<< . .

. 25
( : 97)



. . >>

Choose n so large that D := {x : x ’ 21 a < 21 } ⊆ U , and let g := f on A ∪ D
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];

<< . .

. 25
( : 97)



. . >>