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9.7. Theorem. Linear real analytic mappings. Let E and F be convenient
vector spaces. For any linear mapping » : E ’ F the following assertions are
equivalent.
(1) » is bounded.
(2) » —¦ c : R ’ F is real analytic for all real analytic c : R ’ E.
(3) »—¦c : R ’ F is bornologically real analytic for all bornologically real analytic
curves c : R ’ E
(4) » —¦ c : R ’ F is real analytic for all bornologically real analytic curves
c:R’E

This will be generalized in (10.4) to non-linear mappings.

9.7
10.1 10. Real analytic mappings 101

Proof. (1) ’ (3) ’ (4), and (2) ’ (4) are obvious.
(4) ’ (1) Let » satisfy (4) and suppose that » is unbounded. By composing with
an ∈ E we may assume that » : E ’ R and there is a bounded sequence
(xk ) such that »(xk ) is unbounded. By passing to a subsequence we may suppose
that |»(xk )| > k 2k . Let ak := k ’k xk , then (rk ak ) is bounded and (rk »(ak )) is

unbounded for all r > 0. Hence, the curve c(t) := k=0 tk ak is given by a Mackey
convergent power series. So » —¦ c is real analytic and near 0 we have »(c(t)) =
∞ N
k k N k’N
k=0 bk t for some bk ∈ R. But »(c(t)) = k=0 »(ak )t + t »( k>N ak t )
and t ’ k>N ak tk’N is still a Mackey converging power series in E. Comparing
coe¬cients we see that bk = »(ak ) and consequently »(ak )rk is bounded for some
r > 0, a contradiction.
Proof of (1) ’ (2) Let c : R ’ E be real analytic. By theorem (9.3) the set
1
{ k! c(k) (a) rk : a ∈ K, k ∈ N} is bounded for all compact sets K ‚ R and for all
sequences (rk ) with rk tk ’ 0 for all t > 0. Since c is smooth and bounded linear
mappings are smooth by (2.11), the function » —¦ c is smooth and (» —¦ c)(k) (a) =
»(c(k) (a)). By applying (9.3) we obtain that » —¦ c is real analytic.

9.8. Corollary. For two convenient vector space structures on a vector space E
the following statements are equivalent:
(1) They have the same bounded sets.
(2) They have the same smooth curves.
(3) They have the same real analytic curves.

Proof. (1) ” (2) was shown in (2.11). The implication (1) ’ (3) follows from (9.3),
which shows that real analyticity is a bornological concept, whereas the implication
(1) ⇐ (3) follows from (9.7).

9.9. Corollary. If a cone of linear maps T± : E ’ E± between convenient vector
spaces generates the bornology on E, then a curve c : R ’ E is C ω resp. C ∞
provided all the composites T± —¦ c : R ’ E± are.

Proof. The statement on the smooth curves is shown in (3.8). That on the real
analytic curves follows again from the bornological condition of (9.3).


10. Real Analytic Mappings

10.1. Theorem (Real analytic functions on Fr´chet spaces). Let U ⊆ E
e
be open in a real Fr´chet space E. The following statements on f : U ’ R are
e
equivalent:
(1) f is smooth and is real analytic along topologically real analytic curves.
(2) f is smooth and is real analytic along a¬ne lines.
(3) f is smooth and is locally given by its pointwise converging Taylor series.
(4) f is smooth and is locally given by its uniformly and absolutely converging
Taylor series.

10.1
102 Chapter II. Calculus of holomorphic and real analytic mappings 10.4

(5) f is locally given by a uniformly and absolutely converging power series.
˜˜ ˜
(6) f extends to a holomorphic mapping f : U ’ C for an open subset U in the
˜
complexi¬cation EC with U © E = U .

Proof. (1) ’ (2) is obvious. The implication (2) ’ (3) follows from (7.14), (1)
’ (2), whereas (3) ’ (4) follows from (7.14),(2) ’ (3), and (4) ’ (5) is obvious.
Proof of (5) ’ (6) Locally we can extend converging power series into the complex-
˜
i¬cation by (7.14). Then we take the union U of their domains of de¬nition and
˜
use uniqueness to glue f which is holomorphic by (7.24).
Proof of (6) ’ (1) Obviously, f is smooth. Any topologically real analytic curve c
in E can locally be extended to a holomorphic curve in EC by (9.5). So f —¦ c is real
analytic.

10.2. The assumptions ˜f is smooth™ cannot be dropped in (10.1.1) even in ¬nite
dimensions, as shown by the following example, due to [Boman, 1967].
n+2
xy
Example. The mapping f : R2 ’ R, de¬ned by f (x, y) := x2 +y2 is real analytic
along real analytic curves, is n-times continuous di¬erentiable but is not smooth
and hence not real analytic.

Proof. Take a real analytic curve t ’ (x(t), y(t)) into R2 . The components can be
factored as x(t) = tk u(t), y(t) = tk v(t) for some k and real analytic curves u, v with
uv n+2
u(0)2 +v(0)2 = 0. The composite f —¦(x, y) is then the function t ’ tk(n+1) u2 +v2 (t),
which is obviously real analytic near 0. The mapping f is n-times continuous
di¬erentiable, since it is real analytic on R2 \{0} and the directional derivatives of
order i are (n + 1 ’ i)-homogeneous, hence continuously extendable to R2 . But f
cannot be (n + 1)-times continuous di¬erentiable, otherwise the derivative of order
n + 1 would be constant, and hence f would be a polynomial.

10.3. De¬nition (Real analytic mappings). Let E be a convenient vector
space. Let us denote by C ω (R, E) the space of all real analytic curves.
Let U ⊆ E be c∞ -open, and let F be a second convenient vector space. A mapping
f : U ’ F will be called real analytic or C ω for short, if f is real analytic along
real analytic curves and is smooth (i.e. is smooth along smooth curves); so f —¦
c ∈ C ω (R, F ) for all c ∈ C ω (R, E) with c(R) ⊆ U and f —¦ c ∈ C ∞ (R, F ) for all
c ∈ C ∞ (R, E) with c(R) ⊆ U . Let us denote by C ω (U, F ) the space of all real
analytic mappings from U to F .

10.4. Analogue of Hartogs™ Theorem for real analytic mappings. Let E
and F be convenient vector spaces, let U ⊆ E be c∞ -open, and let f : U ’ F .
Then f is real analytic if and only if f is smooth and » —¦ f is real analytic along
each a¬ne line in E, for all » ∈ F .

Proof. One direction is clear, and by de¬nition (10.3) we may assume that F = R.
Let c : R ’ U be real analytic. We show that f —¦ c is real analytic by using theorem
(9.3). So let (rk ) be a sequence such that rk r ≥ rk+ and rk tk ’ 0 for all t > 0

10.4
10.4 10. Real analytic mappings 103

and let K ‚ R be compact. We have to show, that there is an µ > 0 such that the
set { 1! (f —¦ c)( ) (a) rl ( 2 ) : a ∈ K, ∈ N} is bounded.
µ

1
By theorem (9.3) the set { n! c(n) (a) rn : n ≥ 1, a ∈ K} is contained in some bounded
absolutely convex subset B ⊆ E, such that EB is a Banach space. Clearly, for the
inclusion iB : EB ’ E the function f —¦ iB is smooth and real analytic along a¬ne
lines. Since EB is a Banach space, by (10.1.2) ’ (10.1.4) f —¦ iB is locally given
by its uniformly and absolutely converging Taylor series. Then for each a ∈ K by
1
(7.14.2) ’ (7.14.4) there is an µ > 0 such that the set { k! dk f (c(a))(x1 , . . . , xk ) :
k ∈ N, xj ∈ µB} is bounded. For each y ∈ 1 µB termwise di¬erentiation gives
2
1
p k
d f (c(a) + y)(x1 , . . . , xp ) = k≥p (k’p)! d f (c(a))(x1 , . . . , xp , y, . . . , y), so we may
assume that {dk f (c(a))(x1 , . . . , xk )/k! : k ∈ N, xj ∈ µB, a ∈ K} is contained in
[’C, C] for some C > 0 and some uniform µ > 0.
The Taylor series of f —¦ c at a is given by

k!
dk f (c(a)) ( n! c(n) (a))mn t ,
1 1
(f —¦ c)(a + t) = k!
n mn ! n
≥0 k≥0 (mn )∈NN0
mn =k
n
n mn n=



xmn := (x1 , . . . , x1 , . . . , xn , . . . , xn , . . . ).
where n
n
m1 mn

This follows easily from composing the Taylor series of f and c and ordering by
powers of t. Furthermore, we have

k! ’1
= k’1
mn !
n
(mn )∈NN0
mn =k
n
n mn n=


by the following argument: It is the -th Taylor coe¬cient at 0 of the function

( n≥0 tn ’ 1)k = ( 1’t )k = tk j=0 ’k (’t)j , which turns out to be the binomial
t
j
coe¬cient in question.
By the foregoing considerations we may estimate as follows.

—¦ c)( ) (a)| rl ( 2 ) ¤
1 µ
! |(f
k!
dk f (c(a)) ( n! c(n) (a))mn
1 1 µ
¤ r (2)
k!
n mn ! n
k≥0 N
(mn )∈N0
n mn =k
n mn n=

k!
dk f (c(a)) ( n! c(n) (a) rn µn )mn
1 1 1
¤ k! 2
n mn ! n
k≥0 (mn )∈NN0
mn =k
n
n mn n=

’1
C 2 = 1 C,
1
¤ k’1 2
k≥0


10.4
104 Chapter II. Calculus of holomorphic and real analytic mappings 10.8

because

k! ’1
( n! c(n) (a) µn rn )mn ∈
1
(µB)k ⊆ (EB )k .
k’1
n mn ! n
(mn )∈NN0
mn =k
n
n mn n=




10.5. Corollary. Let E and F be convenient vector spaces, let U ⊆ E be c∞ -open,
and let f : U ’ F . Then f is real analytic if and only if f is smooth and » —¦ f —¦ c
is real analytic for every periodic (topologically) real analytic curve c : R ’ U ⊆ E
and all » ∈ F .

Proof. By (10.4) f is real analytic if and only if f is smooth and » —¦ f is real
analytic along topologically real analytic curves c : R ’ E. Let h : R ’ R be
de¬ned by h(t) = t0 + µ · sin t. Then c —¦ h : R ’ R ’ U is a (topologically)
real analytic, periodic function with period 2π, provided c is (topologically) real
analytic. If c(t0 ) ∈ U we can choose µ > 0 such that h(R) ⊆ c’1 (U ). Since sin is
locally around 0 invertible, real analyticity of » —¦ f —¦ c —¦ h implies that » —¦ f —¦ c is
real analytic near t0 . Hence, the proof is completed.

10.6. Corollary. Reduction to Banach spaces. Let E be a convenient vector
space, let U ⊆ E be c∞ -open, and let f : U ’ R be a mapping. Then f is real
analytic if and only if the restriction f : EB ⊇ U © EB ’ R is real analytic for all
bounded absolutely convex subsets B of E.

So any result valid on Banach spaces can be translated into a result valid on con-
venient vector spaces.

Proof. By theorem (10.4) it su¬ces to check f along bornologically real analytic
curves. These factor by de¬nition (9.4) locally to real analytic curves into some
EB .

10.7. Corollary. Let U be a c∞ -open subset in a convenient vector space E and
let f : U ’ R be real analytic. Then for every bounded B there is some rB > 0
1k k
such that the Taylor series y ’ k! d f (x)(y ) converges to f (x + y) uniformly
and absolutely on rB B.

Proof. Use (10.6) and (10.1.4).

10.8. Scalar analytic functions on convenient vector spaces E are in general not
germs of holomorphic functions from EC to C:

Example. Let fk : R ’ R be real analytic functions with radius of convergence at
zero converging to 0 for k ’ ∞. Let f : R(N) ’ R be the mapping de¬ned on the

countable sum R(N) of the reals by f (x0 , x1 , . . . ) := k=1 xk fk (x0 ). Then f is real

10.8
11.1 11. The real analytic exponential law 105

˜
analytic, but there is no complex valued holomorphic mapping f on some neigh-
borhood of 0 in C(N) which extends f , and the Taylor series of f is not pointwise
convergent on any c∞ -open neighborhood of 0.

Proof. Claim. f is real analytic.
Since the limit R(N) = ’ n Rn is regular, every smooth curve (and hence every real
lim

(N)
is locally smooth (resp. real analytic) into Rn for some n.
analytic curve) in R
Hence, f —¦ c is locally just a ¬nite sum of smooth (resp. real analytic) functions
and is therefore smooth (resp. real analytic).
Claim. f has no holomorphic extension.
˜
Suppose there exists some holomorphic extension f : U ’ C, where U ⊆ C(N) is c∞ -
open neighborhood of 0, and is therefore open in the locally convex Silva topology by
(4.11.2). Then U is even open in the box-topology (52.7), i.e., there exist µk > 0 for
all k, such that {(zk ) ∈ C(N) : |zk | ¤ µk for all k} ⊆ U . Let U0 be the open disk in C
˜ ˜ ˜
with radius µ0 and let fk : U0 ’ C be de¬ned by fk (z) := f (z, 0, . . . , 0, µk , 0, . . . ) µ1 ,
k
˜ is an extension of
where µk is inserted instead of the variable xk . Obviously, fk
fk , which is impossible, since the radius of convergence of fk is less than µ0 for k
su¬ciently large.
Claim. The Taylor series does not converge.
If the Taylor series would be pointwise convergent on some U , then the previous
arguments for R(N) instead of C(N) would show that the radii of convergence of the
fk were bounded from below.



11. The Real Analytic Exponential Law

11.1. Spaces of germs of real-analytic functions. Let M be a real analytic
¬nite dimensional manifold. If f : M ’ M is a mapping between two such
manifolds, then f is real analytic if and only if f maps smooth curves into smooth
ones and real analytic curves into real analytic ones, by (10.1).
For each real analytic manifold M of real dimension m there is a complex manifold
MC of complex dimension m containing M as a real analytic closed submanifold,
whose germ along M is unique ([Whitney, Bruhat, 1959, Prop. 1]), and which can
be chosen even to be a Stein manifold, see [Grauert, 1958, section 3]. The complex
charts are just extensions of the real analytic charts of an atlas of M into the
complexi¬cation of the modeling real vector space.
Real analytic mappings f : M ’ M are the germs along M of holomorphic
mappings W ’ MC for open neighborhoods W of M in MC .
Let C ω (M, F ) be the space of real analytic functions f : M ’ F , for any convenient
vector space F , and let H(MC ⊇ M, C) be the space of germs along M of holomor-
phic functions as in (8.3). Furthermore, for a subset A ⊆ M let C ω (M ⊇ A, R)
denotes the space of germs of real analytic functions along A, de¬ned on some
neighborhood of A.

11.1
106 Chapter II. Calculus of holomorphic and real analytic mappings 11.4

11.2. Lemma. For any subset A of M the complexi¬cation of the real vector space
C ω (M ⊇ A, R) is the complex vector space H(MC ⊇ A, C).

De¬nition. For any A ⊆ M of a real analytic manifold M we will topologize the
space sections C ω (M ⊇ A, R) as subspace of H(MC ⊇ A, C), in fact as the real part
of it.

Proof. Let f, g ∈ C ω (M ⊇ A, R). These are germs of real analytic mappings
de¬ned on some open neighborhood of A in M . Inserting complex numbers into
the locally convergent Taylor series in local coordinates shows, that f and g can be
considered as holomorphic mappings from some neighborhood W of A in MC , which

have real values if restricted to W © M . The mapping h := f + ’1g : W ’ C
gives then an element of H(MC ⊇ A, C).
Conversely, let h ∈ H(MC ⊇ A, C). Then h is the germ of a holomorphic function
h : W ’ C for some open neighborhood W of A in MC . The decomposition of h
¯ ¯
into real and imaginary part f = 1 (h + h) and g = 2√’1 (h ’ h), which are real
1
2
analytic functions if restricted to W © M , gives elements of C ω (M ⊇ A, R).
These correspondences are inverse to each other since a holomorphic germ is deter-
mined by its restriction to a germ of mappings M ⊇ A ’ C.

11.3. Lemma. For a ¬nite dimensional real analytic manifold M the inclusion
C ω (M, R) ’ C ∞ (M, R) is continuous.

Proof. Consider the following diagram, where W is an open neighborhood of M
in its complexi¬cation MC .

y wC ∞
inclusion
C ω (M, R) (M, R)


u u
direct summand (11.2) direct summand


y w C (M, R )
u u

inclusion 2
H(MC ⊇ M, C)

restriction (8.4) restriction


y wC ∞
inclusion
(W, R2 )
H(W, C) (8.2)


11.4. Theorem (Structure of C ω (M ⊇ A, R) for closed subsets A of real
analytic manifolds M ). The inductive cone

C ω (M ⊇ A, R) ← { C ω (W, R) : A ⊆ W ⊆ M }
open

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

C ω (M ⊇ A, R) ’ { C ω (M ⊇ K, R) : K compact in A}

generates the bornology of C ω (M ⊇ A, R).

11.4
11.6 11. The real analytic exponential law 107

If A is even a smooth submanifold, then the following projective cone also generates
the bornology.

C ω (M ⊇ A, R) ’ { C ω (M ⊇ {x}, R) : x ∈ A}

The space C ω (Rm ⊇ {0}, R) is also the regular inductive limit of the spaces p

r (r
Rm ) for all 1 ¤ p ¤ ∞, see (8.1).
+
For general closed A ⊆ N the space C ω (M ⊇ A, R) is Montel (hence quasi-complete
and re¬‚exive), and ultra-bornological (hence a convenient vector space). It is also
webbed and conuclear. If A is compact then it is even a strongly nuclear Silva space
and its dual is a nuclear Fr´chet space and it is smoothly paracompact. It is however
e
not a Baire space.

Proof. This follows using (11.2) from (8.4), (8.6), and (8.8) by passing to the real
parts and from the fact that all properties are inherited by complemented subspaces
as C ω (M ⊇ A, R) of H(MC ⊇ A, C).

11.5. Corollary. A subset B ⊆ C ω (Rm ⊇ {0}, R) is bounded if and only if there
(±)
exists an r > 0 such that { f (0) |±|
: f ∈ B, ± ∈ Nm } is bounded in R.
±! r 0

Proof. The space C ω (Rm ⊇ {0}, R) is the regular inductive limit of the spaces
∞ m
r for r ∈ R+ by (11.4). Hence, B is bounded if and only if it is contained and
bounded in ∞ for some r ∈ Rm , which is the looked for condition.
r +

11.6. Theorem (Special real analytic uniform boundedness principle).
For any closed subset A ⊆ M of a real analytic manifold M , the space C ω (M ⊇
A, R) satis¬es the uniform S-boundedness principle for any point separating set S
of bounded linear functionals.
If A has no isolated points and M is 1-dimensional this applies to the set of all
point evaluations evt , t ∈ A.

Proof. Again this follows from (5.24) using now (11.4). If A has no isolated points
and M is 1-dimensional the point evaluations are separating, by the uniqueness
theorem for holomorphic functions.

Direct proof of a particular case. We show that C ω (R, R) satis¬es the uniform
S-boundedness principle for the set S of all point evaluations.
We check property (5.22.2). Let B ⊆ C ω (R, R) be absolutely convex such that
evt (B) is bounded for all t and such that C ω (R, R)B is complete. We have to show
that B is complete.
By lemma (11.3) the set B satis¬es the conditions of (5.22.2) in the space C ∞ (R, R).
Since C ∞ (R, R) satis¬es the uniform S-boundedness principle, cf. [Fr¨licher, Kriegl,
o
1988], the set B is bounded in C ∞ (R, R). Hence, all iterated derivatives at points
are bounded on B, and a fortiori the conditions of (5.22.2) are satis¬ed for B in
H(R, C). By the particular case of theorem (8.10) the set B is bounded in H(R, C)
and hence also in the direct summand C ω (R, R).


11.6
108 Chapter II. Calculus of holomorphic and real analytic mappings 11.9

11.7. Theorem. The real analytic curves in C ω (R, R) correspond exactly to the
real analytic functions R2 ’ R.

Proof. (’) Let f : R ’ C ω (R, R) be a real analytic curve. Then f : R ’
C ω (R ⊇ {t}, R) is also real analytic. We use theorems (11.4) and (9.6) to conclude
that f is even a topologically real analytic curve in C ω (R ⊇ {t}, R). By lemma
(9.5) for every s ∈ R the curve f can be extended to a holomorphic mapping from
an open neighborhood of s in C to the complexi¬cation (11.2) H(C ⊇ {t}, C) of
C ω (R ⊇ {t}, R).
From (8.4) it follows that H(C ⊇ {t}, C) is the regular inductive limit of all spaces
H(U, C), where U runs through some neighborhood basis of t in C. Lemma (7.7)
shows that f is a holomorphic mapping V ’ H(U, C) for some open neighborhoods
U of t and V of s in C.
By the exponential law for holomorphic mappings (see (7.22)) the canonically asso-
ciated mapping f § : V — U ’ C is holomorphic. So its restriction is a real analytic
function R — R ’ R near (s, t) which coincides with f § for the original f .
(⇐) Let f : R2 ’ R be a real analytic mapping. Then f (t, ) is real analytic, so
the associated mapping f ∨ : R ’ C ω (R, R) makes sense. It remains to show that
it is real analytic. Since the mappings C ω (R, R) ’ C ω (R ⊇ K, R) generate the
bornology, by (11.4), it is by (9.9) enough to show that f ∨ : R ’ C ω (R ⊇ K, R)
is real analytic for each compact K ⊆ R, which may be checked locally near each
s ∈ R.
f : R2 ’ R extends to a holomorphic function on an open neighborhood V — U of
{s} — K in C2 . By cartesian closedness for the holomorphic setting the associated
mapping f ∨ : V ’ H(U, C) is holomorphic, so its restriction V © R ’ C ω (U ©
R, R) ’ C ω (K, R) is real analytic as required.

11.8. Remark. From (11.7) it follows that the curve c : R ’ C ω (R, R) de¬ned in
(9.1) is real analytic, but it is not topologically real analytic. In particular, it does
not factor locally to a real analytic curve into some Banach space C ω (R, R)B for a
bounded subset B and it has no holomorphic extension to a mapping de¬ned on a
neighborhood of R in C with values in the complexi¬cation H(R, C) of C ω (R, R),
cf. (9.5).

11.9. Lemma. For a real analytic manifold M , the bornology on C ω (M, R) is
induced by the following cone:
c—
C (M, R) ’ C ± (R, R) for all C ± -curves c : R ’ M , where ± equals ∞ and ω.
ω


Proof. The maps c— are bornological since C ω (M, R) is convenient by (11.4), and
by the uniform S-boundedness principle (11.6) for C ω (R, R) and by (5.26) for
C ∞ (R, R) it su¬ces to check that evt —¦c— = evc(t) is bornological, which is obvious.
Conversely, we consider the identity mapping i from the space E into C ω (M, R),
where E is the vector space C ω (M, R), but with the locally convex structure in-
duced by the cone.

11.9
11.12 11. The real analytic exponential law 109

Claim. The bornology of E is complete.
The spaces C ω (R, R) and C ∞ (R, R) are convenient by (11.4) and (2.15), respec-
tively. So their product
C ∞ (R, R)
C ω (R, R) —
c∈C ∞ (R,M )
c∈C ω (R,M )

is also convenient. By theorem (10.1.1) ” (10.1.5) the embedding of E into this
product has closed image, hence the bornology of E is complete.
Now we may apply the uniform S-boundedness principle for C ω (M, R) (11.6), since
obviously evp —¦i = ev0 —¦c— is bounded, where cp is the constant curve with value p,
p
for all p ∈ M .

11.10. Structure on C ω (U, F ). Let E be a real convenient vector space and let
U be c∞ -open in E. We equip the space C ω (U, R) of all real analytic functions (cf.
(10.3)) with the locally convex topology induced by the families of mappings
c—
C (U, R) ’ C ω (R, R), for all c ∈ C ω (R, U )
ω

c—
C ω (U, R) ’ C ∞ (R, R), for all c ∈ C ∞ (R, U ).

For a ¬nite dimensional vector spaces E this de¬nition gives the same bornology
as the one de¬ned in (11.1), by lemma (11.9).
If F is another convenient vector space, we equip the space C ω (U, F ) of all real
analytic mappings (cf. (10.3)) with the locally convex topology induced by the
family of mappings
»
C ω (U, F ) ’ — C ω (U, R), for all » ∈ F .
’’

Obviously, the injection C ω (U, F ) ’ C ∞ (U, F ) is bounded and linear.

11.11. Lemma. Let E and F be convenient vector spaces and let U ⊆ E be
c∞ -open. Then C ω (U, F ) is also convenient.

Proof. This follows immediately from the fact that C ω (U, F ) can be considered
as closed subspace of the product of factors C ω (U, R) indexed by all » ∈ F .
And C ω (U, R) can be considered as closed subspace of the product of the fac-
tors C ω (R, R) indexed by all c ∈ C ω (R, U ) and the factors C ∞ (R, R) indexed by
all c ∈ C ∞ (R, U ). Since all factors are convenient so are the closed subspaces.

11.12. Theorem (General real analytic uniform boundedness principle).
Let E and F be convenient vector spaces and U ⊆ E be c∞ -open. Then C ω (U, F )
satis¬es the uniform S-boundedness principle, where S := {evx : x ∈ U }.

Proof. The convenient structure of C ω (U, F ) is induced by the cone of mappings
c— : C ω (U, F ) ’ C ω (R, F ) (c ∈ C ω (R, U )) together with the maps c— : C ω (U, F ) ’
C ∞ (R, F ) (c ∈ C ∞ (R, U )). Both spaces C ω (R, F ) and C ∞ (R, F ) satisfy the uni-
form T -boundedness principle, where T := {evt : t ∈ R}, by (11.6) and (5.26),
respectively. Hence, C ω (U, F ) satis¬es the uniform S-boundedness principle by
lemma (5.25), since evt —¦ c— = evc(t) .


11.12
110 Chapter II. Calculus of holomorphic and real analytic mappings 11.16

11.13. Remark. Let E and F be convenient vector spaces. Then L(E, F ), the
space of bounded linear mappings from E to F, are by (9.7) exactly the real analytic
ones.

11.14. Theorem. Let Ei for i = 1, . . . n and F be convenient vector spaces. Then
the bornology on L(E, . . . , En ; F ) (described in (5.1), see also (5.6)) is induced by
the embedding L(E1 , . . . , En ; F ) ’ C ω (E1 — . . . En , F ).
Thus, mapping f into L(E1 , . . . , En ; F ) is real analytic if and only if the composites
evx —¦ f are real analytic for all x ∈ E1 — . . . En , by (9.9).

Proof. Let S = {evx : x ∈ E1 — . . . — En }. Since C ω (E1 — . . . — En , F ) satis¬es
the uniform S-boundedness principle (11.12), the inclusion is bounded. On the
other hand L(E1 , . . . , En ; F ) also satis¬es the uniform S-boundedness principle by
(5.18), so the identity from L(E1 , . . . , En ; F ) with the bornology induced from
C ω (E1 — . . . — En , F ) into L(E1 , . . . , En ; F ) is bounded as well.
Since to be real analytic depends only on the bornology by (9.4) and since the conve-
nient vector space L(E1 , . . . , En ; F ) satis¬es the uniform S-boundedness principle,
the second assertion follows also.

The following two results will be generalized in (11.20). At the moment we will
make use of the following lemma only in case where E = C ∞ (R, R).

11.15. Lemma. For any convenient vector space E the ¬‚ip of variables induces
an isomorphism L(E, C ω (R, R)) ∼ C ω (R, E ) as vector spaces.
=

Proof. For c ∈ C ω (R, E ) consider c(x) := evx —¦c ∈ C ω (R, R) for x ∈ E. By the
˜
uniform S-boundedness principle (11.6) for S = {evt : t ∈ R} the linear mapping c
˜
is bounded, since evt —¦˜ = c(t) ∈ E .
c
If conversely ∈ L(E, C ω (R, R)), we consider ˜(t) = evt —¦ ∈ E = L(E, R) for
t ∈ R. Since the bornology of E is generated by S := {evx : x ∈ E}, ˜ : R ’ E is
real analytic, for evx —¦ ˜ = (x) ∈ C ω (R, R), by (11.14).

Corollary. We have C ∞ (R, C ω (R, R)) ∼ C ω (R, C ∞ (R, R)) as vector
11.16. =
spaces.

Proof. The dual C ∞ (R, R) is the free convenient vector space over R by (23.11),
and C ω (R, R) is convenient, so we have

C ∞ (R, C ω (R, R)) ∼ L(C ∞ (R, R) , C ω (R, R))
=
∼ C ω (R, C ∞ (R, R) ) by lemma (11.15)
=
∼ C ω (R, C ∞ (R, R)),
=

by re¬‚exivity of C ∞ (R, R), see (6.5.7).


11.16
11.18 11. The real analytic exponential law 111

11.17. Theorem. Let E be a convenient vector space, let U be c∞ -open in E,
let f : R — U ’ R be a real analytic mapping and let c ∈ C ∞ (R, U ). Then
c— —¦ f ∨ : R ’ C ω (U, R) ’ C ∞ (R, R) is real analytic.

This result on the mixing of C ∞ and C ω will become quite essential in the proof
of cartesian closedness. It will be generalized in (11.21), see also (42.15).

Proof. Let I ⊆ R be open and relatively compact, let t ∈ R and k ∈ N. Now
¯
choose an open and relatively compact J ⊆ R containing the closure I of I. There
is a bounded subset B ⊆ E such that c | J : J ’ EB is a Lipk -curve in the Banach
space EB generated by B, by (1.8). Let UB denote the open subset U © EB of the
Banach space EB . Since the inclusion EB ’ E is continuous, f is real analytic as
a function R — UB ’ R — U ’ R. Thus, by (10.1) there is a holomorphic extension
f : V — W ’ C of f to an open set V — W ⊆ C — (EB )C containing the compact
¯
set {t} — c(I). By cartesian closedness of the category of holomorphic mappings
f ∨ : V ’ H(W, C) is holomorphic. Now recall that the bornological structure of
H(W, C) is induced by that of C ∞ (W, C) := C ∞ (W, R2 ). And c— : C ∞ (W, C) ’
Lipk (I, C) is a bounded C-linear map, by the chain rule (12.8) for Lipk -mappings
and by the uniform boundedness principle for the point evaluations (12.9). Thus,
c— —¦ f ∨ : V ’ Lipk (I, C) is holomorphic, and hence its restriction to R © V , which
has values in Lipk (I, R), is (even topologically) real analytic by (9.5). Since t ∈ R
was arbitrary we conclude that c— —¦ f ∨ : R ’ Lipk (I, R) is real analytic. But
the bornology of C ∞ (R, R) is generated by the inclusions into Lipk (I, R), by the
uniform boundedness principles (5.26) for C ∞ (R, R) and (12.9) for Lipk (R, R), and
hence c— —¦ f ∨ : R ’ C ∞ (R, R) is real analytic.

11.18. Theorem. Cartesian closedness. The category of real analytic map-
pings between convenient vector spaces is cartesian closed. More precisely, for con-
venient vector spaces E, F and G and c∞ -open sets U ⊆ E and W ⊆ G a mapping
f : W — U ’ F is real analytic if and only if f ∨ : W ’ C ω (U, F ) is real analytic.

Proof. Step 1. The theorem is true for W = G = F = R.
(⇐) Let f ∨ : R ’ C ω (U, R) be C ω . We have to show that f : R — U ’ R is C ω .
We consider a curve c1 : R ’ R and a curve c2 : R ’ U .
If the ci are C ∞ , then c— —¦ f ∨ : R ’ C ω (U, R) ’ C ∞ (R, R) is C ω by assumption,
2
hence is C , so c2 —¦ f —¦ c1 : R ’ C ∞ (R, R) is C ∞ . By cartesian closedness of
∞ — ∨

smooth mappings, (c— —¦ f ∨ —¦ c1 )§ = f —¦ (c1 — c2 ) : R2 ’ R is C ∞ . By composing
2
with the diagonal mapping ∆ : R ’ R2 we obtain that f —¦ (c1 , c2 ) : R ’ R is C ∞ .
If the ci are C ω , then c— —¦ f ∨ : R ’ C ω (U, R) ’ C ω (R, R) is C ω by assumption,
2
— ∨
so c2 —¦ f —¦ c1 : R ’ C ω (R, R) is C ω . By theorem (11.7) the associated map
(c— —¦ f ∨ —¦ c1 )§ = f —¦ (c1 — c2 ) : R2 ’ R is C ω . So f —¦ (c1 , c2 ) : R ’ R is C ω .
2
(’) Let f : R — U ’ R be C ω . We have to show that f ∨ : R ’ C ω (U, R) is real
analytic. Obviously, f ∨ has values in this space. We consider a curve c : R ’ U .
If c is C ∞ , then by theorem (11.17) the associated mapping c— —¦f ∨ : R ’ C ∞ (R, R)
is C ω .

11.18
112 Chapter II. Calculus of holomorphic and real analytic mappings 11.19

If c is C ω , then f —¦ (Id —c) : R — R ’ R — U ’ R is C ω . By theorem (11.7) the
associated mapping (f —¦ (Id —c))∨ = c— —¦ f ∨ : R ’ C ω (R, R) is C ω .

Step 2. The theorem is true for F = R.
(⇐) Let f ∨ : W ’ C ω (U, R) be C ω . We have to show that f : W — U ’ R is C ω .
We consider a curve c1 : R ’ W and a curve c2 : R ’ U .
If the ci are C ∞ , then c— —¦ f ∨ : W ’ C ω (U, R) ’ C ∞ (R, R) is C ω by assumption,
2
hence is C , so c2 —¦ f —¦ c1 : R ’ C ∞ (R, R) is C ∞ . By cartesian closedness of
∞ — ∨

smooth mappings, the associated mapping (c— —¦ f ∨ —¦ c1 )§ = f —¦ (c1 — c2 ) : R2 ’ R
2
∞ ∞
is C . So f —¦ (c1 , c2 ) : R ’ R is C .
If the ci are C ω , then f ∨ —¦ c1 : R ’ W ’ C ω (U, R) is C ω by assumption, so
by step 1 the mapping (f ∨ —¦ c1 )§ = f —¦ (c1 — IdU ) : R — U ’ R is C ω . Hence,
f —¦ (c1 , c2 ) = f —¦ (c1 — IdU ) —¦ (Id, c2 ) : R ’ R is C ω .
(’) Let f : W — U ’ R be C ω . We have to show that f ∨ : W ’ C ω (U, R) is real
analytic. Obviously, f ∨ has values in this space. We consider a curve c1 : R ’ W .
If c1 is C ∞ , we consider a second curve c2 : R ’ U . If c2 is C ∞ , then f —¦ (c1 — c2 ) :
R — R ’ W — U ’ R is C ∞ . By cartesian closedness the associated mapping
(f —¦ (c1 — c2 ))∨ = c— —¦ f ∨ —¦ c1 : R ’ C ∞ (R, R) is C ∞ . If c2 is C ω , the mapping
2
f —¦ (IdW —c2 ) : W — R ’ R and also the ¬‚ipped one (f —¦ (IdW —c2 ))∼ : R — W ’ R
are C ω , hence by theorem (11.17) c— —¦ ((f —¦ (IdW —c2 ))∼ )∨ : R ’ C ∞ (R, R) is
1
C . By corollary (11.16) the associated mapping (c— —¦ ((f —¦ (IdW —c2 ))∼ )∨ )∼ =
ω
1
— ∨ ∞
ω
c2 —¦ f —¦ c1 : R ’ C (R, R) is C . So for both families describing the structure of
ˇ
C ω (U, R) we have shown that the composite with f —¦ c1 is C ∞ , so f ∨ —¦ c1 is C ∞ .
If c1 is C ω , then f —¦ (c1 — IdU ) : R — U ’ W — U ’ R is C ω . By step 1 the
associated mapping (f —¦ (c1 — IdU ))∨ = f ∨ —¦ c1 : R ’ C ω (U, R) is C ω .

Step 3. The general case.

f : W — U ’ F is C ω
» —¦ f : W — U ’ R is C ω for all » ∈ F

(» —¦ f )∨ = »— —¦ f ∨ : W ’ C ω (U, R) is C ω , by step 2 and (11.10)

f ∨ : W ’ C ω (U, F ) is C ω .



11.19. Corollary. Canonical mappings are real analytic. The following
mappings are C ω :
ev : C ω (U, F ) — U ’ F , (f, x) ’ f (x),
(1)
ins : E ’ C ω (F, E — F ), x ’ (y ’ (x, y)),
(2)
( )§ : C ω (U, C ω (V, G)) ’ C ω (U — V, G),
(3)
( )∨ : C ω (U — V, G) ’ C ω (U, C ω (V, G)),
(4)
comp : C ω (F, G) — C ω (U, F ) ’ C ω (U, G), (f, g) ’ f —¦ g,
(5)
C ω ( , ) : C ω (E2 , E1 ) — C ω (F1 , F2 ) ’
(6)
’ C ω (C ω (E1 , F1 ), C ω (E2 , F2 )), (f, g) ’ (h ’ g —¦ h —¦ f ).

11.19
11.20 11. The real analytic exponential law 113

Proof. Just consider the canonically associated smooth mappings on multiple
products, as in (3.13).

11.20. Lemma. Canonical isomorphisms. One has the following natural iso-
morphisms:
(1) C ω (W1 , C ω (W2 , F )) ∼ C ω (W2 , C ω (W1 , F )),
=
(2) C ω (W1 , C ∞ (W2 , F )) ∼ C ∞ (W2 , C ω (W1 , F )).
=
(3) C ω (W1 , L(E, F )) ∼ L(E, C ω (W1 , F )).
=
(4) C (W1 , (X, F )) ∼ ∞ (X, C ω (W1 , F )).

ω
=
(5) C ω (W1 , Lipk (X, F )) ∼ Lipk (X, C ω (W1 , F )).
=
In (4) the space X is a ∞ -space, i.e. a set together with a bornology induced by a
family of real valued functions on X, cf. [Fr¨licher, Kriegl, 1988, 1.2.4]. In (5) the
o
k
space X is a Lip -space, cf. [Fr¨licher, Kriegl, 1988, 1.4.1]. The spaces ∞ (X, F )
o
and Lipk (W, F ) are de¬ned in [Fr¨licher, Kriegl, 1988, 3.6.1 and 4.4.1].
o

Proof. All isomorphisms, as well as their inverse mappings, are given by the ¬‚ip of
˜ ˜
coordinates: f ’ f , where f (x)(y) := f (y)(x). Furthermore, all occurring function
spaces are convenient and satisfy the uniform S-boundedness theorem, where S is
the set of point evaluations, by (11.11), (11.14), (11.12), and by [Fr¨licher, Kriegl,
o
1988, 3.6.1, 4.4.2, 3.6.6, and 4.4.7].
˜ ˜
That f has values in the corresponding spaces follows from the equation f (x) =
˜
evx —¦ f . One only has to check that f itself is of the corresponding class, since it
˜
follows that f ’ f is bounded. This is a consequence of the uniform boundedness
principle, since

(evx —¦( ˜ ))(f ) = evx (f ) = f (x) = evx —¦f = (evx )— (f ).
˜ ˜


˜
That f is of the appropriate class in (1) and (2) follows by composing with c1 ∈
C β1 (R, W1 ) and C β2 (», c2 ) : C ±2 (W2 , F ) ’ C β2 (R, R) for all » ∈ F and c2 ∈
C β2 (R, W2 ), where βk and ±k are in {∞, ω} and βk ¤ ±k for k ∈ {1, 2}. Then
˜
C β2 (», c2 ) —¦ f —¦ c1 = (C β1 (», c1 ) —¦ f —¦ c2 )∼ : R ’ C β2 (R, R) is C β1 by (11.7) and
(11.16), since C β1 (», c1 ) —¦ f —¦ c2 : R ’ W2 ’ C ±1 (W1 , F ) ’ C β1 (R, R) is C β2 .
˜
That f is of the appropriate class in (3) follows, since L(E, F ) is the c∞ -closed
subspace of C ω (E, F ) formed by the linear C ω -mappings.
˜
That f is of the appropriate class in (4) or (5) follows from (3), using the free
convenient vector spaces 1 (X) or »k (X) over the ∞ -space X or the the Lipk -space
X, see [Fr¨licher, Kriegl, 1988, 5.1.24 or 5.2.3], satisfying ∞ (X, F ) ∼ L( 1 (X), F )
o =
or satisfying Lip (X, F ) ∼ L(»k (X), F ). Existence of these free convenient vector
k
=
spaces can be proved in a similar way as (23.6).

De¬nition. By a C ∞,ω -mapping f : U — V ’ F we mean a mapping f for which
f ∨ ∈ C ∞ (U, C ω (V, F )).

11.20
114 Chapter II. Calculus of holomorphic and real analytic mappings 11.23

11.21. Theorem. Composition of C ∞,ω -mappings. Let f : U — V ’ F and
g : U1 —V1 ’ V be C ∞,ω , and h : U1 ’ U be C ∞ . Then f —¦(h—¦pr1 , g) : U1 —V1 ’ F ,
(x, y) ’ f (h(x), g(x, y)) is C ∞,ω .

Proof. We have to show that the mapping x ’ (y ’ f (h(x), g(x, y))), U1 ’
C ω (V1 , F ) is C ∞ . It is well-de¬ned, since f and g are C ω in the second variable. In
order to show that it is C ∞ we compose with »— : C ω (V1 , F ) ’ C ω (V1 , R), where
» ∈ F is arbitrary. Thus, it is enough to consider the case F = R. Furthermore,
we compose with c— : C ω (V1 , R) ’ C ± (R, R), where c ∈ C ± (R, V1 ) is arbitrary for
± equal to ω and ∞.
In case ± = ∞ the composite with c— is C ∞ , since the associated mapping U1 —R ’
R is f —¦ (h —¦ pr1 , g —¦ (id — c)) which is C ∞ .
Now the case ± = ω. Let I ⊆ R be an arbitrary open bounded interval. Then
c— —¦ g ∨ : U1 ’ C ω (R, G) is C ∞ , where G is the convenient vector space containing
¯
V as an c∞ -open subset, and has values in {γ : γ(I) ⊆ V } ⊆ C ω (R, G). This set is
c∞ -open, since it is open for the topology of uniform convergence on compact sets
which is coarser than the bornological topology on C ∞ (R, E) and hence than the
c∞ -topology on C ω (R, G), see (11.10).
Thus, the composite with c— , comp —¦(f ∨ —¦ h, c— —¦ g ∨ ) is C ∞ , since f ∨ —¦ h : U1 ’
U ’ C ω (V, F ) is C ∞ , c— —¦ g ∨ : U1 ’ C ω (R, G) is C ∞ and comp : C ω (V, R) — {γ ∈
¯
C ω (R, G) : γ(I) ⊆ V } ’ C ω (I, R) is C ω , because it is associated to ev —¦(id — ev) :
¯ ¯
C ω (V, F ) — {γ ∈ C ω (R, G) : γ(I) ⊆ V } — I ’ V . That ev : {γ ∈ C ω (R, G) : γ(I) ⊆
V } — I ’ R is C ω follows, since the associated mapping is the restriction mapping
C ω (R, G) ’ C ω (I, G).

11.22. Corollary. Let w : W1 ’ W be C ω , let u : U ’ U1 be smooth, let v : V ’
V1 be C ω , and let f : U1 — V1 ’ W1 be C ∞,ω . Then w —¦ f —¦ (u — v) : U — V ’ W
is again C ∞,ω .

This is generalization of theorem (11.17).

Proof. Use (11.21) twice.

11.23. Corollary. Let f : E ⊇ U ’ F be C ω , let I ⊆ R be open and bounded,
¯
and ± be ω or ∞. Then f— : C ± (R, E) ⊇ {c : c(I) ⊆ U } ’ C ± (I, F ) is C ω .

Proof. Obviously, f— (c) := f —¦ c ∈ C ± (I, F ) is well-de¬ned for all c ∈ C ± (R, E)
¯
satisfying c(I) ⊆ U .
¯
Furthermore, the composite of f— with any C β -curve γ : R ’ {c : c(I) ⊆ U } ⊆
C ± (R, E) is a C β -curve in C ± (I, F ) for β equal to ω or ∞. For β = ± this follows
from cartesian closedness of the C ± -maps. For ± = β this follows from (11.22).
¯
Finally, {c : c(I) ⊆ U } ⊆ C ± (R, E) is c∞ -open, since it is open for the topology
of uniform convergence on compact sets which is coarser than the bornological and
hence than the c∞ -topology on C ± (R, E). Here is the only place where we make
use of the boundedness of I.

11.23
11.26 11. The real analytic exponential law 115

d
dt |t=0
11.24. Lemma. Derivatives. The derivative d, where df (x)(v) := f (x +
tv), is bounded and linear d : C ω (U, F ) ’ C ω (U, L(E, F )).

Proof. The di¬erential df (x)(v) makes sense and is linear in v, because every real
analytic mapping f is smooth. So it remains to show that (f, x, v) ’ df (x)(v) is
real analytic. So let f , x, and v depend real analytically (resp. smoothly) on a
real parameter s. Since (t, s) ’ x(s) + tv(s) is real analytic (resp. smooth) into
U ⊆ E, the mapping r ’ ((t, s) ’ f (r)(x(s)+tv(s)) is real analytic into C ω (R2 , F )
(resp. smooth into C ∞ (R2 , F ). Composing with ‚t |t=0 : C ω (R2 , F ) ’ C ω (R, F )


(resp. : C ∞ (R2 , F ) ’ C ∞ (R, F )) shows that r ’ (s ’ d(f (r))(x(s))(v(s))), R ’
C ω (R, F ) is real analytic. Considering the associated mapping on R2 composed
with the diagonal map shows that (f, x, v) ’ df (x)(v) is real analytic.

The following examples as well as several others can be found in [Fr¨licher, Kriegl,
o
1988, 5.3.6].

11.25. Example. Let T : C ∞ (R, R) ’ C ∞ (R, R) be given by T (f ) = f . Then the
continuous linear di¬erential equation x (t) = T (x(t)) with initial value x(0) = x0
has a unique smooth solution x(t)(s) = x0 (t + s) which is however not real analytic.

Note the curious form x (t) = x(t) of this di¬erential equation. Beware of careless
notation!

Proof. A smooth curve x : R ’ C ∞ (R, R) is a solution of the di¬erential equation
‚ ‚ d
x (t) = T (x(t)) if and only if ‚t x(t, s) = ‚s x(t, s). Hence, we have dt x(t, r ’ t) = 0,
ˆ ˆ ˆ
i.e. x(t, r ’ t) is constant and hence equal to x(0, r) = x0 (r). Thus, x(t, s) =
ˆ ˆ ˆ
x0 (t + s).
Suppose x : R ’ C ∞ (R, R) were real analytic. Then the composite with ev0 :
C ∞ (R, R) ’ R were a real analytic function. But this composite is just x0 = ev0 —¦x,
which is not in general real analytic.

11.26. Example. Let E be either C ∞ (R, R) or C ω (R, R). Then the mapping
exp— : E ’ E is C ω , has invertible derivative at every point, but the image does
not contain an open neighborhood of exp— (0).

Proof. The mapping exp— is real analytic by (11.23). Its derivative is given by
(exp— ) (f )(g) : t ’ g(t)ef (t) and hence is invertible with g ’ (t ’ g(t)e’f (t) )
as inverse mapping. Now consider the real analytic curve c : R ’ E given by
c(t)(s) = 1 ’ (ts)2 . One has c(0) = 1 = exp— (0), but c(t) is not in the image of
exp— for any t = 0, since c(t)( 1 ) = 0 but exp— (g)(t) = eg(t) > 0 for all g and t.
t




11.26
116 Chapter II. Calculus of holomorphic and real analytic mappings

Historical Remarks on Holomorphic
and Real Analytic Calculus

The notion of holomorphic mappings used in section (15) was ¬rst de¬ned by the
Italian Luigi Fantappi´ in the papers [Fantappi´, 1930] and [Fantappi´, 1933]:
e e e
S.1: “Wenn jeder Funktion y(t) einer Funktionenmenge H eine bestimmte Zahl f entspricht,
d.h. die Zahl f von der Funktion y(t) (unabh¨ngige Ver¨nderliche in der Menge H) abh¨ngt,
a a a
werden wir sagen, daß ein Funktional von y(t):

f = F [y(t)]

ist; H heißt das De¬nitionsfeld des Funktionals F .
[ . . . ] gemischtes Funktional [ . . . ]

f = F [y1 (t1 , . . . ), . . . , yn (t1 , . . . ); z1 , . . . , zm ]”


He also considered the ˜functional transform™ and noticed the relation

f = F [y(t); z] corresponds to y ’ f (z)

S.4: “Sei jetzt F (y(t)) ein Funktional, das in einem Funktionenbereich H (von analytischen
Funktionen) de¬niert ist, und y0 (t) ein Funktion von H, die mit einer Umgebung (r) oder
(r, σ) zu H angeh¨rt. Wenn f¨r jede analytische Mannigfaltigkeit y(t; ±1 , . . . , ±m ), die in
o u
diese Umgebung eindringt (d.h. eine solche, die f¨r alle Wertesysteme ±1 , . . . , ±m ) eines
u
Bereichs “ eine Funktion von t der Umgebung liefert), der Wert des Funktionals

Ft [y(t; ±1 , . . . , ±m )] = f (±1 , . . . , ±m )

immer eine Funktion der Parameter ±1 , . . . , ±m ist, die nicht nur in “ de¬niert, sondern
dort noch eine analytische Funktion ist, werden wir sagen, daß das Funktional F regul¨r ist
a
in der betrachteten Umgebung y0 (t). Wenn ein Funktional F regul¨r ist in einer Umgebung
a
jeder Funktion seines De¬nitionsbereiches, so heißt F analytisch.”

The development in the complex case was much faster than in the smooth case
since one did not have to explain the concept of higher derivatives.
The Portuguese Jos´ Sebasti˜o e Silva showed that analyticity in the sense of
e a
Fantappi´ coincides with other concepts, in his dissertation [Sebasti˜o e Silva,
e a
1948], published as [Sebasti˜o e Silva, 1950a], and in [Sebasti˜o e Silva, 1953].
a a
An overview over various notions of holomorphicity was given by the Brasilian
Domingos Pisanelli in [Pisanelli, 1972a] and [Pisanelli, 1972b].
117




Chapter III
Partitions of Unity


12. Di¬erentiability of Finite Order . . . . . . . . . . . . . . . . . . 118
13. Di¬erentiability of Seminorms ......... . . . . . . . . . 127
14. Smooth Bump Functions . . . . . . . . . . . . . . . . . . . . . 152
15. Functions with Globally Bounded Derivatives .. . . . . . . . . . 159
16. Smooth Partitions of Unity and Smooth Normality . . . . . . . . . 165
The main aim of this chapter is to discuss the abundance or scarcity of smooth
functions on a convenient vector space: E.g. existence of bump functions and parti-
tions of unity. This question is intimately related to di¬erentiability of seminorms
and norms, and in many examples these are, if at all, only ¬nitely often di¬eren-
tiable. So we start this chapter with a short (but complete) account of ¬nite order
di¬erentiability, based on Lipschitz conditions on higher derivatives, since with this
notion we can get as close as possible to exponential laws. A more comprehensive
exposition of ¬nite order Lipschitz di¬erentiability can be found in the monograph
[Fr¨licher, Kriegl, 1988].
o
Then we treat di¬erentiability of seminorms and convex functions, and we have
tried to collect all relevant information from the literature. We give full proofs of
all what will be needed later on or is of central interest. We also collect related
results, mainly on ˜generic di¬erentiability™, i.e. di¬erentiability on a dense Gδ -set.
If enough smooth bump functions exist on a convenient vector space, we call it
˜smoothly regular™. Although the smooth (i.e. bounded) linear functionals separate
points on any convenient vector space, stronger separation properties depend very
much on the geometry. In particular, we show that 1 and C[0, 1] are not even
C 1 -regular. We also treat more general ˜smooth spaces™ here since most results do
not depend on a linear structure, and since we will later apply them to manifolds.
In many problems like E. Borel™s theorem (15.4) that any power series appears
as Taylor series of a smooth function, or the existence of smooth functions with
given carrier (15.3), one uses in ¬nite dimensions the existence of smooth functions
with globally bounded derivatives. These do not exist in in¬nite dimensions in
general; even for bump functions this need not be true globally. Extreme cases
are Hilbert spaces where there are smooth bump functions with globally bounded
derivatives, and c0 which does not even admit C 2 -bump functions with globally
bounded derivatives.
In the ¬nal section of this chapter a space which admits smooth partitions of unity
subordinated to any open cover is called smoothly paracompact. Fortunately, a
118 Chapter III. Partitions of unity 12.2

wide class of convenient vector spaces has this property, among them all spaces of
smooth sections of ¬nite dimensional vector bundles which we shall need later as
modeling spaces for manifolds of mappings. The theorem (16.15) of [Toru´czyk,
n
1973] characterizes smoothly paracompact metrizable spaces, and we will give a
full proof. It is the only tool for investigating whether non-separable spaces are
smoothly paracompact and we give its main applications.



12. Di¬erentiability of Finite Order

12.1. De¬nition. A mapping f : E ⊇ U ’ F , where E and F are convenient
vector spaces, and U ⊆ E is c∞ -open, is called Lipk if f —¦ c is a Lipk -curve (see
(1.2)) for each c ∈ C ∞ (R, U ).
This is equivalent to the property that f —¦c is Lipk on c’1 (U ) for each c ∈ C ∞ (R, E).
This can be seen by reparameterization.

12.2. General curve lemma. Let E be a convenient vector space, and let cn ∈
C ∞ (R, E) be a sequence of curves which converges fast to 0, i.e., for each k ∈ N
the sequence nk cn is bounded. Let sn ≥ 0 be reals with n sn < ∞.
Then there exists a smooth curve c ∈ C ∞ (R, E) and a converging sequence of reals
tn such that c(t + tn ) = cn (t) for |t| ¤ sn , for all n.
rn +rn+1
2
. Let h : R ’ [0, 1] be
Proof. Let rn := k<n ( k2 + 2sk ) and tn := 2
smooth with h(t) = 1 for t ≥ 0 and h(t) = 0 for t ¤ ’1, and put hn (t) := h(n2 (sn +
1
t)).h(n2 (sn ’t)). Then we have hn (t) = 0 for |t| ≥ n2 +sn and hn (t) = 1 for |t| ¤ sn ,
(j)
and for the derivatives we have |hn (t)| ¤ n2j .Hj , where Hj := max{|h(j) | : t ∈ R}.
Thus, in the sum
hn (t ’ tn ).cn (t ’ tn )
c(t) :=
n

at most one summand is non-zero for each t ∈ R, and c is a smooth curve since for
each ∈ E we have

( —¦ c)(t) = fn (t), where fn (t + tn ) := hn (t). (cn (t)),
n

n2 . sup |fn (t)| = n2 . sup |fn (s + tn )| : |s| ¤
(k) (k) 1
+ sn
n2
t
k
k
2
n2j Hj . sup |( —¦ cn )(k’j) (s)| : |s| ¤ 1
¤n + sn
n2
j
j=0
k
k
n2j+2 Hj . sup |( —¦ cn )(i) (s)| : |s| ¤ max( n2 + sn ) and i ¤ k ,
1
¤ j n
j=0


which is uniformly bounded with respect to n, since cn converges to 0 fast.


12.2
12.4 12. Di¬erentiability of ¬nite order 119

12.3. Corollary. Let cn : R ’ E be polynomials of bounded degree with values in
a convenient vector space E. If for each ∈ E the sequence n ’ sup{|( —¦ cn )(t) :
|t| ¤ 1} converges to 0 fast, then the sequence cn converges to 0 fast in C ∞ (R, E),
so the conclusion of (12.2) holds.

Proof. The structure on C ∞ (R, E) is the initial one with respect to the cone
∞ ∞
— : C (R, E) ’ C (R, R) for all ∈ E , by (3.9). So we only have to show the
result for E = R. On the ¬nite dimensional space of all polynomials of degree at
most d the expression in the assumption is a norm, and the inclusion into C ∞ (R, R)
is bounded.

12.4. Di¬erence quotients. For a curve c : R ’ E with values in a vector space
E the di¬erence quotient δ k c of order k is given recursively by

δ 0 c := c,
δ k’1 c(t0 , . . . , tk’1 ) ’ δ k’1 c(t1 , . . . , tk )
k
δ c(t0 , . . . , tk ) := k ,
t 0 ’ tk

for pairwise di¬erent ti . The constant factor k in the de¬nition of δ k is chosen in
such a way that δ k approximates the k-th derivative. By induction, one can easily
see that
k
δ k c(t0 , . . . , tk ) = k! 1
c(ti ) ti ’tj .
i=0 0¤j¤k
j=i

k
We shall mainly need the equidistant di¬erence quotient δeq c of order k, which is
given by

k
k!
k k 1
δeq c(t; v) := δ c(t, t + v, . . . , t + kv) = k c(t + iv) i’j .
v i=0 0¤j¤k
j=i


Lemma. For a convenient vector space E and a curve c : R ’ E the following
conditions are equivalent:
(1) c is Lipk’1 .
(2) The di¬erence quotient δ k c of order k is bounded on bounded sets.
(3) —¦ c is continuous for each ∈ E , and the equidistant di¬erence quotient
k
δeq c of order k is bounded on bounded sets in R — (R \ {0}).

Proof. All statements can be tested by composing with bounded linear functionals
∈ E , so we may assume that E = R.
(3) ’ (2) Let I ‚ R be a bounded interval. Then there is some K > 0 such that
k
|δeq c(x; v)| ¤ K for all x ∈ I and kv ∈ I. Let ti ∈ I be pairwise di¬erent points.
We claim that |δ k c(t0 , . . . , tk )| ¤ K. Since δ k c is symmetric we may assume that
t0 < t1 < · · · < tk , and since it is continuous (c is continuous) we may assume that
’t
all tk’t0 are of the form ni for ni , N ∈ N. Put v := tkN 0 , then δ k c(t0 , . . . , tk ) =
ti
’t0 N


12.4
120 Chapter III. Partitions of unity 12.4

δ k c(t0 , t0 + n1 v, . . . , t0 + nk v) is a convex combination of δeq c(t0 + rv; v) for 0 ¤ r ¤
k

maxi ni ’ k. This follows by recursively inserting intermediate points of the form
t0 + mv, and using

δ k (t0 + m0 v, . . . , t0 + mi v, . . . , t0 + mk+1 v) =
mi ’ m0 k
= δ (t0 + m0 v, . . . , t0 + mk v)
mk+1 ’ m0
mk+1 ’ mi k
+ δ (t1 + m1 v, . . . , t0 + mk+1 v)
mk+1 ’ m0

which itself may be proved by induction on k.
(2) ’ (1) We have to show that c is k times di¬erentiable and that δ 1 c(k) is bounded
on bounded sets. We use induction, k = 0 is clear.
Let T = S be two subsets of R of cardinality j + 1. Then there exist enumerations
T = {t0 , . . . , tj } and S = {s0 , . . . , sj } such that ti = sj for i ¤ j; then we have
j
δ j c(t0 , . . . , tj ) ’ δ j c(s0 , . . . , sj ) = (ti ’ si )δ j+1 c(t0 , . . . , ti , si , . . . , sj ).
1
j+1
i=0

For the enumerations we put the elements of T © S at the end in T and at the
beginning in S. Using the recursive de¬nition of δ j+1 c and symmetry the right
hand side becomes a telescoping sum.
Since δ k c is bounded we see from the last equation that all δ j c are also bounded,
in particular this is true for δ 2 c. Then

c(t + s) ’ c(t) c(t + s ) ’ c(t)
δ 2 c(t, t + s, t + s )
s’s
’ = 2
s s
shows that the di¬erence quotient of c forms a Mackey Cauchy net, and hence the
limit c (t) exists.
Using the easily checked formula
j i’1
(tj ’ tl ) δ j c(t0 , . . . , tj ),
1
c(tj ) = i!
i=0 l=0

induction on j and di¬erentiability of c one shows that
j
δ j c (t0 , . . . , tj ) = δ j+1 c(t0 , . . . , tj , ti ),
1
(4) j+1
i=0

where δ j+1 c(t0 , . . . , tj , ti ) := limt’ti δ j+1 c(t0 , . . . , tj , t). The right hand side of (4)
is bounded, so c is Lipk’2 by induction on k.
(1) ’ (2) For a di¬erentiable function f : R ’ R and t0 < · · · < tj there exist si
with ti < si < ti+1 such that

δ j f (t0 , . . . , tj ) = δ j’1 f (s0 , . . . , sj’1 ).
(5)

12.4
12.5 12. Di¬erentiability of ¬nite order 121

Let p be the interpolation polynomial
j i’1
(t ’ tl ) δ j f (t0 , . . . , tj ).
1
(6) p(t) := i!
i=0 l=0

Since f and p agree on all tj , by Rolle™s theorem the ¬rst derivatives of f and p
agree on some intermediate points si . So p is the interpolation polynomial for
f at these points si . Comparing the coe¬cient of highest order of p and of the
interpolation polynomial (6) for f at the points si (5) follows.
Applying (5) recursively for f = c(k’2) , c(k’3) , . . . , c shows that δ k c is bounded on
bounded sets, and (2) follows.
(2) ’ (3) is obvious.

12.5. Let r0 , . . . , rk be the unique rational solution of the linear equation
k
1 for j = 1
i j ri =
0 for j = 0, 2, 3, . . . , k.
i=0

Lemma. If f : R2 ’ R is Lipk for k ≥ 1 and I is a compact interval then there
exists M such that for all t, v ∈ I we have
k
ri f (t, iv) ¤ M |v|k+1 .

‚s |0 f (t, s).v ’
i=0


Proof. We consider ¬rst the case 0 ∈ I so that v stays away from 0. For this it
/

su¬ces to show that the derivative ‚s |0 f (t, s) is locally bounded. If it is unbounded
near some point x∞ , there are xn with |xn ’x∞ | ¤ 21 such that ‚s |0 f (xn , s) ≥ n.2n .

n

We apply the general curve lemma (12.2) to the curves cn : R ’ R2 given by
cn (t) := (xn , 2tn ) and to sn := 21 in order to obtain a smooth curve c : R ’ R2
n

and scalars tn ’ 0 with c(t + tn ) = cn (t) for |t| ¤ sn . Then (f —¦ c) (tn ) =
1
1‚
2n ‚s |0 f (xn , s) ≥ n, which contradicts that f is Lip .
Now we treat the case 0 ∈ I. If the assertion does not hold there are xn , vn ∈
k

I, such that ‚s |0 f (xn , s).vn ’ i=0 ri f (xn , ivn ) ≥ n.2n(k+1) |vn |k+1 . We may
assume xn ’ x∞ , and by the case 0 ∈ I we may assume that vn ’ 0, even with
/
|xn ’ x∞ | ¤ 21 and |vn | ¤ 21 . We apply the general curve lemma (12.2) to the
n n

curves cn : R ’ R2 given by cn (t) := (xn , 2tn ) and to sn := 21 to obtain a smooth
n
2
curve c : R ’ R and scalars tn ’ 0 with c(t + tn ) = cn (t) for |t| ¤ sn . Then we
have
k
(f —¦ c) (tn )2n vn ’ ri (f —¦ c)(tn + i2n vn ) =
i=0
k
n
ri (f —¦ cn )(i2n vn )
= (f —¦ cn ) (0)2 vn ’
i=0
k
n
ri f (xn , ivn ) ≥ n(2n |vn |)k+1 .
1‚
2n ‚s |0 f (xn , s)2 vn ’
=
i=0

12.5
122 Chapter III. Partitions of unity 12.6

This contradicts the next claim for g = f —¦ c.
Claim. If g : R ’ R is Lipk for k ≥ 1 and I is a compact interval then there is
k
M > 0 such that for t, v ∈ I we have g (t).v ’ i=0 ri g(t + iv) ¤ M |v|k+1 .
k
Consider gt (v) := g (t).v ’ i=0 ri g(t + iv). Then the derivatives up to order k at
v = 0 of gt vanish by the choice of the ri . Since g (k) is locally Lipschitzian there
(k)
exists an M such that |gt (v)| ¤ M |v| for all t, v ∈ I, which we may integrate in
|v|k+1
turn to obtain |gt (v)| ¤ M (k+1)! .

12.6. Lemma. Let f : R2 ’ R be Lipk+1 . Then t ’ is Lipk .

‚s |0 f (t, s)

Proof. Suppose that g : t ’ ‚s |0 f (t, s) is not Lipk . Then by lemma (12.4) the

k+1
equidistant di¬erence quotient δeq g is not locally bounded at some point which we
may assume to be 0. Then there are xn and vn with |xn | ¤ 1/4n and 0 < vn < 1/4n
such that

|δeq g(xn ; vn )| > n.2n(k+2) .
k+1
(1)

We apply the general curve lemma (12.2) to the curves cn : R ’ R2 given by
cn (t) := en ( 2tn + xn ) := ( 2tn + xn ’ vn , 2tn ) and to sn := k+2 in order to obtain a
2n
2
smooth curve c : R ’ R and scalars tn ’ 0 with c(t + tn ) = cn (t) for 0 ¤ t ¤ sn .
k
Put f0 (t, s) := i=0 ri f (t, is) for ri as in (12.5), put f1 (t, s) := g(t)s, ¬nally put
f2 := f1 ’f0 . Then f0 in Lipk+1 , so f0 —¦c is Lipk+1 , hence the equidistant di¬erence
quotient δeq (f0 —¦ c)(xn ; 2n vn ) is bounded.
k+2

By lemma (12.5) there exists M > 0 such that |f2 (t, s)| ¤ M |s|k+2 for all t, s ∈
[’(k + 1), k + 1], so we get

|δeq (f2 —¦ c)(xn ; 2n vn )| = |δeq (f2 —¦ cn )(0; 2n vn )|
k+2 k+2


|δ k+2 (f2 —¦ en )(xn ; vn )|
1
= 2n(k+2) eq
k+2
i(k+2)
|f2 ((i ’ 1)vn + xn , ivn )|
(k+2)!
¤ 2n(k+2) |ivn |(k+2) j=i |i ’ j|
i=1
k+2
i(k+2)
(k+2)!
¤ M .
2n(k+2) |i ’ j|
j=i
i=1


This is bounded, and so for f1 = f0 + f2 the expression |δeq (f1 —¦ c)(xn ; 2n vn )| is
k+2

also bounded, with respect to n. However, on the other hand we get

δeq (f1 —¦ c)(xn ; 2n vn ) = δeq (f1 —¦ cn )(0; 2n vn )
k+2 k+2


δ k+2 (f1 —¦ en )(xn ; vn )
1
= 2n(k+2) eq
k+2
f1 ((i ’ 1)vn + xn , ivn )
(k+2)! 1
= 2n(k+2) i’j
(k+2)
vn
i=0 0¤j¤k+2
j=i


12.6
12.8 12. Di¬erentiability of ¬nite order 123

k+2
g((i ’ 1)vn + xn )ivn
(k+2)! 1
= 2n(k+2) i’j
(k+2)
vn
i=0 0¤j¤k+2
j=i
k+1
g(lvn + xn )
(k+2)! 1
= 2n(k+2) l’j
(k+1)
vn
l=0 0¤j¤k+1
j=l

δ k+1 g(xn ; vn ),
k+2
= 2n(k+2) eq


which in absolute value is larger than (k + 2)n by (1), a contradiction.

12.7. Lemma. Let E be a normed space and F be a convenient vector space, U
open in E. Then, a mapping f : U ’ F is Lip0 if and only if f is locally Lipschitz,
i.e., f (x)’f (y) is locally bounded.
x’y


Proof. (’) If f is Lip0 but not locally Lipschitz near z ∈ U , there are ∈ F
and points xn = yn in U with xn ’ z ¤ 1/2n and yn ’ z ¤ 1/2n , such that
(f (yn ) ’ f (xn )) ≥ n. yn ’ xn . Now we apply the general curve lemma (12.2)
with sn := yn ’ xn and cn (t) := xn ’ z + t(yn ’ xn ) to get a smooth curve c with
c(t + tn ) = cn (t) for 0 ¤ t ¤ sn . Then s1 (( —¦ f —¦ c)(tn + sn ) ’ ( —¦ f —¦ c)(tn )) =
n
1
yn ’xn (f (yn ) ’ f (xn )) ≥ n.

(⇐) This is obvious, since the composition of locally Lipschitzian mappings is again
locally Lipschitzian.

12.8. Theorem. Let f : E ⊇ U ’ F be a mapping, where E and F are convenient
vector spaces, and U ⊆ E is c∞ -open. Then the following assertions are equivalent
for each k ≥ 0:
(1) f is Lipk+1 .
(2) The directional derivative


‚t |t=0 (f (x
(dv f )(x) := + tv))

exists for x ∈ U and v ∈ E and de¬nes a Lipk -mapping U — E ’ F .

Note that this result gives a di¬erent (more algebraic) proof of Boman™s theorem
(3.4) and (3.14).

Proof. (1) ’ (2) Clearly, t ’ f (x+tv) is Lipk+1 , and so the directional derivative
exists and is the Mackey-limit of the di¬erence quotients, by lemma (1.7). In order
to show that df : (x, v) ’ dv f (x) is Lipk we take a smooth curve (x, v) : R ’
U — E and ∈ F , and we consider g(t, s) := x(t) + s.v(t), g : R2 ’ E. Then
—¦ f —¦ g : R2 ’ R is Lipk+1 , so by lemma (12.6) the curve

‚ ‚
t ’ (df (x(t), v(t))) = ‚s |0 f (g(t, s)) ‚s |0
= (f (g(t, s)))

is of class Lipk .

12.8
124 Chapter III. Partitions of unity 12.10

(2) ’ (1) If c ∈ C ∞ (R, U ) then
f (c(t)) ’ f (c(0))
’ df (c(0), c (0)) =
t
1
df (c(0) + s(c(t) ’ c(0)), c(t)’c(0) ) ’ df (c(0), c (0)) ds
= t
0

converges to 0 for t ’ 0 since g : (t, s) ’ df (c(0) + s(c(t) ’ c(0)), c(t)’c(0) ) ’
t
k
df (c(0), c (0)) is Lip , thus by lemma (12.7) g is locally Lipschitz, so the set of all
1
g(t1 ,s)’g(t2 ,s)
is locally bounded, and ¬nally t ’ 0 g(t, s)ds is locally Lipschitz.
t1 ’t2
Thus, f —¦ c is di¬erentiable with derivative (f —¦ c) (0) = df (c(0), c (0)).
Since df is Lipk and (c, c ) is smooth we get that (f —¦ c) is Lipk , hence f —¦ c is
Lipk+1 .

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