ńņš. 12 |

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

ńņš. 12 |