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(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

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