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U (z + »t1 )
d» »=0

existe quel que soit t1 . On l™appelle la variation premi`re de U au point z: δU (z, t1 ). C™est
une fonctionnelle de z et de t1 , qu™on suppose habituellement lin´aire, en chaque point z,
par rapport ` t1 .”

Several mathematicians gave conditions implying the linearity of the Gˆteaux-de-
rivative. In [Daniell, 1919] is was shown that this holds for a Lipschitz function
whose Gˆteaux-derivative exists locally. Another student of Hadamard assumed
linearity in [L´vy, 1922], see again [Fr´chet, 1937]:
e e
S.51: “Une fonction abstraite X = F (x) sera dite di¬´rentiable au sens de M. Paul Levy
pour x = x0 , s™il existe une transformation vectorielle lin´aire Ψ(∆x) de l™accroissement ∆x
telle que, pour chaque vecteur ∆x,
’’ ’ ’ ’ ’ ’’
F (x0 )F (x0 + »∆x)
lim existe et = Ψ(∆x).”

Hadamard di¬erentiability. In [Hadamard, 1923] a function f : R2 ’ R was
called di¬erentiable if all compositions with di¬erentiable curves are again di¬er-
entiable and satisfy the chain rule. He refers to a lecture of Poincar´ in 1904.
In [Fr´chet, 1937] it was shown that Hadamard™s notion is equivalent to that of
S.244: “Une fonctionelle U [f ] sera dite di¬´rentiable pour f ≡ f0 au sens de M. Hadamard
g´n´ralis´, s™il existe une fonctionnelle W [df, f0 ], lin´aire par rapport ` df , telle que si l™on
ee e e a
consid`re une fonction f (t, ») d´rivable par rapport ` » pour » = 0, avec f (t, 0) = f0 (t), la
e e a
fonction de », U [f (t, »)] soit d´rivable en » pour » = 0 et qu™on ait pour » = 0

d df
U f (t, ») = W , f0
d» d»

ou avec les notations des “variations”

δU [f ] = W [δf, f0 ].”
Historical remarks on the development of smooth calculus 75

S.245: “la di¬´rentielle au sens de M. Hadamard g´n´ralis´ qui est ´quivalente ` la nˆtre
e ee e e a o
dans l™Analyse classique est plus g´n´rale dans l™Analyse fonctionnelle.”

He also realized the importance of Hadamard™s de¬nition:
S.249: “L™int´rˆt de la d´¬nition de M. Hadamard n™est pas ´puis´ par son utilization en
ee e e e
Analyse fonctionnelle. Il est peut-ˆtre plus encore dans la possibilit´ de son extension en
e e
Analyse g´n´rale.
Dans ce domaine, on peut g´n´raliser la notion de fonctionnelle et consid´rer des transfor-
ee e
mations X = F [x] d˜un ´l´ment abstrait x en un ´l´ment abstrait X. Nous avons pu en 1925
ee ee
[Fr´chet, 1925b] ´tendre notre d´¬nition (rappel´e plus haut p.241 et 242) de la di¬´rentielle
e e e e e
d˜une fonctionnelle, d´¬nir la di¬´rentielle de F [x] quand X et x appartiennent ` des espaces
e e a
“vectoriels abstraits distanci´s” et en etablir les propri´t´s les plus importantes.
e ee
La d´¬nition au sens de M. Hadamard g´n´ralis´ pr´sente sur notre d´¬nition l™avantage de
e ee ee e
garder un sens pour des espaces abstraits vectoriels non distanci´s o` notre d´¬nition ne
eu e
s˜applique pas. [ . . . ]
Il reste ` voir si elle conserve les propri´tes les plus importantes de la di¬´rentielle classique
a e e
en dehors de la propri´t´ (g´n´ralisant le th´or`me des fonctions compos´es) qui lui sert de
ee e e ee e
d´¬nition. C™est un point sur lequel nous reviendrons ult´rieurement.”
e e

Hadamard™s notion of di¬erentiability was later extended to in¬nite dimensions by
[Michal, 1938] who de¬ned a mapping f : E ’ F between topological vector spaces
to be di¬erentiable at x if there exists a continuous linear mapping : E ’ F
such that f —¦ c : R ’ F is di¬erentiable at 0 with derivative ( —¦ c )(0) for each
everywhere di¬erentiable curve c : R ’ E with c(0) = x.
Independently, a student of Fr´chet extended in [Ky Fan, 1942] di¬erentiability in
the sense of Hadamard to normed spaces, and proved the basic properties like the
chain rule:
S.307: “M. Fr´chet a eu l™obligeance de me conseiller d™´tudier cette question qu™il avait
e e
d™abord l™intention de traiter lui-mˆme.”

Hadamard di¬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 “di¬erentiable implies continuous” is equivalent to the property that
the closure of a set coincides with the sequential adherence.

Di¬erentiability via bornology. Here the basic observation is that convergence
which appears in questions of di¬erentiability is much better than just topological,
cf. (1.7). The relevant notion of convergence was introduced by [Mackey, 1945].
Di¬erentiability based on the von Neumann bornology was ¬rst considered in [Se-
basti˜o e Silva, 1956a, 1956b, 1957]. In [Sebasti˜o e Silva, 1961] he extended this
a a
to bornological vector spaces and referred to Waelbroeck and Fantappi´ for these
“ . . . de g´n´raliser aux espaces localement convexes, r´els ou complexes, la notion de fonc-
ee e
tion di¬´rentiable, ainsi que les th´or`mes fondamentaux du calcul di¬´rentiel et int´gral,
e ee e e
et de la th´orie des fonctions analytiques de plusieurs variables complexes.
Je me suis persuad´ que, pour cette g´n´ralisation, c™est la notion d™ensemble born´, plutˆt
e ee e o
que celle de voisinage, qui doit jouer un rˆle essentiel.”
76 Chapter I. Calculus of smooth mappings

In [Waelbroeck, 1967a, 1967b] the notion of ˜b-space™ was introduced, and di¬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) ’ (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 di¬erential calculus in them was further developed by [Lazet, 1971], and
[Colombeau, 1973], see also [Colombeau, 1982]. The importance of di¬erentiability
with respect to the bornology generated by the compact subsets was realized in
[Sova, 1966b].
An overview on di¬erentiability of ¬rst order can be found in [Averbukh, Smol-
yanov, 1968]. One ¬nds there 25 inequivalent de¬nitions of the ¬rst derivative in
a single point, and one sees how complicated ¬nite order di¬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)—E ’
E is jointly continuous, and [Keller, 1965] generalized this. We have given the
archetypical argument in the introduction.
Thus, a ˜satisfactory™ calculus seemed to stop at the level of Banach spaces, where
an elaborated theory including existence theorems was presented already in the
very in¬‚uential text book [Dieudonn´, 1960].
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], [Fr¨licher, Bucher, 1966], and by [Binz, 1966]. A review is [Keller,
1974], where the following was shown: Continuity of the derivative implied stronger
remainder convergence conditions. So for continuously di¬erentiable mappings the
many possible notions collapse to 9 inequivalent ones (fewer for Fr´chet spaces).
And if one looks for in¬nitely often di¬erentiable mappings, then one ends up with
6 inequivalent notions (only 3 for Fr´chet spaces). Further work in this direction
culminated in the two huge volumes [G¨hler, 1977, 1978], and in the historically
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 ’ C(X, Y ). The ˜exponential law™

Z X—Y ∼ (Z Y )X , or C(X — Y, Z) ∼ 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 ˜a convenient category of topological spaces™,
which is the source of the widespread use of ˜convenient™, also in this book. See
also [Vogt, 1971].
Historical remarks on the development of smooth calculus 77

Following the advise of A. Fr¨licher, [Seip, 1972] used compactly generated vector
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 modi¬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 ˜smooth manifolds™ 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 Seip™s work in the thesis [Kriegl, 1980], supervised by Peter Mi-
chor, smooth mappings between arbitrary subsets ˜Vektormengen™ of locally convex
spaces were supposed to respect smooth curves and to induce ˜tangent mappings™
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 ’ 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, [Fr¨licher, 1980] considered categories generated by monoids of real
valued functions and characterized cartesian closedness in terms of the monoid.
[Fr¨licher, 1981] used the result of [Boman, 1967] to show that on Fr´chet spaces
o e
usual smoothness is equivalent to smoothness in the sense of the category generated
by the monoid C ∞ (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 ¬nite order dif-
ferentiability, since it fails to be true for C n -functions. Motivated by this, ¬nite
di¬erentiability based on Lipschitz conditions has then been developed by [Fr¨licher,
Gisin, Kriegl, 1983]. A careful presentation can be found in the monograph [Fr¨li-o
cher, Kriegl, 1988]. Finite di¬erentiability based on H¨lder conditions were studied
by [Faure, 1989] and [Faure, 1991].

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 in¬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 ‚ C
into a complex convenient vector space E, where all possible de¬nitions of being
holomorphic coincide, see (7.4). This replaces the set of all smooth curves in the
smooth theory. A mapping between c∞ -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 satis¬es
the Cauchy-Riemann di¬erential equation); or that f is holomorphic along each
a¬ne complex line and is c∞ -continuous (generalized Hartog™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 c∞ -open set U ⊆ E into
a convenient vector space F carries a natural structure of a complex convenient
vector space (7.21), and satis¬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 )) ∼ H(U — 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 c∞ -open
subset in general, and not on open subsets.
The second part of this chapter is devoted to real analytic mappings in in¬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 modi¬cations: In (9.1)
we shall see that for the real analytic mapping f : R2 (s, t) ’ (st)1 +1 ∈ R there is

no reasonable topology on C ω (R, R), such that the mapping f ∨ : R ’ C ω (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 ω (R, E) of all curves c such that
—¦ c : R ’ 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 ’ R (9.3).
Now a mapping f : U ’ F is called real analytic if f —¦ c is smooth for smooth c
and is real analytic for real analytic c : R ’ U . The second condition alone is not
su¬cient, even for f : R2 ’ R. Then a version of Hartog™s theorem is true: f is real
analytic if and only if it is smooth and real analytic along each a¬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 ω (U, F ). We start with C ω (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 = ’ 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

its real part Re —¦ , by (x) = (Re —¦ )(x) ’ ’1(Re —¦ )(Jx). So for the respective
spaces of bounded linear functionals we have
ER = LR (ER , R) ∼ LC (E, C) =: E — ,

7.4 7. Calculus of holomorphic mappings 81

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

7.2. De¬nition. Let D be the the open unit disk {z ∈ C : |z| < 1}. A mapping
c : D ’ E into a locally convex space E is called complex di¬erentiable, if

c(z + w) ’ c(z)
c (z) = lim

exists for all z ∈ D.

7.3. Lemma. Let E be convenient and an ∈ E. Then the following statements
are equivalent:
(1) {rn an : n ∈ N} is bounded for all |r| < 1.
(2) The power series n≥0 z n an is Mackey convergent in E, uniformly on each
compact subset of D, i.e., the Mackey coe¬cient sequence and the bounded
set can be chosen valid in the whole compact subset.
(3) The power series converges weakly for all z ∈ D.

Proof. (1) ’ (2) Any compact set is contained in rD for some 0 < r < 1, the
set {Rn an : n ∈ 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
1 1
z n an ∈ B.
(r/R)N ’ (r/R)M +1 1 ’ (r/R)

(2) ’ (3) is clear.
Proof of (3) ’ (1) The summands are weakly bounded, thus bounded.

7.4. Theorem. If E is convenient then the following statements for a curve c :
D ’ E are equivalent:
(1) c is complex di¬erentiable.
(2) —¦ c : D ’ C is holomorphic for all ∈ E —
(3) c is continuous and γ c = 0 in the completion of E for all closed smooth
(Lip0 -) curves in D.
∞ 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.

(5) For each z ∈ D all c(n) (z) exist and c(z + w) = n=0 w c(n) (z) is Mackey
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 ’ E
satisfying these equivalent conditions will be called a holomorphic curve.

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

Proof. (2) ’ (1) By assumption, the di¬erence quotient c(z+w)’c(z) , composed
with a linear functional, extends to a complex valued holomorphic function of w,
hence it is locally Lipschitz. So the di¬erence quotient is a Mackey Cauchy net. So
it has a limit for w ’ 0.
Proof of (1) ’ (2) Suppose that is bounded. Let c : D ’ E be a complex
di¬erentiable curve. Then c1 : z ’ z c(z)’c(0) ’ c (0) is a complex di¬erentiable
curve (test with linear functionals), hence

(c(z)) ’ (c(0))
( —¦ c1 )(z) = ’ (c (0))
z z

is locally bounded in z. So —¦ c is complex di¬erentiable with derivative —¦ 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 c∞ -completeness as it does in the smooth case. The complex
di¬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 c∞ -completeness (see (2.14)) we refer to [Kriegl, Nel,
1985, pp. 2.16].

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

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