ńņš. 18 |

o u

diese Umgebung eindringt (d.h. eine solche, die fĀØr alle Wertesysteme Ī±1 , . . . , Ī±m ) eines

u

Bereichs Ī“ eine Funktion von t der Umgebung liefert), der Wert des Funktionals

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

immer eine Funktion der Parameter Ī±1 , . . . , Ī±m ist, die nicht nur in Ī“ deļ¬niert, sondern

dort noch eine analytische Funktion ist, werden wir sagen, daĆ das Funktional F regulĀØr ist

a

in der betrachteten Umgebung y0 (t). Wenn ein Funktional F regulĀØr ist in einer Umgebung

a

jeder Funktion seines Deļ¬nitionsbereiches, so heiĆt F analytisch.ā

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 JosĀ“ SebastiĖo e Silva showed that analyticity in the sense of

e a

FantappiĀ“ coincides with other concepts, in his dissertation [SebastiĖo e Silva,

e a

1948], published as [SebastiĖo e Silva, 1950a], and in [SebastiĖo 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. Diļ¬erentiability of Finite Order . . . . . . . . . . . . . . . . . . 118

13. Diļ¬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 diļ¬erentiability of seminorms

and norms, and in many examples these are, if at all, only ļ¬nitely often diļ¬eren-

tiable. So we start this chapter with a short (but complete) account of ļ¬nite order

diļ¬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 ļ¬nite order Lipschitz diļ¬erentiability can be found in the monograph

[FrĀØlicher, Kriegl, 1988].

o

Then we treat diļ¬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 ā˜generic diļ¬erentiabilityā™, i.e. diļ¬erentiability on a dense GĪ“ -set.

If enough smooth bump functions exist on a convenient vector space, we call it

ā˜smoothly regularā™. 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 ā˜smooth spacesā™ 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. Borelā™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 ļ¬nite dimensions the existence of smooth functions

with globally bounded derivatives. These do not exist in inļ¬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 ļ¬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 ļ¬nite dimensional vector bundles which we shall need later as

modeling spaces for manifolds of mappings. The theorem (16.15) of [ToruĀ“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. Diļ¬erentiability of Finite Order

12.1. Deļ¬nition. A mapping f : E ā U ā’ F , where E and F are convenient

vector spaces, and U ā E is cā -open, is called Lipk if f ā—¦ c is a Lipk -curve (see

(1.2)) for each c ā C ā (R, U ).

This is equivalent to the property that f ā—¦c is Lipk on cā’1 (U ) for each c ā C ā (R, E).

This can be seen by reparameterization.

12.2. General curve lemma. Let E be a convenient vector space, and let cn ā

C ā (R, E) be a sequence of curves which converges fast to 0, i.e., for each k ā N

the sequence nk cn is bounded. Let sn ā„ 0 be reals with n sn < ā.

Then there exists a smooth curve c ā C ā (R, E) and a converging sequence of reals

tn such that c(t + tn ) = cn (t) for |t| ā¤ sn , for all n.

rn +rn+1

2

. Let h : R ā’ [0, 1] be

Proof. Let rn := k<n ( k2 + 2sk ) and tn := 2

smooth with h(t) = 1 for t ā„ 0 and h(t) = 0 for t ā¤ ā’1, and put hn (t) := h(n2 (sn +

1

t)).h(n2 (sn ā’t)). Then we have hn (t) = 0 for |t| ā„ n2 +sn and hn (t) = 1 for |t| ā¤ sn ,

(j)

and for the derivatives we have |hn (t)| ā¤ n2j .Hj , where Hj := max{|h(j) | : t ā R}.

Thus, in the sum

hn (t ā’ tn ).cn (t ā’ tn )

c(t) :=

n

at most one summand is non-zero for each t ā R, and c is a smooth curve since for

each ā E we have

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

n

n2 . sup |fn (t)| = n2 . sup |fn (s + tn )| : |s| ā¤

(k) (k) 1

+ sn

n2

t

k

k

2

n2j Hj . sup |( ā—¦ cn )(kā’j) (s)| : |s| ā¤ 1

ā¤n + sn

n2

j

j=0

k

k

n2j+2 Hj . sup |( ā—¦ cn )(i) (s)| : |s| ā¤ max( n2 + sn ) and i ā¤ k ,

1

ā¤ j n

j=0

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

12.2

12.4 12. Diļ¬erentiability of ļ¬nite order 119

12.3. Corollary. Let cn : R ā’ E be polynomials of bounded degree with values in

a convenient vector space E. If for each ā E the sequence n ā’ sup{|( ā—¦ cn )(t) :

|t| ā¤ 1} converges to 0 fast, then the sequence cn converges to 0 fast in C ā (R, E),

so the conclusion of (12.2) holds.

Proof. The structure on C ā (R, E) is the initial one with respect to the cone

ā ā

ā— : C (R, E) ā’ C (R, R) for all ā E , by (3.9). So we only have to show the

result for E = R. On the ļ¬nite dimensional space of all polynomials of degree at

most d the expression in the assumption is a norm, and the inclusion into C ā (R, R)

is bounded.

12.4. Diļ¬erence quotients. For a curve c : R ā’ E with values in a vector space

E the diļ¬erence quotient Ī“ k c of order k is given recursively by

Ī“ 0 c := c,

Ī“ kā’1 c(t0 , . . . , tkā’1 ) ā’ Ī“ kā’1 c(t1 , . . . , tk )

k

Ī“ c(t0 , . . . , tk ) := k ,

t 0 ā’ tk

for pairwise diļ¬erent ti . The constant factor k in the deļ¬nition of Ī“ k is chosen in

such a way that Ī“ k approximates the k-th derivative. By induction, one can easily

see that

k

Ī“ k c(t0 , . . . , tk ) = k! 1

c(ti ) ti ā’tj .

i=0 0ā¤jā¤k

j=i

k

We shall mainly need the equidistant diļ¬erence quotient Ī“eq c of order k, which is

given by

k

k!

k k 1

Ī“eq c(t; v) := Ī“ c(t, t + v, . . . , t + kv) = k c(t + iv) iā’j .

v i=0 0ā¤jā¤k

j=i

Lemma. For a convenient vector space E and a curve c : R ā’ E the following

conditions are equivalent:

(1) c is Lipkā’1 .

(2) The diļ¬erence quotient Ī“ k c of order k is bounded on bounded sets.

(3) ā—¦ c is continuous for each ā E , and the equidistant diļ¬erence quotient

k

Ī“eq c of order k is bounded on bounded sets in R Ć— (R \ {0}).

Proof. All statements can be tested by composing with bounded linear functionals

ā E , so we may assume that E = R.

(3) ā’ (2) Let I ā‚ R be a bounded interval. Then there is some K > 0 such that

k

|Ī“eq c(x; v)| ā¤ K for all x ā I and kv ā I. Let ti ā I be pairwise diļ¬erent points.

We claim that |Ī“ k c(t0 , . . . , tk )| ā¤ K. Since Ī“ k c is symmetric we may assume that

t0 < t1 < Ā· Ā· Ā· < tk , and since it is continuous (c is continuous) we may assume that

ā’t

all tkā’t0 are of the form ni for ni , N ā N. Put v := tkN 0 , then Ī“ k c(t0 , . . . , tk ) =

ti

ā’t0 N

12.4

120 Chapter III. Partitions of unity 12.4

Ī“ k c(t0 , t0 + n1 v, . . . , t0 + nk v) is a convex combination of Ī“eq c(t0 + rv; v) for 0 ā¤ r ā¤

k

maxi ni ā’ k. This follows by recursively inserting intermediate points of the form

t0 + mv, and using

Ī“ k (t0 + m0 v, . . . , t0 + mi v, . . . , t0 + mk+1 v) =

mi ā’ m0 k

= Ī“ (t0 + m0 v, . . . , t0 + mk v)

mk+1 ā’ m0

mk+1 ā’ mi k

+ Ī“ (t1 + m1 v, . . . , t0 + mk+1 v)

mk+1 ā’ m0

which itself may be proved by induction on k.

(2) ā’ (1) We have to show that c is k times diļ¬erentiable and that Ī“ 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 ā¤ j; then we have

j

Ī“ j c(t0 , . . . , tj ) ā’ Ī“ j c(s0 , . . . , sj ) = (ti ā’ si )Ī“ j+1 c(t0 , . . . , ti , si , . . . , sj ).

1

j+1

i=0

For the enumerations we put the elements of T ā© S at the end in T and at the

beginning in S. Using the recursive deļ¬nition of Ī“ j+1 c and symmetry the right

hand side becomes a telescoping sum.

Since Ī“ k c is bounded we see from the last equation that all Ī“ j c are also bounded,

in particular this is true for Ī“ 2 c. Then

c(t + s) ā’ c(t) c(t + s ) ā’ c(t)

Ī“ 2 c(t, t + s, t + s )

sā’s

ā’ = 2

s s

shows that the diļ¬erence quotient of c forms a Mackey Cauchy net, and hence the

limit c (t) exists.

Using the easily checked formula

j iā’1

(tj ā’ tl ) Ī“ j c(t0 , . . . , tj ),

1

c(tj ) = i!

i=0 l=0

induction on j and diļ¬erentiability of c one shows that

j

Ī“ j c (t0 , . . . , tj ) = Ī“ j+1 c(t0 , . . . , tj , ti ),

1

(4) j+1

i=0

where Ī“ j+1 c(t0 , . . . , tj , ti ) := limtā’ti Ī“ j+1 c(t0 , . . . , tj , t). The right hand side of (4)

is bounded, so c is Lipkā’2 by induction on k.

(1) ā’ (2) For a diļ¬erentiable function f : R ā’ R and t0 < Ā· Ā· Ā· < tj there exist si

with ti < si < ti+1 such that

Ī“ j f (t0 , . . . , tj ) = Ī“ jā’1 f (s0 , . . . , sjā’1 ).

(5)

12.4

12.5 12. Diļ¬erentiability of ļ¬nite order 121

Let p be the interpolation polynomial

j iā’1

(t ā’ tl ) Ī“ j f (t0 , . . . , tj ).

1

(6) p(t) := i!

i=0 l=0

Since f and p agree on all tj , by Rolleā™s theorem the ļ¬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 coeļ¬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(kā’2) , c(kā’3) , . . . , c shows that Ī“ k c is bounded on

bounded sets, and (2) follows.

(2) ā’ (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 ā’ R is Lipk for k ā„ 1 and I is a compact interval then there

exists M such that for all t, v ā I we have

k

ri f (t, iv) ā¤ M |v|k+1 .

ā‚

ā‚s |0 f (t, s).v ā’

i=0

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

/

ā‚

suļ¬ces to show that the derivative ā‚s |0 f (t, s) is locally bounded. If it is unbounded

near some point xā , there are xn with |xn ā’xā | ā¤ 21 such that ā‚s |0 f (xn , s) ā„ n.2n .

ā‚

n

We apply the general curve lemma (12.2) to the curves cn : R ā’ R2 given by

cn (t) := (xn , 2tn ) and to sn := 21 in order to obtain a smooth curve c : R ā’ R2

n

and scalars tn ā’ 0 with c(t + tn ) = cn (t) for |t| ā¤ sn . Then (f ā—¦ c) (tn ) =

1

1ā‚

2n ā‚s |0 f (xn , s) ā„ n, which contradicts that f is Lip .

Now we treat the case 0 ā I. If the assertion does not hold there are xn , vn ā

k

ā‚

I, such that ā‚s |0 f (xn , s).vn ā’ i=0 ri f (xn , ivn ) ā„ n.2n(k+1) |vn |k+1 . We may

assume xn ā’ xā , and by the case 0 ā I we may assume that vn ā’ 0, even with

/

|xn ā’ xā | ā¤ 21 and |vn | ā¤ 21 . We apply the general curve lemma (12.2) to the

n n

curves cn : R ā’ R2 given by cn (t) := (xn , 2tn ) and to sn := 21 to obtain a smooth

n

2

curve c : R ā’ R and scalars tn ā’ 0 with c(t + tn ) = cn (t) for |t| ā¤ sn . Then we

have

k

(f ā—¦ c) (tn )2n vn ā’ ri (f ā—¦ c)(tn + i2n vn ) =

i=0

k

n

ri (f ā—¦ cn )(i2n vn )

= (f ā—¦ cn ) (0)2 vn ā’

i=0

k

n

ri f (xn , ivn ) ā„ n(2n |vn |)k+1 .

1ā‚

2n ā‚s |0 f (xn , s)2 vn ā’

=

i=0

12.5

122 Chapter III. Partitions of unity 12.6

This contradicts the next claim for g = f ā—¦ c.

Claim. If g : R ā’ R is Lipk for k ā„ 1 and I is a compact interval then there is

k

M > 0 such that for t, v ā I we have g (t).v ā’ i=0 ri g(t + iv) ā¤ M |v|k+1 .

k

Consider gt (v) := g (t).v ā’ 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)| ā¤ M |v| for all t, v ā I, which we may integrate in

|v|k+1

turn to obtain |gt (v)| ā¤ M (k+1)! .

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

ā‚

ā‚s |0 f (t, s)

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

ā‚

k+1

equidistant diļ¬erence quotient Ī“eq g is not locally bounded at some point which we

may assume to be 0. Then there are xn and vn with |xn | ā¤ 1/4n and 0 < vn < 1/4n

such that

|Ī“eq g(xn ; vn )| > n.2n(k+2) .

k+1

(1)

We apply the general curve lemma (12.2) to the curves cn : R ā’ R2 given by

cn (t) := en ( 2tn + xn ) := ( 2tn + xn ā’ vn , 2tn ) and to sn := k+2 in order to obtain a

2n

2

smooth curve c : R ā’ R and scalars tn ā’ 0 with c(t + tn ) = cn (t) for 0 ā¤ t ā¤ 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, ļ¬nally put

f2 := f1 ā’f0 . Then f0 in Lipk+1 , so f0 ā—¦c is Lipk+1 , hence the equidistant diļ¬erence

quotient Ī“eq (f0 ā—¦ c)(xn ; 2n vn ) is bounded.

k+2

By lemma (12.5) there exists M > 0 such that |f2 (t, s)| ā¤ M |s|k+2 for all t, s ā

[ā’(k + 1), k + 1], so we get

|Ī“eq (f2 ā—¦ c)(xn ; 2n vn )| = |Ī“eq (f2 ā—¦ cn )(0; 2n vn )|

k+2 k+2

|Ī“ k+2 (f2 ā—¦ en )(xn ; vn )|

1

= 2n(k+2) eq

k+2

i(k+2)

|f2 ((i ā’ 1)vn + xn , ivn )|

(k+2)!

ā¤ 2n(k+2) |ivn |(k+2) j=i |i ā’ j|

i=1

k+2

i(k+2)

(k+2)!

ā¤ M .

2n(k+2) |i ā’ j|

j=i

ńņš. 18 |