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. 4
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e
exposer la th´orie dans un autre volume.”
e

Gˆteaux derivative. Another student of Hadamard de¬ned the derivative in
a
[Gˆteaux, 1913] with proofs in [Gˆteaux, 1922] as follows, see also [Gˆteaux, 1922]:
a a a
“Consid´rons U (z + »t1 ) (t1 fonction analogue ` z). Supposons que
e a

d
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
e
une fonctionnelle de z et de t1 , qu™on suppose habituellement lin´aire, en chaque point z,
e
par rapport ` t1 .”
a

Several mathematicians gave conditions implying the linearity of the Gˆteaux-de-
a
rivative. In [Daniell, 1919] is was shown that this holds for a Lipschitz function
whose Gˆteaux-derivative exists locally. Another student of Hadamard assumed
a
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
e
pour x = x0 , s™il existe une transformation vectorielle lin´aire Ψ(∆x) de l™accroissement ∆x
e
telle que, pour chaque vecteur ∆x,
’’ ’ ’ ’ ’ ’’
’’’’’’’
F (x0 )F (x0 + »∆x)
lim existe et = Ψ(∆x).”
»
»’0



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.
e
In [Fr´chet, 1937] it was shown that Hadamard™s notion is equivalent to that of
e
Stolz-Pierpoint-Young:
S.244: “Une fonctionelle U [f ] sera dite di¬´rentiable pour f ≡ f0 au sens de M. Hadamard
e
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
e

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.”
ee

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.
ee
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
e
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.”
e

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
e
spaces:
“ . . . 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.
e
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.”
o
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].
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], [Fr¨licher, 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 di¬erentiable mappings the
many possible notions collapse to 9 inequivalent ones (fewer for Fr´chet spaces).
e
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
e
culminated in the two huge volumes [G¨hler, 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 ’ 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
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 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
o
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,
o
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
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 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
2

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 — ,
=

7.1
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
w
w’0
C

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
M
1 1
z n an ∈ B.
(r/R)N ’ (r/R)M +1 1 ’ (r/R)
n=N


(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.
n

(5) For each z ∈ 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 ’ 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) ’ (1) By assumption, the di¬erence quotient c(z+w)’c(z) , composed
w
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
1
z
curve (test with linear functionals), hence

(c(z)) ’ (c(0))
1
( —¦ 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
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 ⊃
D ’ 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 su¬ces to show that for the absolutely convex
closed hull B of c(W ) the Taylor series of c at each z ∈ W converges in EB , i.e.
that c|W : W ’ 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) ∈ B
k! c

where B is the closed absolutely convex hull of { c(w) : |w ’ z| = r}. (By the
Hahn-Banach theorem this follows directly from the scalar valued case.)
Thus, we get
r k (k)
m m
w’z k w’z k
· ∈ ·B
k=n ( r ) k! c (z) k=n ( r )

c(k) (z)
’ z)k is convergent in EB for |w ’ 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 E± ) inductive limit of complex locally convex spaces E± ⊆ E, let
c : C ⊇ U ’ E be a holomorphic mapping, and let W ⊆ C be open and such that
the closure W is compact and contained in U . Then there exists some ±, such that
c|W : W ’ E± is well de¬ned and holomorphic.

Proof. By lemma (7.6) the restriction of c to W factors to a holomorphic curve
c|W : W ’ EB for a suitable bounded absolutely convex set B ⊆ E. Since B is
contained and bounded in some E± one has c|W : W ’ EB = (E± )B ’ E± is
holomorphic.

7.8. De¬nition. Let E and F be convenient vector spaces and let U ⊆ E be
c∞ -open. A mapping f : U ’ F is called holomorphic, if it maps holomorphic
curves in U to holomorphic curves in F .
It is remarkable that [Fantappi´, 1930] already gave this de¬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 —¦f —¦c : D ’ C is a holomorphic function
for all ∈ F — and holomorphic curve c.
Clearly, any composition of holomorphic mappings is again holomorphic.
For ¬nite dimensions this coincides with the usual notion of holomorphic mappings,
by the ¬nite dimensional Hartogs™ theorem.

7.9. Hartogs™ Theorem. Let E1 , E2 , and F be convenient vector spaces with U
c∞ -open in E1 — E2 . Then a mapping f : U ’ 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 c∞ -open set E1 —
{y} © U = incl’1 (U ), likewise for f (x, ).
y
If f is separately holomorphic, for any holomorphic curve (c1 , c2 ) : D ’ U ⊆ E1 —E2
we consider the holomorphic mapping c1 — c2 : D2 ’ E1 — E2 . Since the ck are
smooth by (7.4.8) also c1 — c2 is smooth and thus (c1 — c2 )’1 (U ) is open in C2 .
For each » ∈ F — the mapping » —¦ f —¦ (c1 — c2 ) : (c1 — c2 )’1 (U ) ’ C is separately
holomorphic and so holomorphic by the usual Hartogs™ theorem. By composing
with the diagonal mapping we see that » —¦ f —¦ (c1 , c2 ) is holomorphic, thus f is
holomorphic.

7.10. Lemma. Let f : E ⊇ U ’ F be holomorphic from a c∞ -open subset in
a convenient vector space to another convenient vector space. Then the derivative
(df )§ : U — E ’ F is again holomorphic and complex linear in the second variable.

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

and linear functionals and see that (v, w) ’ ‚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 ’ df (v)w is a holomorphic and C-homogeneous mapping E ’ 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 ¬nite
dimensional assertion.

7.11. Remark. In the de¬nition of holomorphy (7.8) one could also have admitted
subsets U which are only open in the ¬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 c∞ -complete vector spaces, see [Kriegl, Nel, 1985, 1.4].

Proof. Since both conditions can be tested in each factor separately by Hartogs™
theorem (7.9) and by (5.19), and by testing with linear functionals, we may restrict
our attention to linear mappings f : E ’ C only.
By theorem (7.4.2) a bounded linear mapping is holomorphic. Conversely, suppose
that f : E ’ 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

series n=0 (an ’ an’1 )(2z)n . This describes a holomorphic curve c in E, by (7.3)
and (7.4.2). Then f —¦ c is holomorphic and thus has a power series expansion

f (c(z)) = n=0 bn z n . On the other hand
N
(f (an ) ’ f (an’1 ))(2z)n + (2z)N f (an ’ an’1 )(2z)n’N
f (c(z)) = .
n=0 n>N

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

(f (an ) ’ f (an’1 )) = lim f (an ).
0 = f (0) = f (c(1/2)) =
n’∞
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) ∈ E k .

7.13. Lemma. Polarization formulas. Let f : E — · · · — E ’ F be an k-linear
symmetric mapping between vector spaces. Then we have:
1
k
(’1)k’Σµj f (x0 +
1
(1) f (x1 , . . . , xk ) = µj xj ) .
k!
µ1 ,...,µk =0
k
k
k
(’1)k’j
1
f ((a + jx)k ).
(2) f (x ) = k! j
j=0
k
kk j
k
k
(’1)k’j f ((a + k x)k ).
(3) f (x ) = k! j
j=0
1
»Σµj f (xµ1 , . . . , xµk ).
f (x0 »x1 , . . . , x0 »x1 )
(4) + + =
1 1 k k 1 k
µ1 ,...,µk =0


7.13
7.14 7. Calculus of holomorphic mappings 85


’1 in the passage to the complexi¬cation.
Formula (4) will mainly be used for » =

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

Aj0 ,...,jk
f (x0 , . . . , x0 , . . . , xk , . . . , xk ),
j0 ! · · · jk !
j0 +···+jk =k j0 jk


where the coe¬cients are given by

1
(’1)k’Σµj µj1 · · · µjk .
Aj0 ,...,jk = 1 k
µ1 ,...,µk =0


The only nonzero coe¬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 Fr´chet space and let
e
fk be a k-linear symmetric scalar valued bounded functional on E, for each k ∈ 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 ∈ N, x ∈ U } is bounded for some neighborhood U of 0.
(4) {fk (x1 , . . . , xk ) : k ∈ N, xj ∈ U } is bounded for some neighborhood U of 0.
If any of these statements are satis¬ed over the reals, then also for the complexi¬-
cation of the functionals fk .

Proof. (1) ’ (3) The set AK,r := {x ∈ E : |fk (xk )| ¤ 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 ∈ U and let V be an absolutely convex neighborhood of 0 contained
in U ’ x0
From (7.13) (3) we get for all x ∈ V the following estimate:

k
k j
k
k k
|f ((x0 + k x)k )|
|f (x )| ¤ k! j
j=0
kk k k
¤ K(2re)k .
¤ 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) ’ (4) From (7.13) (1) we get for all xj ∈ U the estimate:

1
k
1
|f (x1 , . . . , xk )| ¤ |f ( |
µj xj )
k!
µ1 ,...,µk =0
1 k
µj xj
k
1
= ( µj ) f
k! µj
µ1 ,...,µk =0
1
k
1
¤ ( µj ) C
k!
µ1 ,...,µk =0
k
k
1
j k C ¤ C(2e)k .
¤ k! j
j=0

1
Now we replace U by U and get (4).
2e
Proof of (4) ’ (2) The series converges on rU uniformly and absolutely for any
0 < r < 1.
(2) ’ (1) is clear.

(4), real case, ’ (4), complex case, by (7.13.4) for » = ’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 ∈ N. If k fk (xk )
converges pointwise on E and x ’ 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 ∈ 2B we apply the
vector valued Cauchy inequalities from (7.6) to the holomorphic curve z ’ 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 ∈ 2B, k ∈ 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 < ∞}. 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) — 2 ’ C given by g(x, y) := k xk f (kx1 y) is holomor-
phic since it is a ¬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 | ¤
µk for all k, y 2 ¤ µ} and so it contains points (x, y) with |xk f (kx1 y)| ≥ 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 Fr´chet space and let fk be a k-
e
linear symmetric scalar valued bounded functional on E, for each k ∈ 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 k≥1
a power series in E which converges to a(z) ∈ E for z near 0 in C.
Then the composite

fn (ak1 , . . . , akn ) z k
k≥0 n≥0 k1 ,...,kn ∈N
k1 +···+kn =k

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

Proof. By (7.14) there exists a 0-neighborhood U in E such that {fk (x1 , . . . , xk ) :
k ∈ N, xj ∈ U } is bounded. Since the series for a converges there is r > 0 such that
r
ak rk ∈ U for all k. For |z| < 2 we have

ak1 z k1 , . . . , akn z kn
f (a(z)) = fn
k1 ≥1 kn ≥1
n≥0

fn ak1 , . . . , akn z k1 +···+kn
···
=
n≥0 k1 ≥1 kn ≥1

fn (ak1 , . . . , akn ) z k ,
=
k≥0 n≥0 k1 ,...,kn ∈N
k1 +···+kn =k

since the last complex series converges absolutely: the coe¬cient of z k is a sum of
2k ’ 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 ∈ X such that there exists N ∈ N and a
1
neighborhood U of x with |fk (y) ’ f (y)| < n for all k ≥ N and all y ∈ 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 ∈ N the set VN := {x ∈ V : |fk (x) ’ f (x)| ¤ 2n for all k, ≥ 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 ∈ W we have
1
|fk (y) ’ f (y)| ¤ 2n for all k, ≥ N . We take the pointwise limit for ’ ∞ and
see that W ⊆ V © 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 ⊇ U ’ F be a mapping from a c∞ -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 ∈ F — and absolutely convex closed bounded sets B the mapping
—¦ f : EB ’ C is holomorphic.
(3) f is holomorphic along all a¬ne (complex) lines and is c∞ -continuous.
(4) f is holomorphic along all a¬ne (complex) lines and is bounded on bornolog-
ically compact sets (i.e. those compact in some EB ).
(5) f is holomorphic along all a¬ne (complex) lines and at each point the ¬rst
derivative is a bounded linear mapping.
(6) f is c∞ -locally a convergent series of bounded homogeneous complex poly-
nomials.
(7) f is holomorphic along all a¬ne (complex) lines and in every connected
component for the c∞ -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) ” (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 ¬rst for the case where E is a Banach space.
(1) ’ (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) ’ (6) Choose a ¬xed point z ∈ U . Since f is holomorphic along each complex
line through z it is given there by a pointwise convergent power series. By the
classical Hartogs™ theorem f is holomorphic along each ¬nite dimensional linear
subspace. The mapping f : E ⊇ U ’ E is well de¬ned by assumption and is also
holomorphic along each a¬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 ’ 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 + »v ∈
U for all |»| ¤ 1}. Now (6) follows from lemma (7.14).
(6) ’ (3) By lemma (7.14) the series converges uniformly and hence f is continuous.

(3) ’ (4) is obvious.

7.19
7.19 7. Calculus of holomorphic mappings 89

(4) ’ (5) By the (1-dimensional) Cauchy integral formula we have
1 f (z + »v)

f (z)v = d».
»2
2π ’1 |»|=1

So f (z) is a linear functional which is bounded on compact sets K for which
{z + »v : |»| ¤ 1, v ∈ K} ⊆ U , thus it is bounded, by lemma (5.4).
(6) ’ (1) follows by composing the two locally uniformly converging power series,
see corollary (7.17).

Sublemma. Let E be a Fr´chet space and let U ⊆ E be open. Let f : U ’ C be
e
holomorphic along a¬ne lines which is also the pointwise limit on U of a power
series with bounded homogeneous composants. Then f is holomorphic on U .

Proof. By assumption, and the lemma in (7.18) the function f is almost con-
tinuous, since it is the pointwise limit of polynomials. For each z the derivative
f (z) : E ’ C as pointwise limit of di¬erence quotients is also almost continuous
on {v : z + »v ∈ U for |»| ¤ 1}, thus continuous on E since it is linear and by the
Baire property.
By (5) ’ (1) the function f is holomorphic on U .

(6) ’ (7) is obvious.
(7) ’ (1) [Zorn, 1945] We treat each connected component of U separately and
assume thus that U is connected. The set U0 := {z ∈ U : f is holomorphic near z}
is open. By (6) ’ (1) f is holomorphic near the point, where all derivatives are
bounded, so U0 is not empty. From the sublemma above we see that for any point
z in U0 the whole star {z + v : z + »v ∈ U for all |»| ¤ 1} is contained in U . Since
U is in particular polygonally connected, we have U0 = U .
(8) ’ (9) is trivial.
(9) ’ (3) Clearly, f is holomorphic along a¬ne lines and c∞ -continuous.
(1) ’ (8) All derivatives are again holomorphic by (7.10) and thus locally bounded.
So f is smooth by (5.20).
Now we treat the case where E is a general convenient vector space. Restricting to
suitable spaces EB transforms each of the statements into the weaker corresponding
one where E is a Banach space. These pairs of statements are equivalent: This is
obvious except the following two cases.
For (6) we argue as follows. The function f |(U © EB ) satis¬es condition (6) (so
all the others) for each bounded closed absolutely convex B ⊆ E. By (5.20) f is
smooth and it remains to show that the Taylor series at z converges pointwise on a
c∞ -open neighborhood of z. The star {z + v : z + »v ∈ U for all |»|le1} with center
z in U is again c∞ -open by (4.17) and on it the Taylor series of f at z converges
pointwise.
For (7) replace on both sides the condition ”at least one point” by the condition
”for all points”.


7.19
90 Chapter II. Calculus of holomorphic and real analytic mappings 7.24

7.20. Chain rule. The composition of holomorphic mappings is holomorphic and
the usual formula for the derivative of the composite holds.

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

7.21. De¬nition. For convenient vector spaces E and F and for a c∞ -open subset
U ⊆ E we denote by H(U, F ) the space of all holomorphic mappings U ’ F . It
is a closed linear subspace of C ∞ (U, F ) by (7.19.8) and we give it the induced
convenient vector space structure.

7.22. Theorem. Cartesian closedness. For convenient vector spaces E1 , E2 ,
and F , and for c∞ -open subsets Uj ⊆ Ej a mapping f : U1 — U2 ’ F is holo-
morphic if and only if the canonically associated mapping f ∨ : U1 ’ H(U2 , F ) is
holomorphic.

Proof. Obviously, f ∨ has values in H(U2 , F ) and is smooth by smooth cartesian
closedness (3.12). Since its derivative is canonically associated to the ¬rst partial
derivative of f , it is complex linear. So f ∨ is holomorphic by (7.19.8).
If conversely f ∨ is holomorphic, then it is smooth into H(U2 , F ) by (7.19), thus
also smooth into C ∞ (U2 , F ). Thus, f : U1 — U2 ’ F is smooth by smooth carte-
sian closedness. The derivative df (x, y)(u, v) = (df ∨ (x)v)(y) + (d —¦ f ∨ )(x)(y)w is
obviously complex linear, so f is holomorphic.

7.23. Corollary. Let E etc. be convenient vector spaces and let U etc. be c∞ -open
subsets of such. Then the following canonical mappings are holomorphic.

ev : H(U, F ) — U ’ F, ev(f, x) = f (x)
ins : E ’ H(F, E — F ), ins(x)(y) = (x, y)
)§ : H(U, H(V, G)) ’ H(U — V, G)
(
)∨ : H(U — V, G) ’ H(U, H(V, G))
(
comp : H(F, G) — H(U, F ) ’ H(U, G)
H( ) : H(F, F ) — H(U , E) ’ H(H(E, F ), H(U , F ))
,
(f, g) ’ (h ’ f —¦ h —¦ g)
H(Ei , Fi ) ’ H(
: Ei , Fi )

Proof. Just consider the canonically associated holomorphic mappings on multiple
products.

7.24. Theorem (Holomorphic functions on Fr´chet spaces).
e
Let U ⊆ E be open in a complex Fr´chet space E. The following statements on
e
f : U ’ C are equivalent:
(1) f is holomorphic.
(2) f is smooth and is locally given by its uniformly and absolutely converging
Taylor series.
(3) f is locally given by a uniformly and absolutely converging power series.

7.24
8.2 8. Spaces of holomorphic mappings and germs 91

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

(2) ’ (3) is obvious.

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



8. Spaces of Holomorphic Mappings and Germs

8.1. Spaces of holomorphic functions. For a complex manifold N (always
assumed to be separable) let H(N, C) be the space of all holomorphic functions on
N with the topology of uniform convergence on compact subsets of N .
Let Hb (N, C) denote the Banach space of bounded holomorphic functions on N
equipped with the supremum norm.
For any open subset W of N let Hbc (N ⊇ W, C) be the closed subspace of Hb (W, C)
of all holomorphic functions on W which extend to continuous functions on the
closure W .
p
For a poly-radius r = (r1 , . . . , rn ) with ri > 0 and for 1 ¤ p ¤ ∞ let denote the
r
n
real Banach space x ∈ RN : (x± r± )±∈Nn p < ∞ .

8.2. Theorem (Structure of H(N, C) for complex manifolds N ).
The space H(N, C) of all holomorphic functions on N with the topology of uniform
convergence on compact subsets of N is a (strongly) nuclear Fr´chet space and
e
embeds bornologically as a closed subspace into C ∞ (N, R)2 .

Proof. By taking a countable covering of N with compact sets, one obtains a
countable neighborhood basis of 0 in H(N, C). Hence, H(N, C) is metrizable.
That H(N, C) is complete, and hence a Fr´chet space, follows since the limit of a
e
sequence of holomorphic functions with respect to the topology of uniform conver-
gence on compact sets is again holomorphic.
The vector space H(N, C) is a subspace of C ∞ (N, R2 ) = C ∞ (N, R)2 since a function
N ’ C is holomorphic if and only if it is smooth and the derivative at every point
is C-linear. It is a closed subspace, since it is described by the continuous linear
√ √
equations df (x)( ’1 · v) = ’1 · df (x)(v). Obviously, the identity from H(N, C)
with the subspace topology to H(N, C) is continuous, hence by the open mapping
theorem (52.11) for Fr´chet spaces it is an isomorphism.
e
That H(N, C) is nuclear and unlike C ∞ (N, R) even strongly nuclear can be shown
as follows. For N equal to the open polycylinder Dn ⊆ Cn this result can be found in
(52.36). For an arbitrary N the space H(N, C) carries the initial topology induced
by the linear mappings (u’1 )— : H(N, C) ’ H(u(U ), C) for all charts (u, U ) of
N , for which we may assume u(U ) = Dn , and hence by the stability properties of
strongly nuclear spaces, cf. (52.34), H(N, C) is strongly nuclear.


8.2
92 Chapter II. Calculus of holomorphic and real analytic mappings 8.4

8.3. Spaces of germs of holomorphic functions. For a subset A ⊆ N let
H(N ⊇ A, C) be the space of germs along A of holomorphic functions W ’ C for
open sets W in N containing A. We equip H(N ⊇ A, C) with the locally convex
topology induced by the inductive cone H(W, C) ’ H(N ⊇ A, C) for all W . This
is Hausdor¬, since iterated derivatives at points in A are continuous functionals
and separate points. In particular, H(N ⊇ W, C) = H(W, C) for W open in N .
For A1 ⊆ A2 ⊆ N the ”restriction” mappings H(N ⊇ A2 , C) ’ H(N ⊇ A1 , C) are
continuous.
The structure of H(S 2 ⊇ A, C), where A ⊆ S 2 is a subset of the Riemannian sphere,
has been studied by [Toeplitz, 1949], [Sebasti˜o e Silva, 1950b,] [Van Hove, 1952],
a
[K¨the, 1953], and [Grothendieck, 1953].
o

8.4. Theorem (Structure of H(N ⊇ K, C) for compact subsets K of com-
plex manifolds N ). The following inductive cones are co¬nal to each other.

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

If K = {z} these inductive cones and the following ones for 1 ¤ p ¤ ∞ are co¬nal
to each other.
H(N ⊇ {z}, C) ← { p — C, r ∈ Rn }
r +


So all inductive limit topologies coincide. Furthermore, the space H(N ⊇ K, C) is a
Silva space, i.e. a countable inductive limit of Banach spaces, where the connecting
mappings between the steps are compact, i.e. mapping bounded sets to relatively
compact ones. The connecting mappings are even strongly nuclear. In particular,
the limit is regular, i.e. every bounded subset is contained and bounded in some
step, and H(N ⊇ K, C) is complete and (ultra-)bornological (hence a convenient
vector space), webbed, strongly nuclear and thus re¬‚exive, and its dual is a nuclear
Fr´chet space. The space H(N ⊇ K, C) is smoothly paracompact. It is however not
e
a Baire space.

Proof. Let K ⊆ V ⊆ V ⊆ W ⊆ N , where W and V are open and V is compact.
Then the obvious mappings
Hbc (N ⊇ W, C) ’ Hb (W, C) ’ H(W, C) ’ Hbc (N ⊇ V, C)
are continuous. This implies the ¬rst co¬nality assertion. For q ¤ p and multiradii
s < r the obvious maps q ’ p , ∞ ’ 1 , and 1 — C ’ Hb ({w ∈ Cn : |wi ’ zi | <
r r r s r

ri }, C) ’ s —C are continuous, by the Cauchy inequalities from the proof of (7.7).
So the remaining co¬nality assertion follows.
Let us show next that the connecting mapping Hb (W, C) ’ Hb (V, C) is strongly
nuclear (hence nuclear and compact). Since the restriction mapping from E :=
H(W, C) to Hb (V, C) is continuous, it factors over E ’ E(U ) for some zero neigh-
borhood U in E, where E(U ) is the completed quotient of E with the Minkowski

8.4
8.6 8. Spaces of holomorphic mappings and germs 93

functional of U as norm, see (52.15). Since E is strongly nuclear by (8.2), there ex-
ists by de¬nition some larger 0-neighborhood U in E such that the natural mapping
E(U ) ’ E(U ) is strongly nuclear. So the claimed connecting mapping is strongly
nuclear, since it can be factorized as

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

So H(N ⊇ K, C) is a Silva space. It is strongly nuclear by the permanence proper-
ties of strongly nuclear spaces (52.34). By (16.10) this also shows that H(N ⊇ K, C)
is smoothly paracompact. The remaining properties follow from (52.37).

Completeness of H(Cn ⊇ K, C) was shown in [Van Hove, 1952, th´or`me II], and
ee
for regularity of the inductive limit H(C ⊇ K, C) see e.g. [K¨the, 1953, Satz 12].
o

8.5. Lemma. For a closed subset A ⊆ C the spaces H(A ⊆ S 2 , C) and the space
H∞ (S 2 ⊇ S 2 \ A, C) of all germs vanishing at ∞ are strongly dual to each other.

Proof. This is due to [K¨the, 1953, Satz 12] and has been generalized by [Grothen-
o
dieck, 1953,] th´or`me 2 bis, to arbitrary subsets A ⊆ S 2 .
ee

Compare also the modern theory of hyperfunctions, cf. [Kashiwara, Kawai, Kimura,
1986].

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

H(N ⊇ A, C) ← { H(W, C) : A ⊆ W ⊆ N }
open


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

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

generates the bornology of H(N ⊇ A, C).
The space H(N ⊇ A, C) is Montel (hence quasi-complete and re¬‚exive), and ultra-
bornological (hence a convenient vector space). Furthermore, it is webbed and conu-
clear.

Proof. Compare also with the proof of the more general theorem (30.6).
We choose a continuous function f : N ’ R which is positive and proper. Then
(f ’1 ([n, n + 1]))n∈N0 is an exhaustion of N by compact subsets and (Kn := A ©
f ’1 ([n, n + 1])) is a compact exhaustion of A.
Let B ⊆ H(N ⊇ A, C) be bounded. Then B|K is also bounded in H(N ⊇ K, C) for
each compact subset K of A. Since the cone

{H(W, C) : K ⊆ W ⊆ N } ’ H(N ⊇ K, C)
open


8.6
94 Chapter II. Calculus of holomorphic and real analytic mappings 8.8

is regular by (8.4), there exist open subsets WK of N containing K such that B|K is
contained (so that the extension of each germ is unique) and bounded in H(WK , C).
In particular, we choose WKn ©Kn+1 ⊆ WKn © WKn+1 © f ’1 ((n, n + 2)). Then we
let W be the union of those connected components of

(WKn © f ’1 ((n, n + 1))) ∪
W := WKn ©Kn+1
n n


which meet A. Clearly, W is open and contains A. Each f ∈ B has an extension to
W : Extend f |Kn uniquely to fn on WKn . The function f |(Kn © Kn+1 ) has also
a unique extension fn,n+1 on WKn ©Kn+1 , so we have fn |WKn ©Kn+1 = fn,n+1 . This
extension of f ∈ B has a unique restriction to W . B is bounded in H(W, C) if it is
uniformly bounded on each compact subset K of W . Each K is covered by ¬nitely
many WKn and B|Kn is bounded in H(WKn , C), so B is bounded as required.
The space H(N ⊇ A, C) is ultra-bornological, Montel and in particular quasi-
complete, and conuclear, as regular inductive limit of the nuclear Fr´chet spaces
e
H(W, C).
And it is webbed because it is the (ultra-)bornologi¬cation of the countable pro-
jective limit of webbed spaces H(N ⊇ K, C), see (52.14) and (52.13).

8.7. Lemma. Let A be closed in C. Then the dual generated by the projective
cone
H(C ⊇ A, C) ’ { H(C ⊇ K, C), K compact in A }

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

Proof. The induced topology is obviously coarser than the given one. So let »
be a continuous linear functional on H(C ⊇ A, C). Then we have » ∈ H∞ (S 2 ⊇
S 2 \ A, C) by (8.5). Hence, » ∈ H∞ (U, C) for some open neighborhood U of S 2 \ A,
so again by (8.5) » is a continuous functional on H(S 2 ⊇ K, C), where K = S 2 \ U
is compact in A. So » is continuous for the induced topology.

Problem. Does this cone generate even the topology of H(C ⊇ A, C)? This would
imply that the bornological topology on H(C ⊇ A, C) is complete and nuclear.

8.8. Lemma (Structure of H(N ⊇ A, C) for smooth closed submanifolds
A of complex manifolds N ). The projective cone

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

generates the bornology.

Proof. Let B ⊆ H(N ⊇ A, C) be such that the set B is bounded in H(N ⊇ {z}, C)
for all z ∈ A. By the regularity of the inductive cone H(Cn ⊇ {0}, C) ← H(W, C)
we ¬nd arbitrary small open neighborhoods Wz such that the set Bz of the germs
at z of all germs in B is contained and bounded in H(Wz , C).

8.8
8.10 8. Spaces of holomorphic mappings and germs 95

Now choose a tubular neighborhood p : U ’ A of A in N . We may assume that Wz
is contained in U , has ¬bers which are star shaped with respect to the zero-section
and the intersection with A is connected. The union W of all the Wz , is therefore
an open subset of U containing A. And it remains to show that the germs in B
extend to W . For this it is enough to show that the extensions of the germs at
z1 and z2 agree on the intersection of Wz1 with Wz2 . So let w be a point in the
intersection. It can be radially connected with the base point p(w), which itself can
be connected by curves in A with z1 and z2 . Hence, the extensions of both germs
to p(w) coincide with the original germ, and hence their extensions to w are equal.
That B is bounded in H(W, C), follows immediately since every compact subset
K ⊆ W can be covered by ¬nitely many Wz .

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

1
Example. Let A := { n : n ∈ N} ∪ {0}. Then A is compact in C but the projective
cone H(C ⊇ A, C) ’ {H(C ⊇ {z}, C) : z ∈ A} does not generate the bornology.

Proof. Let B ⊆ H(C ⊇ A, C) be the set of germs of the following locally constant
functions fn : {x + iy ∈ C : x = rn } ’ C, with fn (x + iy) equal to 0 for x < rn
2
and equal to 1 for x > rn , where rn := 2n+1 , for n ∈ N. Then B ⊆ H(C ⊇ A, C)
is not bounded, otherwise there would exist a neighborhood W of A such that the
germ of fn extends to a holomorphic mapping on W for all n. Since every fn is 0
on some neighborhood of 0, these extensions have to be zero on the component of
1
W containing 0, which is not possible, since fn ( n ) = 1.
But on the other hand the set Bz ⊆ H(C ⊇ {z}, C) of germs at z of all germs in B
is bounded, since it contains only the germs of the constant functions 0 and 1.

8.10. Theorem (Holomorphic uniform boundedness principle).
Let E and F be complex convenient vector spaces, and let U ⊆ E be a c∞ -open
subset. Then H(U, F ) satis¬es the uniform boundedness principle for the point
evaluations evx , x ∈ U .
For any closed subset A ⊆ N of a complex manifold N the locally convex space
H(N ⊇ A, C) satis¬es the uniform S-boundedness principle for every point sepa-
rating set S of bounded linear functionals.

Proof. By de¬nition (7.21) H(U, F ) carries the structure induced from the embed-
ding into C ∞ (U, F ) and hence satis¬es the uniform boundedness principle (5.26)
and (5.25).
The second part is an immediate consequence of (5.24) and (8.6), and (8.4).

Direct proof of a particular case of the second part. We prove the theorem
for a closed smooth submanifold A ⊆ C and the set S of all iterated derivatives at
points in A.

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

Let us suppose ¬rst that A is the point 0. We will show that condition (5.22.3) is
satis¬ed. Let (bn ) be an unbounded sequence in H({0}, C) such that each Taylor
1 (k)
coe¬cient bn,k = k! bn (0) is bounded with respect to n:

sup{ |bn,k | : n ∈ N } < ∞.
(1)

1
We have to ¬nd (tn ) ∈ such that n tn b n is no longer the germ of a holomorphic
function at 0.
Each bn has positive radius of convergence, in particular there is an rn > 0 such
that

k
sup{ |bn,k rn | : k ∈ N } < ∞.
(2)

By theorem (8.4) the space H({0}, C) is a regular inductive limit of spaces ∞ . r
Hence, a subset B is bounded in H({0}, C) if and only if there exists an r > 0 such
1
that { k! b(k) (0) rk : b ∈ B, k ∈ N } is bounded. That the sequence (bn ) is unbounded
thus means that for all r > 0 there are n and k such that |bn,k | > ( 1 )k . We can
r
k
even choose k > 0 for otherwise the set { bn,k r : n, k ∈ N, k > 0 } is bounded, so
only { bn,0 : n ∈ N } can be unbounded. This contradicts (1).
Hence, for each m there are km > 0 such that Nm := { n ∈ N : |bn,km | > mkm }
is not empty. We can choose (km ) strictly increasing, for if they were bounded,
|bn,km | < C for some C and all n by (1), but |bnm ,km | > mkm ’ ∞ for some nm .
Since by (1) the set { bn,km : n ∈ N } is bounded, we can choose nm ∈ Nm such
that
1
|bnm ,km | ≥ 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 ’ 0.
We pass now to the subsequence (bnm ) which we denote again by (bm ). We put
« 
1 1
tj bj,km  ·
(4) tm := sign  .
4m
bm,km j<m


Assume now that b∞ = m tm bm converges weakly with respect to S to a holomor-
phic germ. Then its Taylor series is b∞ (z) = k≥0 b∞,k z k , where the coe¬cients
are given by b∞,k = m≥0 tm bm,k . But we may compute as follows, using (3) and
(4) :


|b∞,km | ≥ tj bj,km ’ |tj bj,km |
j>m
j¤m


8.10
9.1 9. Real analytic curves 97


tj bj,km + |tm bm,km |
= (same sign)
j<m

’ |tj bj,km | ≥
j>m
« 

≥ 0 + |bm,km | · |tm | ’ 2 |tj |
j>m

mkm
1
= |bm,km | · ≥ .
3 · 4m 3 · 4m

So |b∞,km |1/km goes to ∞, hence b∞ 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 ⊆ C. By (8.8) the
projective cone H(N ⊇ A, C) ’ {H(N ⊇ {z}, C), z ∈ 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 ’ R is real analytic in the classical sense if and only if f ∨ : R ’ C ω (R, R)
is real analytic in some appropriate sense.
The following example shows that there are some subtleties involved.

(s, t) ’ (st)1 +1 ∈ R is real analytic, whereas
Example. The mapping f : R2 2

there is no reasonable topology on C ω (R, R), such that the mapping f ∨ : R ’
C ω (R, R) is locally given by its convergent Taylor series.

Proof. For a topology on C ω (R, R) to be reasonable we require only that all eval-
uations evt : C ω (R, R) ’ R are bounded linear functionals. Now suppose that

f ∨ (s) = k=0 fk sk converges in C ω (R, R) for small s, where fk ∈ C ω (R, R). Then
the series converges even bornologically, see (9.5) below, so f (s, t) = evt (f ∨ (s)) =

fk (t) sk for all t and small s. On the other hand f (s, t) = k=0 (’1)k (st)2k for
|s| < 1/|t|. So for all t we have fk (t) = (’1)m tk for k = 2m, and 0 otherwise, since
for ¬xed t we have a real analytic function in one variable. Moreover, the series
fk z k (t) = (’1)k t2k z 2k has to converge in C ω (R, R) — C for |z| ¤ δ and all

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

There is, however, another notion of real analytic curves.
Example. Let f : R ’ R be a real analytic function with ¬nite radius of conver-
gence at 0. Now consider the curve c : R ’ RN de¬ned by c(t) := (f (k · t))k∈N .
Clearly, the composite of c with any continuous linear functional is real analytic,
since these functionals depend only on ¬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 coe¬cients the fol-
9.2. Lemma. For a formal power series k≥0
lowing conditions are equivalent.
(1) The series has positive radius of convergence.
ak rk converges absolutely for all sequences (rk ) with rk tk ’ 0 for all
(2)
t > 0.
(3) The sequence (ak rk ) is bounded for all (rk ) with rk tk ’ 0 for all t > 0.
(4) For each sequence (rk ) satisfying rk > 0, rk r ≥ rk+ , and rk tk ’ 0 for all
t > 0 there exists an µ > 0 such that (ak rk µk ) 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
coe¬cients of a formal power series enforcing local convergence.

(ak tk )(rk t’k ) converges absolutely for
Proof. (1) ’ (2) The series ak rk =
some small t.

(2) ’ (3) ’ (4) is clear.
1
|ak | ( n2 )k = ∞ for all
(4) ’ (1) If the series has radius of convergence 0, then k
∞ with
n. There are kn

kn ’1
|ak | ( n2 )k ≥ 1.
1

k=kn’1


1 1
We put rk := ( n )k for kn’1 ¤ k < kn , then k |ak | rk ( n )k = ∞ 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 kn’1 ¤ 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 ’ R the following statements are equivalent.
(1) The function f is real analytic.
(2) For each sequence (rk ) with rk tk ’ 0 for all t > 0, and each compact set
1
K in R, the set { k! c(k) (a) rk : a ∈ K, k ∈ N} is bounded.
(3) For each sequence (rk ) satisfying rk > 0, rk r ≥ rk+ , and rk tk ’ 0 for
all t > 0, and each compact set K in R, there exists an µ > 0 such that
1
{ k! c(k) (a) rk µk : a ∈ K, k ∈ N} is bounded.
(4) For each compact set K ‚ R there exist constants M, ρ > 0 with the property
1
that | k! c(k) (a)| < M ρk for all k ∈ N and a ∈ K.

Proof. (1) ’ (4) Clearly, c is smooth. Since the Taylor series of c converges at a
there are constants Ma , ρa satisfying the claimed inequality for ¬xed a. For a with

9.3
9.5 9. Real analytic curves 99

()
c (a)
1
|a ’ a | ¤ 2ρa we obtain by di¬erentiating c(a ) = ! (a ’ a) with respect
≥0
to a the estimate
c(k) (a ) k
k1 ‚ 1
¤ M a ρa ,
k! ‚tk t= 1 1 ’ t
k!
2

hence the condition is satis¬ed locally with some new constants Ma , ρa incorporat-
1
ing the estimates for the coe¬cients of 1’t . Since K is compact the claim follows.
1
(4) ’ (2) We have | k! c(k) (a) rk | ¤ M rk (ρ)k which is bounded since rk ρk ’ 0, as
required.
(2) ’ (3) follows by choosing µ = 1.
1
(3) ’ (1) Let ak := supa∈K | k! c(k) (a)|. Using (9.2).(4’1) these are the coe¬cients
of a power series with positive radius ρ of convergence. Hence, the remainder
1 (k+1)
(a + θ(a ’ a))(a ’ 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 ’ E in a convenient
vector space E are equivalent:
—¦ c : R ’ R is real analytic for all in some family of bounded linear
(1)
functionals, which generates the bornology of E.
(2) —¦ c : R ’ R is real analytic for all ∈ E
A curve satisfying these equivalent conditions will be called real analytic.

Proof. The non-trivial implication is (1 ’ 2). So assume (1). By (2.14.6) the
curve c is smooth and hence —¦ c is smooth for all bounded linear : E ’ R and
satis¬es ( —¦ c)(k) (t) = (c(k) (t)). In order to show that —¦ c is real analytic, we have
to prove boundedness of
1 (k) 1
( —¦ c)(k) (a)rk : a ∈ K, k ∈ N
c (a)rk : a ∈ K, k ∈ N =
k! k!
for all compact K ‚ R and appropriate rk , by (9.3). Since is bounded it su¬ces
1
to show that { k! c(k) (a)rk : a ∈ K, k ∈ 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 ’ 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 ⊆ E.
(3) The curve c extends to a holomorphic curve from some open neighborhood
U of R in C into the complexi¬cation (EC , EC ).
Where a curve satisfying condition (1) will be called topologically real analytic. One
that satis¬es condition (2) will be called bornologically real analytic.

Proof. (1) ’ (3) For every t ∈ R one has for some δ > 0 and all |s| < δ a
∞ 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| < δ 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 ¬t together to give a holomorphic
extension as required.
(3) ’ (2) A holomorphic curve factors locally over (EC )B by (7.6), where B can

be chosen of the form B — ’1B. Hence, the restriction of this factorization to R
is real analytic into EB .
(2) ’ (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 ∈ E are
continuous. Then any real analytic curve c : R ’ 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 ¬x t0 ∈ R, let an := n! c(n) (t0 ). It su¬ces to ¬nd some r > 0 for which
{rn an : n ∈ 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 := {» ∈ E : |»(an )| ¤ rn for all n ∈ 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 »0 ∈ U , then for all » in the open
neighborhood U ’ »0 of 0 we have |»(an )| ¤ |(» + »0 )(an )| + |»0 (an )| ¤ 2rn . The
set U ’ »0 is absorbing, thus for every » ∈ E some multiple µ» is in U ’ »0 and so
»(an ) ¤ 2 rn as required.

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