounded subset

is still bounded, since there is a neighborhood basis of 0 consisting of C-absolutely

convex sets. So in this section absolutely convex will refer always to the complex

notion. For absolutely convex bounded sets B the real normed spaces EB (see

(1.5)) inherit the complex structure.

A complex linear functional on a convex vector space is uniquely determined by

ā

its real part Re ā—¦ , by (x) = (Re ā—¦ )(x) ā’ ā’1(Re ā—¦ )(Jx). So for the respective

spaces of bounded linear functionals we have

ER = LR (ER , R) ā¼ LC (E, C) =: E ā— ,

=

7.1

7.4 7. Calculus of holomorphic mappings 81

where the complex structure on the left hand side is given by Ī» ā’ Ī» ā—¦ J.

7.2. Deļ¬nition. Let D be the the open unit disk {z ā C : |z| < 1}. A mapping

c : D ā’ E into a locally convex space E is called complex diļ¬erentiable, if

c(z + w) ā’ c(z)

c (z) = lim

w

wā’0

C

exists for all z ā D.

7.3. Lemma. Let E be convenient and an ā E. Then the following statements

are equivalent:

(1) {rn an : n ā N} is bounded for all |r| < 1.

(2) The power series nā„0 z n an is Mackey convergent in E, uniformly on each

compact subset of D, i.e., the Mackey coeļ¬cient sequence and the bounded

set can be chosen valid in the whole compact subset.

(3) The power series converges weakly for all z ā D.

Proof. (1) ā’ (2) Any compact set is contained in rD for some 0 < r < 1, the

set {Rn an : n ā N} is contained in some absolutely convex bounded B for some

r < R < 1. So the partial sums of the series form a Mackey Cauchy sequence

uniformly on rD since

M

1 1

z n an ā B.

(r/R)N ā’ (r/R)M +1 1 ā’ (r/R)

n=N

(2) ā’ (3) is clear.

Proof of (3) ā’ (1) The summands are weakly bounded, thus bounded.

7.4. Theorem. If E is convenient then the following statements for a curve c :

D ā’ E are equivalent:

(1) c is complex diļ¬erentiable.

(2) ā—¦ c : D ā’ C is holomorphic for all ā E ā—

(3) c is continuous and Ī³ c = 0 in the completion of E for all closed smooth

(Lip0 -) curves in D.

ā z n (n)

(4) All c(n) (0) exist and c(z) = n=0 n! c (0) is Mackey convergent, uni-

formly on each compact subset of D.

n

ā

(5) For each z ā D all c(n) (z) exist and c(z + w) = n=0 w c(n) (z) is Mackey

n!

convergent, uniformly on each compact set in the largest disk with center z

contained in D.

(6) c(z)dz is a closed Lip1 1-form with values in ER .

(7) c is the complex derivative of some complex curve in E.

(8) c is smooth (Lip1 ) with complex linear derivative dc(z) for all z.

From now on all locally convex spaces will be convenient. A curve c : D ā’ E

satisfying these equivalent conditions will be called a holomorphic curve.

7.4

82 Chapter II. Calculus of holomorphic and real analytic mappings 7.6

Proof. (2) ā’ (1) By assumption, the diļ¬erence quotient c(z+w)ā’c(z) , composed

w

with a linear functional, extends to a complex valued holomorphic function of w,

hence it is locally Lipschitz. So the diļ¬erence quotient is a Mackey Cauchy net. So

it has a limit for w ā’ 0.

Proof of (1) ā’ (2) Suppose that is bounded. Let c : D ā’ E be a complex

diļ¬erentiable curve. Then c1 : z ā’ z c(z)ā’c(0) ā’ c (0) is a complex diļ¬erentiable

1

z

curve (test with linear functionals), hence

(c(z)) ā’ (c(0))

1

( ā—¦ c1 )(z) = ā’ (c (0))

z z

is locally bounded in z. So ā—¦ c is complex diļ¬erentiable with derivative ā—¦ c .

Composition with a complex continuous linear functional translates all statements

to one dimensional versions which are all equivalent by complex analysis. Moreover,

each statement is equivalent to its weak counterpart, where for (4) and (5) we use

lemma (7.3).

7.5. Remarks. In the holomorphic case the equivalence of (7.4.1) and (7.4.2)

does not characterize cā -completeness as it does in the smooth case. The complex

diļ¬erentiable curves do not determine the bornology of the space, as do the smooth

ones. See [Kriegl, Nel, 1985, 1.4]. For a discussion of the holomorphic analogues of

smooth characterizations for cā -completeness (see (2.14)) we refer to [Kriegl, Nel,

1985, pp. 2.16].

7.6. Lemma. Let c : D ā’ E be a holomorphic curve in a convenient space. Then

locally in D the curve factors to a holomorphic curve into EB for some bounded

absolutely convex set B.

First Proof. By the obvious extension of lemma (1.8) for smooth mappings R2 ā

D ā’ E the curve c factors locally to a Lip1 -curve into some complete EB . Since it

has complex linear derivative, by theorem (7.4) it is holomorphic.

Second direct proof. Let W be a relatively compact neighborhood of some point

in D. Then c(W ) is bounded in E. It suļ¬ces to show that for the absolutely convex

closed hull B of c(W ) the Taylor series of c at each z ā W converges in EB , i.e.

that c|W : W ā’ EB is holomorphic. This follows from the

Vector valued Cauchy inequalities. If r > 0 is smaller than the radius of

convergence at z of c then

r k (k)

(z) ā B

k! c

where B is the closed absolutely convex hull of { c(w) : |w ā’ z| = r}. (By the

Hahn-Banach theorem this follows directly from the scalar valued case.)

Thus, we get

r k (k)

m m

wā’z k wā’z k

Ā· ā Ā·B

k=n ( r ) k! c (z) k=n ( r )

c(k) (z)

ā’ z)k is convergent in EB for |w ā’ z| < r.

and so k! (w

k

7.6

7.10 7. Calculus of holomorphic mappings 83

This proof also shows that holomorphic curves with values in complex convenient

vector spaces are topologically and bornologically holomorphic in the sense analo-

gous to (9.4).

7.7. Lemma. Let E be a regular (i.e. every bounded set is contained and bounded

in some step EĪ± ) inductive limit of complex locally convex spaces EĪ± ā E, let

c : C ā U ā’ E be a holomorphic mapping, and let W ā C be open and such that

the closure W is compact and contained in U . Then there exists some Ī±, such that

c|W : W ā’ EĪ± is well deļ¬ned and holomorphic.

Proof. By lemma (7.6) the restriction of c to W factors to a holomorphic curve

c|W : W ā’ EB for a suitable bounded absolutely convex set B ā E. Since B is

contained and bounded in some EĪ± one has c|W : W ā’ EB = (EĪ± )B ā’ EĪ± is

holomorphic.

7.8. Deļ¬nition. Let E and F be convenient vector spaces and let U ā E be

cā -open. A mapping f : U ā’ F is called holomorphic, if it maps holomorphic

curves in U to holomorphic curves in F .

It is remarkable that [FantappiĀ“, 1930] already gave this deļ¬nition. Connections to

e

other concepts of holomorphy are discussed in [Kriegl, Nel, 1985, 2.19].

So by (7.4) f is holomorphic if and only if ā—¦f ā—¦c : D ā’ C is a holomorphic function

for all ā F ā— and holomorphic curve c.

Clearly, any composition of holomorphic mappings is again holomorphic.

For ļ¬nite dimensions this coincides with the usual notion of holomorphic mappings,

by the ļ¬nite dimensional Hartogsā™ theorem.

7.9. Hartogsā™ Theorem. Let E1 , E2 , and F be convenient vector spaces with U

cā -open in E1 Ć— E2 . Then a mapping f : U ā’ F is holomorphic if and only if it

is separately holomorphic, i.e. f ( , y) and f (x, ) are holomorphic.

Proof. If f is holomorphic then f ( , y) is holomorphic on the cā -open set E1 Ć—

{y} ā© U = inclā’1 (U ), likewise for f (x, ).

y

If f is separately holomorphic, for any holomorphic curve (c1 , c2 ) : D ā’ U ā E1 Ć—E2

we consider the holomorphic mapping c1 Ć— c2 : D2 ā’ E1 Ć— E2 . Since the ck are

smooth by (7.4.8) also c1 Ć— c2 is smooth and thus (c1 Ć— c2 )ā’1 (U ) is open in C2 .

For each Ī» ā F ā— the mapping Ī» ā—¦ f ā—¦ (c1 Ć— c2 ) : (c1 Ć— c2 )ā’1 (U ) ā’ C is separately

holomorphic and so holomorphic by the usual Hartogsā™ theorem. By composing

with the diagonal mapping we see that Ī» ā—¦ f ā—¦ (c1 , c2 ) is holomorphic, thus f is

holomorphic.

7.10. Lemma. Let f : E ā U ā’ F be holomorphic from a cā -open subset in

a convenient vector space to another convenient vector space. Then the derivative

(df )ā§ : U Ć— E ā’ F is again holomorphic and complex linear in the second variable.

Proof. (z, v, w) ā’ f (v + zw) is holomorphic. We test with all holomorphic curves

ā‚

and linear functionals and see that (v, w) ā’ ā‚z |z=0 f (v + zw) =: df (v)w is again

holomorphic, C-homogeneous in w by (7.4).

7.10

84 Chapter II. Calculus of holomorphic and real analytic mappings 7.13

Now w ā’ df (v)w is a holomorphic and C-homogeneous mapping E ā’ F . But

any such mapping is automatically C-linear: Composed with a bounded linear

functional on F and restricted to any two dimensional subspace of E this is a ļ¬nite

dimensional assertion.

7.11. Remark. In the deļ¬nition of holomorphy (7.8) one could also have admitted

subsets U which are only open in the ļ¬nal topology with respect to holomorphic

curves. But then there is a counterexample to (7.10), see [Kriegl, Nel, 1985, 2.5].

7.12. Theorem. A multilinear mapping between convenient vector spaces is holo-

morphic if and only if it is bounded.

This result is false for not cā -complete vector spaces, see [Kriegl, Nel, 1985, 1.4].

Proof. Since both conditions can be tested in each factor separately by Hartogsā™

theorem (7.9) and by (5.19), and by testing with linear functionals, we may restrict

our attention to linear mappings f : E ā’ C only.

By theorem (7.4.2) a bounded linear mapping is holomorphic. Conversely, suppose

that f : E ā’ C is a holomorphic but unbounded linear functional. So there exists

a sequence (an ) in E with |f (an )| > 1 and {2n an } bounded. Consider the power

ā

series n=0 (an ā’ anā’1 )(2z)n . This describes a holomorphic curve c in E, by (7.3)

and (7.4.2). Then f ā—¦ c is holomorphic and thus has a power series expansion

ā

f (c(z)) = n=0 bn z n . On the other hand

N

(f (an ) ā’ f (anā’1 ))(2z)n + (2z)N f (an ā’ anā’1 )(2z)nā’N

f (c(z)) = .

n=0 n>N

So bn = 2n (f (an ) ā’ f (anā’1 )) and we get the contradiction

ā

(f (an ) ā’ f (anā’1 )) = lim f (an ).

0 = f (0) = f (c(1/2)) =

nā’ā

n=0

Parts of the following results (7.13) to (10.2) can be found in [Bochnak, Siciak,

1971]. For x in any vector space E let xk denote the element (x, . . . , x) ā E k .

7.13. Lemma. Polarization formulas. Let f : E Ć— Ā· Ā· Ā· Ć— E ā’ F be an k-linear

symmetric mapping between vector spaces. Then we have:

1

k

(ā’1)kā’Ī£Īµj f (x0 +

1

(1) f (x1 , . . . , xk ) = Īµj xj ) .

k!

Īµ1 ,...,Īµk =0

k

k

k

(ā’1)kā’j

1

f ((a + jx)k ).

(2) f (x ) = k! j

j=0

k

kk j

k

k

(ā’1)kā’j f ((a + k x)k ).

(3) f (x ) = k! j

j=0

1

Ī»Ī£Īµj f (xĪµ1 , . . . , xĪµk ).

f (x0 Ī»x1 , . . . , x0 Ī»x1 )

(4) + + =

1 1 k k 1 k

Īµ1 ,...,Īµk =0

7.13

7.14 7. Calculus of holomorphic mappings 85

ā

ā’1 in the passage to the complexiļ¬cation.

Formula (4) will mainly be used for Ī» =

Proof. (1). (see [Mazur, Orlicz, 1935]). By multilinearity and symmetry the right

hand side expands to

Aj0 ,...,jk

f (x0 , . . . , x0 , . . . , xk , . . . , xk ),

j0 ! Ā· Ā· Ā· jk !

j0 +Ā·Ā·Ā·+jk =k j0 jk

where the coeļ¬cients are given by

1

(ā’1)kā’Ī£Īµj Īµj1 Ā· Ā· Ā· Īµjk .

Aj0 ,...,jk = 1 k

Īµ1 ,...,Īµk =0

The only nonzero coeļ¬cient is A0,1,...,1 = 1.

(2). In formula (1) we put x0 = a and all xj = x.

(3). In formula (2) we replace a by ka and pull k out of the k-linear expression

f ((ka + jx)k ).

(4) is obvious.

7.14. Lemma. Power series. Let E be a real or complex FrĀ“chet space and let

e

fk be a k-linear symmetric scalar valued bounded functional on E, for each k ā N.

Then the following statements are equivalent:

k

(1) k fk (x ) converges pointwise on an absorbing subset of E.

k

(2) k fk (x ) converges uniformly and absolutely on some neighborhood of 0.

(3) {fk (xk ) : k ā N, x ā U } is bounded for some neighborhood U of 0.

(4) {fk (x1 , . . . , xk ) : k ā N, xj ā U } is bounded for some neighborhood U of 0.

If any of these statements are satisļ¬ed over the reals, then also for the complexiļ¬-

cation of the functionals fk .

Proof. (1) ā’ (3) The set AK,r := {x ā E : |fk (xk )| ā¤ Krk for all k} is closed

in E since every bounded multilinear mapping is continuous. The countable union

K,r AK,r is E, since the series converges pointwise on an absorbing subset. Since

E is Baire there are K > 0 and r > 0 such that the interior U of AK,r is non

void. Let x0 ā U and let V be an absolutely convex neighborhood of 0 contained

in U ā’ x0

From (7.13) (3) we get for all x ā V the following estimate:

k

k j

k

k k

|f ((x0 + k x)k )|

|f (x )| ā¤ k! j

j=0

kk k k

ā¤ K(2re)k .

ā¤ k! 2 Kr

1

Now we replace V by V and get the result.

2re

7.14

86 Chapter II. Calculus of holomorphic and real analytic mappings 7.17

(3) ā’ (4) From (7.13) (1) we get for all xj ā U the estimate:

1

k

1

|f (x1 , . . . , xk )| ā¤ |f ( |

Īµj xj )

k!

Īµ1 ,...,Īµk =0

1 k

Īµj xj

k

1

= ( Īµj ) f

k! Īµj

Īµ1 ,...,Īµk =0

1

k

1

ā¤ ( Īµj ) C

k!

Īµ1 ,...,Īµk =0

k

k

1

j k C ā¤ C(2e)k .

ā¤ k! j

j=0

1

Now we replace U by U and get (4).

2e

Proof of (4) ā’ (2) The series converges on rU uniformly and absolutely for any

0 < r < 1.

(2) ā’ (1) is clear.

ā

(4), real case, ā’ (4), complex case, by (7.13.4) for Ī» = ā’1.

7.15. Lemma. Let E be a complex convenient vector space and let fk be a k-linear

symmetric scalar valued bounded functional on E, for each k ā N. If k fk (xk )

converges pointwise on E and x ā’ f (x) := k fk (xk ) is bounded on bounded sets,

then the power series converges uniformly on bounded sets.

Proof. Let B be an absolutely convex bounded set in E. For x ā 2B we apply the

vector valued Cauchy inequalities from (7.6) to the holomorphic curve z ā’ f (zx)

at z = 0 for r = 1 and get that fk (xk ) is contained in the closed absolutely convex

hull of {f (zx) : |z| = 1}. So {fk (xk ) : x ā 2B, k ā N} is bounded and the series

converges uniformly on B.

7.16. Example. We consider the power series k k(xk )k on the Hilbert space

2

= {x = (xk ) : k |xk |2 < ā}. This series converges pointwise everywhere, it

yields a holomorphic function f on 2 by (7.19.5) which however is unbounded on

the unit sphere, so convergence cannot be uniform on the unit sphere.

The function g : C(N) Ć— 2 ā’ C given by g(x, y) := k xk f (kx1 y) is holomor-

phic since it is a ļ¬nite sum locally along each holomorphic curve by (7.7), but its

Taylor series at 0 does not converge uniformly on any neighborhood of 0 in the

locally convex topology: A typical neighborhood is of the form {(x, y) : |xk | ā¤

Īµk for all k, y 2 ā¤ Īµ} and so it contains points (x, y) with |xk f (kx1 y)| ā„ 1, for all

large k. This shows that lemma (7.14) is not true for arbitrary convenient vector

spaces.

7.17. Corollary. Let E be a real or complex FrĀ“chet space and let fk be a k-

e

linear symmetric scalar valued bounded functional on E, for each k ā N such that

7.17

7.18 7. Calculus of holomorphic mappings 87

fk (xk ) converges to f (x) for x near 0 in E. Let ak z k be

the power series kā„1

a power series in E which converges to a(z) ā E for z near 0 in C.

Then the composite

fn (ak1 , . . . , akn ) z k

kā„0 nā„0 k1 ,...,kn āN

k1 +Ā·Ā·Ā·+kn =k

of the power series converges to f ā—¦ a near 0.

Proof. By (7.14) there exists a 0-neighborhood U in E such that {fk (x1 , . . . , xk ) :

k ā N, xj ā U } is bounded. Since the series for a converges there is r > 0 such that

r

ak rk ā U for all k. For |z| < 2 we have

ak1 z k1 , . . . , akn z kn

f (a(z)) = fn

k1 ā„1 kn ā„1

nā„0

fn ak1 , . . . , akn z k1 +Ā·Ā·Ā·+kn

Ā·Ā·Ā·

=

nā„0 k1 ā„1 kn ā„1

fn (ak1 , . . . , akn ) z k ,

=

kā„0 nā„0 k1 ,...,kn āN

k1 +Ā·Ā·Ā·+kn =k

since the last complex series converges absolutely: the coeļ¬cient of z k is a sum of

2k ā’ 1 terms which are bounded when multiplied by rk . The second equality follows

from boundedness of all fk .

7.18. Almost continuous functions. In the proof of the next theorem we will

need the following notion: A (real valued) function on a topological space is called

almost continuous if removal of a meager set yields a continuous function on the

remainder.

Lemma. [Hahn, 1932, p. 221] A pointwise limit of a sequence of almost continuous

functions on a Baire space is almost continuous.

Proof. Let (fk ) be a sequence of almost continuous real valued functions on a Baire

space X which converges pointwise to f . Since the complement of a meager set in

a Baire space is again Baire we may assume that each function fk is continuous

on X. We denote by Xn the set of all x ā X such that there exists N ā N and a

1

neighborhood U of x with |fk (y) ā’ f (y)| < n for all k ā„ N and all y ā U . The set

Xn is clearly open.

We claim that each Xn is dense: Let V be a nonempty open subset of X. For

1

N ā N the set VN := {x ā V : |fk (x) ā’ f (x)| ā¤ 2n for all k, ā„ N } is closed

in V and V = N VN since the sequence (fk ) converges pointwise. Since V is a

Baire space, some VN contains a nonempty open set W . For each y ā W we have

1

|fk (y) ā’ f (y)| ā¤ 2n for all k, ā„ N . We take the pointwise limit for ā’ ā and

see that W ā V ā© Xn .

Since X is Baire, the set n Xn has a meager complement and obviously the re-

striction of f on this set is continuous.

7.18

88 Chapter II. Calculus of holomorphic and real analytic mappings 7.19

7.19. Theorem. Let f : E ā U ā’ F be a mapping from a cā -open subset in

a convenient vector space to another convenient vector space. Then the following

assertions are equivalent:

(1) f is holomorphic.

(2) For all ā F ā— and absolutely convex closed bounded sets B the mapping

ā—¦ f : EB ā’ C is holomorphic.

(3) f is holomorphic along all aļ¬ne (complex) lines and is cā -continuous.

(4) f is holomorphic along all aļ¬ne (complex) lines and is bounded on bornolog-

ically compact sets (i.e. those compact in some EB ).

(5) f is holomorphic along all aļ¬ne (complex) lines and at each point the ļ¬rst

derivative is a bounded linear mapping.

(6) f is cā -locally a convergent series of bounded homogeneous complex poly-

nomials.

(7) f is holomorphic along all aļ¬ne (complex) lines and in every connected

component for the cā -topology there is at least one point where all deriva-

tives are bounded multilinear mappings.

(8) f is smooth and the derivative is complex linear at every point.

(9) f is Lip1 in the sense of (12.1) and the derivative is complex linear at every

point.

Proof. (1) ā” (2) By (7.6) every holomorphic curve factors locally over some EB

and we test with linear functionals on F .

So for the rest of the proof we may assume that F = C. We prove the rest of the

theorem ļ¬rst for the case where E is a Banach space.

(1) ā’ (5) By lemma (7.10) the derivative of f is holomorphic and C-linear in the

second variable. By (7.12) f (z) is bounded.

(5) ā’ (6) Choose a ļ¬xed point z ā U . Since f is holomorphic along each complex

line through z it is given there by a pointwise convergent power series. By the

classical Hartogsā™ theorem f is holomorphic along each ļ¬nite dimensional linear

subspace. The mapping f : E ā U ā’ E is well deļ¬ned by assumption and is also

holomorphic along each aļ¬ne line since we may test this by all point evaluations:

using (5.18) we see that it is smooth and by (7.4.8) it is a holomorphic curve. So

the mapping

v ā’ f (n+1) (z)(v, v1 , . . . , vn ) = (f ( )(v))(n) (z)(v1 , . . . , vn )

= (f )(n) (z)(v1 , . . . , vn )(v).

is bounded, and by symmetry of higher derivatives at z they are thus separately

bounded in all variables. By (5.19) f is given by a power series of bounded homo-

geneous polynomials which converges pointwise on the open set {z + v : z + Ī»v ā

U for all |Ī»| ā¤ 1}. Now (6) follows from lemma (7.14).

(6) ā’ (3) By lemma (7.14) the series converges uniformly and hence f is continuous.

(3) ā’ (4) is obvious.

7.19

7.19 7. Calculus of holomorphic mappings 89

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

1 f (z + Ī»v)

ā

f (z)v = dĪ».

Ī»2

2Ļ ā’1 |Ī»|=1

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

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

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

see corollary (7.17).

Sublemma. Let E be a FrĀ“chet space and let U ā E be open. Let f : U ā’ C be

e

holomorphic along aļ¬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 ā’ C as pointwise limit of diļ¬erence quotients is also almost continuous

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

Baire property.

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

(6) ā’ (7) is obvious.

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

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

is open. By (6) ā’ (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 + Ī»v ā U for all |Ī»| ā¤ 1} is contained in U . Since

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

(8) ā’ (9) is trivial.

(9) ā’ (3) Clearly, f is holomorphic along aļ¬ne lines and cā -continuous.

(1) ā’ (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 ā© EB ) satisļ¬es condition (6) (so

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

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

cā -open neighborhood of z. The star {z + v : z + Ī»v ā U for all |Ī»|le1} with center

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

pointwise.

For (7) replace on both sides the condition āat least one pointā by the condition

āfor all pointsā.

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) ā” (7.19.8), and the real chain rule (3.18).

7.21. Deļ¬nition. For convenient vector spaces E and F and for a cā -open subset

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

is a closed linear subspace of C ā (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 cā -open subsets Uj ā Ej a mapping f : U1 Ć— U2 ā’ F is holo-

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

holomorphic.

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

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

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

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

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

sian closedness. The derivative df (x, y)(u, v) = (df āØ (x)v)(y) + (d ā—¦ f āØ )(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 cā -open

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

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

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

)ā§ : H(U, H(V, G)) ā’ H(U Ć— V, G)

(

)āØ : H(U Ć— V, G) ā’ H(U, H(V, G))

(

comp : H(F, G) Ć— H(U, F ) ā’ H(U, G)

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

,

(f, g) ā’ (h ā’ f ā—¦ h ā—¦ g)

H(Ei , Fi ) ā’ H(

: Ei , Fi )

Proof. Just consider the canonically associated holomorphic mappings on multiple

products.

7.24. Theorem (Holomorphic functions on FrĀ“chet spaces).

e

Let U ā E be open in a complex FrĀ“chet space E. The following statements on

e

f : U ā’ 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) ā’ (2) follows from (7.14.1) ā’ (7.14.2) and (7.19.1) ā’ (7.19.6).

(2) ā’ (3) is obvious.

(3) ā’ (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 ā 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 ā¤ p ā¤ ā let denote the

r

n

real Banach space x ā RN : (xĪ± rĪ± )Ī±āNn p < ā .

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 FrĀ“chet space and

e

embeds bornologically as a closed subspace into C ā (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 FrĀ“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 ā (N, R2 ) = C ā (N, R)2 since a function

N ā’ 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

ā ā

equations df (x)( ā’1 Ā· v) = ā’1 Ā· 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 FrĀ“chet spaces it is an isomorphism.

e

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

as follows. For N equal to the open polycylinder Dn ā 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 (uā’1 )ā— : H(N, C) ā’ 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 ā N let

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

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

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

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

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

For A1 ā A2 ā N the ārestrictionā mappings H(N ā A2 , C) ā’ H(N ā A1 , C) are

continuous.

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

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

a

[KĀØthe, 1953], and [Grothendieck, 1953].

o

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

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

H(N ā K, C) ā {H(W, C), N ā W ā K}

H(N ā K, C) ā {Hb (W, C), N ā W ā K}

H(N ā K, C) ā {Hbc (N ā W, C), N ā W ā K}

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

to each other.

H(N ā {z}, C) ā { p ā— C, r ā Rn }

r +

So all inductive limit topologies coincide. Furthermore, the space H(N ā 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 ā K, C) is complete and (ultra-)bornological (hence a convenient

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

FrĀ“chet space. The space H(N ā K, C) is smoothly paracompact. It is however not

e

a Baire space.

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

Then the obvious mappings

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

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

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

r r r s r

ā

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

So the remaining coļ¬nality assertion follows.

Let us show next that the connecting mapping Hb (W, C) ā’ 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 ā’ 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 deļ¬nition some larger 0-neighborhood U in E such that the natural mapping

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

nuclear, since it can be factorized as

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

So H(N ā 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 ā K, C)

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

Completeness of H(Cn ā K, C) was shown in [Van Hove, 1952, thĀ“or`me II], and

ee

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

o

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

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

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

o

dieck, 1953,] thĀ“or`me 2 bis, to arbitrary subsets A ā S 2 .

ee

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

1986].

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

manifolds N ). The inductive cone

H(N ā A, C) ā { H(W, C) : A ā W ā N }

open

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

The projective cone

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

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

The space H(N ā A, C) is Montel (hence quasi-complete and reļ¬‚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 ā’ R which is positive and proper. Then

(f ā’1 ([n, n + 1]))nāN0 is an exhaustion of N by compact subsets and (Kn := A ā©

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

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

each compact subset K of A. Since the cone

{H(W, C) : K ā W ā N } ā’ H(N ā 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 ā©Kn+1 ā WKn ā© WKn+1 ā© f ā’1 ((n, n + 2)). Then we

let W be the union of those connected components of

(WKn ā© f ā’1 ((n, n + 1))) āŖ

W := WKn ā©Kn+1

n n

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

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

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

extension of f ā 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 ļ¬nitely

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

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

complete, and conuclear, as regular inductive limit of the nuclear FrĀ“chet spaces

e

H(W, C).

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

jective limit of webbed spaces H(N ā 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 ā A, C) ā’ { H(C ā K, C), K compact in A }

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

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

be a continuous linear functional on H(C ā A, C). Then we have Ī» ā Hā (S 2 ā

S 2 \ A, C) by (8.5). Hence, Ī» ā Hā (U, C) for some open neighborhood U of S 2 \ A,

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

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

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

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

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

A of complex manifolds N ). The projective cone

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

generates the bornology.

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

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

we ļ¬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 ā’ A of A in N . We may assume that Wz

is contained in U , has ļ¬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 ā W can be covered by ļ¬nitely many Wz .

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

subsets A ā N .

1

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

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

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

functions fn : {x + iy ā C : x = rn } ā’ 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 ā N. Then B ā H(C ā 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 ā H(C ā {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 ā E be a cā -open

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

evaluations evx , x ā U .

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

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

rating set S of bounded linear functionals.

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

ding into C ā (U, F ) and hence satisļ¬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 ā 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 ļ¬rst that A is the point 0. We will show that condition (5.22.3) is

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

1 (k)

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

sup{ |bn,k | : n ā N } < ā.

(1)

1

We have to ļ¬nd (tn ) ā 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 ā N } < ā.

(2)

By theorem (8.4) the space H({0}, C) is a regular inductive limit of spaces ā . 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 ā B, k ā 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 ā N, k > 0 } is bounded, so

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

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

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

|bn,km | < C for some C and all n by (1), but |bnm ,km | > mkm ā’ ā for some nm .

Since by (1) the set { bn,km : n ā N } is bounded, we can choose nm ā Nm such

that

1

|bnm ,km | ā„ 2 |bj,km | for j > nm

(3)

|bnm ,km | > mkm

We can choose also (nm ) strictly increasing, for if they were bounded we would get

1

|bnm ,km rkm | < C for some r > 0 and C by (2). But ( m )km ā’ 0.

We pass now to the subsequence (bnm ) which we denote again by (bm ). We put

ļ£« ļ£¶

1 1

tj bj,km ļ£ø Ā·

(4) tm := sign ļ£ .

4m

bm,km j<m

Assume now that bā = m tm bm converges weakly with respect to S to a holomor-

phic germ. Then its Taylor series is bā (z) = kā„0 bā,k z k , where the coeļ¬cients

are given by bā,k = mā„0 tm bm,k . But we may compute as follows, using (3) and

(4) :

|bā,km | ā„ tj bj,km ā’ |tj bj,km |

j>m

jā¤m

8.10

9.1 9. Real analytic curves 97

tj bj,km + |tm bm,km |

= (same sign)

j<m

ā’ |tj bj,km | ā„

j>m

ļ£« ļ£¶

ā„ 0 + |bm,km | Ā· ļ£|tm | ā’ 2 |tj |ļ£ø

j>m

mkm

1

= |bm,km | Ā· ā„ .

3 Ā· 4m 3 Ā· 4m

So |bā,km |1/km goes to ā, hence bā cannot have a positive radius of convergence,

a contradiction. So the theorem follows for the space H({t}, C).

Let us consider now an arbitrary closed smooth submanifold A ā C. By (8.8) the

projective cone H(N ā A, C) ā’ {H(N ā {z}, C), z ā A} generates the bornology.

Hence, the result follows from the case where A = {0} by (5.25).

9. Real Analytic Curves

9.1. As for smoothness and holomorphy we would like to obtain cartesian closed-

ness for real analytic mappings. Thus, one should have at least the following:

f : R2 ā’ R is real analytic in the classical sense if and only if f āØ : R ā’ C Ļ (R, R)

is real analytic in some appropriate sense.

The following example shows that there are some subtleties involved.

(s, t) ā’ (st)1 +1 ā R is real analytic, whereas

Example. The mapping f : R2 2

there is no reasonable topology on C Ļ (R, R), such that the mapping f āØ : R ā’

C Ļ (R, R) is locally given by its convergent Taylor series.

Proof. For a topology on C Ļ (R, R) to be reasonable we require only that all eval-

uations evt : C Ļ (R, R) ā’ R are bounded linear functionals. Now suppose that

ā

f āØ (s) = k=0 fk sk converges in C Ļ (R, R) for small s, where fk ā C Ļ (R, R). Then

the series converges even bornologically, see (9.5) below, so f (s, t) = evt (f āØ (s)) =

ā

fk (t) sk for all t and small s. On the other hand f (s, t) = k=0 (ā’1)k (st)2k for

|s| < 1/|t|. So for all t we have fk (t) = (ā’1)m tk for k = 2m, and 0 otherwise, since

for ļ¬xed t we have a real analytic function in one variable. Moreover, the series

fk z k (t) = (ā’1)k t2k z 2k has to converge in C Ļ (R, R) ā— C for |z| ā¤ Ī“ and all

ā

t, see (9.5). This is not the case: use z = ā’1 Ī“, t = 1/Ī“.

There is, however, another notion of real analytic curves.

Example. Let f : R ā’ R be a real analytic function with ļ¬nite radius of conver-

gence at 0. Now consider the curve c : R ā’ RN deļ¬ned by c(t) := (f (k Ā· t))kāN .

Clearly, the composite of c with any continuous linear functional is real analytic,

since these functionals depend only on ļ¬nitely many coordinates. But the Taylor

9.1

98 Chapter II. Calculus of holomorphic and real analytic mappings 9.3

series of c at 0 does not converge on any neighborhood of 0, since the radii of con-

vergence of the coordinate functions go to 0. For an even more natural example see

(11.8).

ak tk with real coeļ¬cients the fol-

9.2. Lemma. For a formal power series kā„0

lowing conditions are equivalent.

(1) The series has positive radius of convergence.

ak rk converges absolutely for all sequences (rk ) with rk tk ā’ 0 for all

(2)

t > 0.

(3) The sequence (ak rk ) is bounded for all (rk ) with rk tk ā’ 0 for all t > 0.

(4) For each sequence (rk ) satisfying rk > 0, rk r ā„ rk+ , and rk tk ā’ 0 for all

t > 0 there exists an Īµ > 0 such that (ak rk Īµk ) is bounded.

This bornological description of real analytic curves will be rather important for

the theory presented here, since condition (3) and (4) are linear conditions on the

coeļ¬cients of a formal power series enforcing local convergence.

(ak tk )(rk tā’k ) converges absolutely for

Proof. (1) ā’ (2) The series ak rk =

some small t.

(2) ā’ (3) ā’ (4) is clear.

1

|ak | ( n2 )k = ā for all

(4) ā’ (1) If the series has radius of convergence 0, then k

ā with

n. There are kn

kn ā’1

|ak | ( n2 )k ā„ 1.

1

k=knā’1

1 1

We put rk := ( n )k for knā’1 ā¤ k < kn , then k |ak | rk ( n )k = ā for all n, so

1 t

(ak rk ( 2n )k )k is not bounded for any n, but rk tk , which equals ( n )k for knā’1 ā¤ k <

kn , converges to 0 for all t > 0, and the sequence (rk ) is subadditive as required.

9.3. Theorem (Description of real analytic functions). For a smooth func-

tion c : R ā’ R the following statements are equivalent.

(1) The function f is real analytic.

(2) For each sequence (rk ) with rk tk ā’ 0 for all t > 0, and each compact set

1

K in R, the set { k! c(k) (a) rk : a ā K, k ā N} is bounded.

(3) For each sequence (rk ) satisfying rk > 0, rk r ā„ rk+ , and rk tk ā’ 0 for

all t > 0, and each compact set K in R, there exists an Īµ > 0 such that

1

{ k! c(k) (a) rk Īµk : a ā K, k ā N} is bounded.

(4) For each compact set K ā‚ R there exist constants M, Ļ > 0 with the property

1

that | k! c(k) (a)| < M Ļk for all k ā N and a ā K.

Proof. (1) ā’ (4) Clearly, c is smooth. Since the Taylor series of c converges at a

there are constants Ma , Ļa satisfying the claimed inequality for ļ¬xed a. For a with

9.3

9.5 9. Real analytic curves 99

()

c (a)

1

|a ā’ a | ā¤ 2Ļa we obtain by diļ¬erentiating c(a ) = ! (a ā’ a) with respect

ā„0

to a the estimate

c(k) (a ) k

k1 ā‚ 1

ā¤ M a Ļa ,

k! ā‚tk t= 1 1 ā’ t

k!

2

hence the condition is satisļ¬ed locally with some new constants Ma , Ļa incorporat-

1

ing the estimates for the coeļ¬cients of 1ā’t . Since K is compact the claim follows.

1

(4) ā’ (2) We have | k! c(k) (a) rk | ā¤ M rk (Ļ)k which is bounded since rk Ļk ā’ 0, as

required.

(2) ā’ (3) follows by choosing Īµ = 1.

1

(3) ā’ (1) Let ak := supaāK | k! c(k) (a)|. Using (9.2).(4ā’1) these are the coeļ¬cients

of a power series with positive radius Ļ of convergence. Hence, the remainder

1 (k+1)

(a + Īø(a ā’ a))(a ā’ a)k+1 of the Taylor series goes locally to zero.

(k+1)! c

9.4. Corollary. Real analytic curves. For a curve c : R ā’ E in a convenient

vector space E are equivalent:

ā—¦ c : R ā’ R is real analytic for all in some family of bounded linear

(1)

functionals, which generates the bornology of E.

(2) ā—¦ c : R ā’ R is real analytic for all ā E

A curve satisfying these equivalent conditions will be called real analytic.

Proof. The non-trivial implication is (1 ā’ 2). So assume (1). By (2.14.6) the

curve c is smooth and hence ā—¦ c is smooth for all bounded linear : E ā’ R and

satisļ¬es ( ā—¦ c)(k) (t) = (c(k) (t)). In order to show that ā—¦ c is real analytic, we have

to prove boundedness of

1 (k) 1

( ā—¦ c)(k) (a)rk : a ā K, k ā N

c (a)rk : a ā K, k ā N =

k! k!

for all compact K ā‚ R and appropriate rk , by (9.3). Since is bounded it suļ¬ces

1

to show that { k! c(k) (a)rk : a ā K, k ā N} is bounded, we follows since its image

under all mentioned in (1) is bounded, again by (9.3).

9.5. Lemma. Let E be a convenient vector space and let c : R ā’ E be a curve.

Then the following conditions are equivalent.

(1) The curve c is locally given by a power series converging with respect to the

locally convex topology.

(2) The curve c factors locally over a topologically real analytic curve into EB

for some bounded absolutely convex set B ā E.

(3) The curve c extends to a holomorphic curve from some open neighborhood

U of R in C into the complexiļ¬cation (EC , EC ).

Where a curve satisfying condition (1) will be called topologically real analytic. One

that satisļ¬es condition (2) will be called bornologically real analytic.

Proof. (1) ā’ (3) For every t ā R one has for some Ī“ > 0 and all |s| < Ī“ a

ā k

converging power series representation c(t + s) = k=1 xk s . For any complex

9.5

100 Chapter II. Calculus of holomorphic and real analytic mappings 9.7

number z with |z| < Ī“ the series converges for z = s in EC , hence c can be locally

extended to a holomorphic curve into EC . By the 1-dimensional uniqueness theorem

for holomorphic maps, these local extensions ļ¬t together to give a holomorphic

extension as required.

(3) ā’ (2) A holomorphic curve factors locally over (EC )B by (7.6), where B can

ā

be chosen of the form B Ć— ā’1B. Hence, the restriction of this factorization to R

is real analytic into EB .

(2) ā’ (1) Let c be bornologically real analytic, i.e. c is locally real analytic into some

EB , which we may assume to be complete. Hence, c is locally even topologically

real analytic in EB by (9.6) and so also in E.

Although topological real analyticity is a strictly stronger than real analyticity,

cf. (9.4), sometimes the converse is true as the following slight generalization of

[Bochnak, Siciak, 1971, Lemma 7.1] shows.

9.6. Theorem. Let E be a convenient vector space and assume that a Baire

vector space topology on E exists for which the point evaluations evx for x ā E are

continuous. Then any real analytic curve c : R ā’ E is locally given by its Mackey

convergent Taylor series, and hence is bornologically real analytic and topologically

real analytic for every locally convex topology compatible with the bornology.

Proof. Since c is real analytic, it is smooth and all derivatives exist in E, since E

is convenient, by (2.14.6).

1

Let us ļ¬x t0 ā R, let an := n! c(n) (t0 ). It suļ¬ces to ļ¬nd some r > 0 for which

{rn an : n ā N0 } is bounded; because then tn an is Mackey-convergent for |t| < r,

and its limit is c(t0 + t) since we can test this with functionals.

Consider the sets Ar := {Ī» ā E : |Ī»(an )| ā¤ rn for all n ā N}. These Ar are closed

in the Baire topology, since the point evaluations at an are continuous. Since c

is real analytic, r>0 Ar = E , and by the Baire property there is an r > 0 such

that the interior U of Ar is not empty. Let Ī»0 ā U , then for all Ī» in the open

neighborhood U ā’ Ī»0 of 0 we have |Ī»(an )| ā¤ |(Ī» + Ī»0 )(an )| + |Ī»0 (an )| ā¤ 2rn . The

set U ā’ Ī»0 is absorbing, thus for every Ī» ā E some multiple ĪµĪ» is in U ā’ Ī»0 and so

Ī»(an ) ā¤ 2 rn as required.

ńņš. 4 |