n n

and 0 on E \ D. Then f ∈ C n and f (n) is bounded. Up to a¬ne transformations

this is the required bump function.

∞

15.4. Borel™s theorem. [Wells, 1973]. Suppose a Banach space E has Cb -

bump functions. Then every formal power series with coe¬cients in Ln (E; F )

sym

for another Banach space F is the Taylor-series of a smooth mapping E ’ F .

Moreover, if G is a second Banach space, and if for some open set U ⊆ G we are

given bk ∈ Cb (U, Lk (E, F )), then there is a smooth f ∈ C ∞ (E — U, F ) with

∞

sym

k

d (f ( , y))(0) = bk (y) for all y ∈ U and k ∈ N. In particular, smooth curves can

be lifted along the mapping C ∞ (E, F ) ’ k Lk (E; F ).

sym

∞ ∞

Proof. Let ρ ∈ Cb (E, R) be a Cb -bump function, which equals 1 locally at 0.

We shall use the notation bk (x, y) := bk (y)(xk ). De¬ne

1

fk (x, y) := bk (x, y) ρ(x)

k!

and

1

fk (tk · x, y)

f (x, y) :=

tk

k≥0 k

with appropriately chosen tk > 0. Then fk ∈ C ∞ (E — U, F ) and fk has carrier

inside of carr(ρ) — U , i.e. inside {x : x < 1} — U . For the derivatives of bk we

have

ji

‚1 ‚2 bk (x, y)(ξ, ·) = k (k ’ 1) . . . (k ’ j) (di bk (y)(·))(xk’j , ξ j ).

Hence, for x ¤ 1 this derivative is bounded by

(k)j sup di bk (y) L(F,Lk (E;G)) ,

sym

y∈U

where (k)j := k(k ’ 1) . . . (k ’ j). Using the product rule we see that for j ≥ k the

ji

derivative ‚1 ‚2 fk of fk is globally bounded by

j

sup{ ρ(j’l) (x) : x ∈ E} (k)l sup di bk (y) < ∞.

l y∈U

l¤k

15.4

15.4 15. Functions with globally bounded derivatives 161

The partial derivatives of f would be

tj j i

ji k

‚1 ‚2 fk (x, y) = ‚ ‚ fk (tk x, y).

tk 1 2

k

k

We now choose the tk ≥ 1 such that these series converge uniformly. This is the

case if,

1 ji

sup{ ‚1 ‚2 fk (x, y) : x ∈ E, y ∈ U } ¤

tk’j

k

1 1

ji

¤ sup{ ‚1 ‚2 fk (x, y) : x ∈ E, y ∈ U } ¤ ,

k’(j+i) 2k’(j+i)

tk

and thus if

1

ji

tk ≥ 2. sup{ ‚1 ‚2 fk (x, y) : x ∈ E, y ∈ U, j + i < k}.

k’(j+i)

j j

1

Since we have ‚1 fk (0, y)(ξ) = k! (k)j bk (y)(0k’j , ξ j ) ρ(0) = δk bk (y), we conclude

j

the desired result ‚1 f (0, y) = bk (y).

Remarks on Borel™s theorem.

(1) [Colombeau, 1979]. Let E be a strict inductive limit of a non-trivial se-

quence of Fr´chet spaces En . Then Borel™s theorem is wrong for f : R ’ E.

e

The idea is to choose bn = f (n) (0) ∈ En+1 \ En and to use that locally every

smooth curve has to have values in some En .

(2) [Colombeau, 1979]. Let E = RN . Then Borel™s theorem is wrong for f :

E ’ R. In fact, let bn : E — . . . — E ’ R be given by bn := prn — · · · — prn .

Assume f ∈ C ∞ (E, R) exists with f (n) (0) = bn . Let fn be the restriction of

(n)

f to the n-th factor R in E. Then fn ∈ C ∞ (R, R) and fn (0) = 1. Since f :

Rn ’ (Rn ) = R(N) is continuous, the image of B := {x : |xn | ¤ 1 for all n}

in R(N) is bounded, hence contained in some RN ’1 . Since fN is not constant

on the interval (’1, 1) there exists some |tN | < 1 with fN (tN ) = 0. For

xN := (0, . . . , 0, tN , 0, . . . ) we obtain

f (xN )(y) = fN (tN )(yN ) + ai yi ,

i=N

a contradiction to f (xn ) ∈ RN ’1 .

(3) [Colombeau, 1979] showed that Borel™s theorem is true for mappings f :

E ’ F , where E has a basis of Hilbert-seminorms and for any countable

∞

family of 0-neighborhoods Un there exist tn > 0 such that n=1 tn Un is a

0-neighborhood.

(4) If theorem (15.4) would be true for G = k Lk (E; F ) and bk = prk , then

sym

∞

the quotient mapping C (E, F ) ’ G = k Lk (E; F ) would admit a

sym

smooth and hence a linear section. This is well know to be wrong even for

E = F = R, see (21.5).

15.4

162 Chapter III. Partitions of unity 15.6

∞

15.5. Proposition. Hilbert spaces have Cb -bump functions. [Wells, 1973]

If the norm is given by the n-th root of a homogeneous polynomial b of even degree

∞

n, then x ’ ρ(b(xn )) is a Cb -bump function, where ρ : R ’ R is smooth with

ρ(t) = 1 for t ¤ 0 and ρ(t) = 0 for t ≥ 1.

Proof. As before in the proof of (15.4) we see that the j-th derivative of x ’ b(xn )

is bounded by (n)j on the closed unit ball. Hence, by the chain-rule and the

global boundedness of all derivatives of ρ separately, the composite has bounded

derivatives on the unit ball, and since it is zero outside, even everywhere. Obviously,

ρ(b(0)) = ρ(0) = 1.

In [Bonic, Frampton, 1966] it is shown that Lp is Lipn

global -smooth for all n if p is

[p’1]

an even integer and is Lipglobal -smooth otherwise. This follows from the fact (see

loc. cit., p. 140) that d(p+1) x p = 0 for even integers p and

p!

dk x + h p

’ dk x p

h||p’k

¤

k!

otherwise, cf. (13.13).

15.6. Estimates for the remainder in the Taylor-expansion. The Taylor

formula of order k of a C k+1 -function is given by

k 1

(1 ’ t)k (k+1)

1 (j)

f (x)(hj ) + (x + th)(hk+1 ) dt,

f (x + h) = f

j! k!

0

j=0

1

which can easily be seen by repeated partial integration of f (x + th)(h) dt =

0

f (x + h) ’ f (x).

2

For a CB function we have

1

1

(1 ’ t) f (2) (x + th) 2

h 2.

|f (x + h) ’ f (x) ’ f (x)(h)| ¤ dt ¤ B

h

2!

0

If we take the Taylor formula of f up to order 0 instead, we obtain

1

f (x + h) = f (x) + f (x + th)(h) dt

0

1

and usage of f (x)(h) = f (x)(h) dt gives

0

1

f (x + th) ’ f (x) 1

2

h 2,

|f (x + h) ’ f (x) ’ f (x)(h)| ¤ dt ¤ B

h

th 2!

0

so it is in fact enough to assume f ∈ C 1 with f satisfying a Lipschitz-condition

with constant B.

15.6

15.7 15. Functions with globally bounded derivatives 163

3

For a CB function we have

1

|f (x + h) ’ f (x) ’ f (x)(h) ’ f (x)(h2 )| ¤

2

1

(1 ’ t)2 (3) 1

3

h 3.

¤ dt ¤ B

f (x + th) h

2! 3!

0

If we take the Taylor formula of f up to order 1 instead, we obtain

1

(1 ’ t) f (x + th)(h2 ) dt,

f (x + h) = f (x) + f (x)(h) +

0

1

and using 1 f (x)(h2 ) = ’ t) f (x)(h2 ) dt we get

(1

2 0

1

|f (x + h) ’ f (x) ’ f (x)(h) ’ f (x)(h2 )| ¤

2

1

f (x + th) ’ f (x) 1

3

h 3.

¤ (1 ’ t)t dt ¤ B

h

th 3!

0

Hence, it is in fact enough to assume f ∈ C 2 with f satisfying a Lipschitz-condition

with constant B.

1

k

Let f ∈ CB be ¬‚at of order k at 0. Applying f (h) ’ f (0) = 0 f (th)(h) dt ¤

sup{ f (th) : t ∈ [0, 1]} h to f (j) ( )(h1 , . . . , hj ) gives using f (k) (x) ¤ B

inductively

f (k’1) (x) ¤ B · x

1 1

B

(k’2) (k’1) 2 2

(x) ¤ (tx)(x, . . . ) dt ¤ B

f f t dt x = x

2

0 0

.

.

.

B

f (j) (x) ¤ k’j

x .

(k ’ j)!

15.7. Lemma. Lip1 n N

global -functions on R . [Wells, 1973]. Let n := 2 and

E = Rn with the ∞-norm. Suppose f ∈ Lip1 (E, R) with f (0) = 0 and f (x) ≥ 1

M

for x ≥ 1. Then M ≥ 2N .

The idea behind the proof is to construct recursively a sequence of points xk :=

k’1

j<k σj hj of norm N starting at x0 = 0, such that the increment along the

segment is as small as possible. In order to evaluate this increment one uses the

Taylor-formula and chooses the direction hk such that the derivative at xk vanishes.

Proof. Let A be the set of all edges of a hyper-cube, i.e.

A := {x : xi = ±1 for all i except one i0 and |xi0 | ¤ 1}.

15.7

164 Chapter III. Partitions of unity 15.9

Then A is symmetric. Let x ∈ E be arbitrary. We want to ¬nd h ∈ A with

f (x)(h) = 0. By permuting the coordinates we may assume that i ’ |f (x)(ei )|

is monotone decreasing. For 2 ¤ i ¤ n we choose recursively hi ∈ {±1} such that

i i j 1

j=2 hj f (x)(ej ) is an alternating sum. Then | j=2 f (x)(e )hj | ¤ |f (x)(e )|.

Finally, we choose h1 ¤ 1 such that f (x)(h) = 0.

1

Now we choose inductively hi ∈ N A and σi ∈ {±1} such that f (xi )(hi ) = 0 for

N ’i i

x := j<i σj hj and xi has at least 2 coordinates equal to N . For the last

statement we have that xi+1 = xi + σi hi and at least 2N ’i coordinates of xi are

i 1

N . Among those coordinates all but at most 1 of the hi are ± N . Now let σi be

the sign which occurs more often and hence at least 2N ’i /2 times. Then those

2N ’(i+1) many coordinates of xi+1 are i+1 .

N

i

for i ¤ N , since at least one coordinate has this value. Furthermore

Thus xi = N

we have

N ’1

1 = |f (xN ) ’ f (x0 )| ¤ |f (xk+1 ) ’ f (xk ) ’ f (xk )(hk )|

k=0

N

M M1

2

¤ ¤N

hk ,

2 N2

2

k=1

hence M ≥ 2N .

15.8. Corollary. c0 is not Lip1

global -regular. [Wells, 1973]. The space c0 is not

Lip1

global -smooth.

Proof. Suppose there exists an f ∈ Lip1 global with f (0) = 1 and f (x) = 0 for all

x ≥ 1. Then the previous lemma applied to 1 ’ f restricted to ¬nite dimensional

subspaces shows that the Lipschitz constant M of the derivative has to be greater

or equal to N for all N , a contradiction.

This shows even that there exist no di¬erentiable bump functions on c0 (A) which

have uniformly continuous derivative. Since otherwise there would exist an N ∈ N

such that

1

1

f (x + h) ’ f (x) ’ f (x)h ¤ f (x + t h) ’ f (x) h dt ¤ h,

2

0

for h ¤ N . Hence, the estimation in the proof of (15.7) would give 1 ¤ N 1 N = 1 ,

1 1

2 2

a contradiction.

15.9. Positive results on Lip1

global -functions. [Wells, 1973].

(1) Every closed subset of a Hilbert space is the zero-set of a Lip1 global -function.

(2) For every two closed subsets of a Hilbert space which have distance d > 0

there exists a Lip1 2 -function which has value 0 on one set and 1 on the

4/d

other.

(3) Whitney™s extension theorem is true for Lip1 global -functions on closed subsets

of Hilbert spaces.

15.9

165

16. Smooth Partitions of Unity and Smooth Normality

16.1. De¬nitions. We say that a Hausdor¬ space X is smoothly normal with

respect to a subalgebra S ⊆ C(X, R) or S-normal, if for two disjoint closed subsets

A0 and A1 of X there exists a function f : X ’ R in S with f |Ai = i for i = 0, 1.

If an algebra S is speci¬ed, then by a smooth function we will mean an element of

S. Otherwise it is a C ∞ -function.

A S-partition of unity on a space X is a set F of smooth functions f : X ’ R

which satisfy the following conditions:

(1) For all f ∈ F and x ∈ X one has f (x) ≥ 0.

(2) The set {carr(f ) : f ∈ F} of all carriers is a locally ¬nite covering of X.

(3) The sum f ∈F f (x) equals 1 for all x ∈ X.

Since a family of open sets is locally ¬nite if and only if the family of the closures

is locally ¬nite, the foregoing condition (2) is equivalent to:

(2™) The set {supp(f ) : f ∈ F} of all supports is a locally ¬nite covering of X.

The partition of unity is called subordinated to an open covering U of X, if for

every f ∈ F there exists an U ∈ U with carr(f ) ⊆ U .

We say that X is smoothly paracompact with respect to S or S-paracompact if every

open cover U admits a S-partition F of unity subordinated to it. This implies that

X is S-normal.

The partition of unity can then even be chosen in such a way that for every f ∈ F

there exists a U ∈ U with supp(f ) ⊆ U . This is seen as follows. Since the family of

carriers is a locally ¬nite open re¬nement of U, the topology of X is paracompact.

˜ ˜

So we may ¬nd a ¬ner open cover {U : U ∈ U} such that the closure of U is

contained in U for all U ∈ U, see [Bourbaki, 1966, IX.4.3]. The partition of unity

subordinated to this ¬ner cover has the support property for the original one.

Lemma. Let S be an algebra which is closed under sums of locally ¬nite families

of functions. If F is an S-partition of unity subordinated to an open covering U,

then we may ¬nd an S-partition of unity (fU )U ∈U with carr(fU ) ⊆ U .

Proof. For every f ∈ F we choose a Uf ∈ U with carr(f ) ∈ Uf . For U ∈ U put

FU := {f : Uf = U } and let fU := f ∈FU f ∈ S.

16.2. Proposition. Characterization of smooth normality. Let X be a

Hausdor¬ space with S ⊆ C(X, R) as in (14.1) Consider the following statements:

(1) X is S-normal;

(2) For any two closed disjoint subsets Ai ⊆ X there is a function f ∈ S with

f |A0 = 0 and 0 ∈ f (A1 );

/

(3) Every locally ¬nite open covering admits S-partitions of unity subordinated

to it.

(4) For any two disjoint zero-sets A0 and A1 of continuous functions there exists

a function g ∈ S with g|Aj = j for j = 0, 1 and g(X) ⊆ [0, 1];