16.2

166 Chapter III. Partitions of unity 16.2

(5) For any continuous function f : X ’ R there exists a function g ∈ S with

f ’1 (0) ⊆ g ’1 (0) ⊆ f ’1 (R \ {1}).

(6) The set S is dense in the algebra of continuous functions with respect to the

topology of uniform convergence;

(7) The set of all bounded functions in S is dense in the algebra of continuous

bounded functions on X with respect to the supremum norm;

ˇ

(8) The bounded functions in S separate points in the Stone-Cech-compacti¬ca-

tion βX of X.

The statements (1)-(3) are equivalent, and (4)-(8) are equivalent as well. If X is

metrizable all statements are equivalent.

If every open set is the carrier set of a smooth function then X is S-normal. If X

is S-normal, then it is S-regular.

A space is S-paracompact if and only if it is paracompact and S-normal.

Proof. (2) ’ (1). By assumption, there is a smooth function f0 with f0 |A1 = 0

and 0 ∈ f0 (A0 ), and again by assumption, there is a smooth function f1 with

/

f1 |A0 = 0 and 0 ∈ f1 ({x : f0 (x) = 0}). The function f = f0f1 1 has the required

/ +f

properties.

(1) ’ (2) is obvious.

(3) ’ (1) Let A0 and A1 be two disjoint closed subset. Then U := {X \ A1 , X \ A0 }

admits a S-partition of unity F subordinated to it, and

{f ∈ F : carr f ⊆ X \ A0 }

is the required bump function.

(1) ’ (3) Let U be a locally ¬nite covering of X. The space X is S-normal, so

its topology is also normal, and therefore for every U ∈ U there exists an open set

VU such that VU ⊆ U and {VU : U ∈ U} is still an open cover. By assumption,

there exist smooth functions gU ∈ S such that VU ⊆ carr(gU ) ⊆ U , cf. (16.1). The

function g := U gU is well de¬ned, positive, and smooth since U is locally ¬nite,

and {fU := gU /g : U ∈ U} is the required partition of unity.

’1

(5) ’ (4) Let Aj := fj (aj ) for j = 0, 1. By replacing fj by (fj ’ aj )2 we may

’1

assume that fj ≥ 0 and Aj = fj (0). Then (f1 + f2 )(x) > 0 for all x ∈ X, since

A1 ©A2 = …. Thus, f := f0f0 1 is a continuous function in C(X, [0, 1]) with f |Aj = j

+f

for j = 0, 1.

Now we reason as in ((2) ’ (1)): By (4) there exists a g0 ∈ S with A0 ⊆ f ’1 (0) ⊆

’1

g0 (0) ⊆ f ’1 (R \ {1}) = X \ f ’1 (1) ⊆ X \ A1 . By replacing g0 by g0 we may

2

assume that g0 ≥ 0.

’1

Applying the same argument to the zero-sets A1 and g0 (0) we obtain a g1 ∈ S

’1 ’1

with A1 ⊆ g1 (0) ⊆ X \ g0 (0). Thus, (g0 + g1 )(x) > 0, and hence g := g0g0 1 ∈ S

+g

satis¬es g|Aj = j for j = 0, 1 and g(X) ⊆ [0, 1].

(4) ’ (6) Let f be continuous. Without loss of generality we may assume f ≥ 0

(decompose f = f+ ’ f’ ). Let µ > 0. Then choose gk ∈ S with image in [0, 1], and

gk (x) = 0 for all x with f (x) ¤ k µ, and gk (x) = 1 for all x with f (x) ≥ (k + 1) µ.

16.2

16.2 16. Smooth partitions of unity and smooth normality 167

Let k be the largest integer less or equal to f (x) . Then gj (x) = 1 for all j < k, and

µ

gj (x) = 0 for all j > k. Hence, the sum g := µ k∈N gk ∈ S is locally ¬nite, and

|f (x) ’ g(x)| < 2 µ.

(6) ’ (7) This is obvious, since for any given bounded continuous f and for any

µ > 0, by (6) there exists g ∈ S with |f (x) ’ g(x)| < µ for all x ∈ X, hence

f ’ g ∞ ¤ µ and g ∞ ¤ f ∞ + f ’ g ∞ < ∞.

(7) ” (8) This follows from the Stone-Weierstraß theorem, since obviously the

bounded functions in S form a subalgebra in Cb (X) = C(βX). Hence, it is dense

if and only if it separates points in the compact space βX.

(7) ’ (4) By cutting o¬ f at 0 and at 1, we may assume that f is bounded. By

(7) there exists a bounded g0 ∈ S with f ’ g0 ∞ < 1 . Let h ∈ C ∞ (R, R) be

2

1

such that h(t) = 0 ” t ¤ 2 . Then g := h —¦ g0 ∈ S, and f (x) = 0 ’ g0 (x) ¤

|g0 (x)| ¤ |f (x)| + f ’ g0 ∞ ¤ 1 ’ g(x) = h(g0 (x)) = 0 and also f (x) = 1 ’

2

g0 (x) ≥ f (x) ’ f ’ g0 ∞ > 1 ’ 2 = 1 ’ g(x) = 0.

1

2

If X is metrizable and A ⊆ X is closed, then dist( , A) : x ’ sup{dist(x, a) : a ∈

A} is a continuous function with f ’1 (0) = A. Thus, (1) and (4) are equivalent.

Let every open subset be the carrier of a smooth mapping, and let A0 and A1 be

closed disjoint subsets of X. By assumption, there is a smooth function f with

carr(f ) = X \ A0 .

Obviously, every S-normal space is S-regular. Take as second closed set in (2) a

single point. If we take instead the other closed set as single point, then we have

what has been called small zero-sets in (19.8).

That a space is S-paracompact if and only if it is paracompact and S-normal can

be shown as in the proof that a paracompact space admits continuous partitions of

unity, see [Engelking, 1989, 5.1.9].

In [Kriegl, Michor, Schachermayer, 1989] it is remarked that in an uncountable

product of real lines there are open subsets, which are not carrier sets of continuous

functions.

Corollary. Denseness of smooth functions. Let X be S-paracompact, let F

be a convenient vector space, and let U ⊆ X — F be open such that for all x ∈ X

the set ι’1 (U ) ⊆ F is convex and non-empty, where ιx : F ’ X — F is given by

x

y ’ (x, y). Then there exists an f ∈ S whose graph is contained in U .

Under the following assumption this result is due to [Bonic, Frampton, 1966]: For

U := {(x, y) : p(y ’ g(x)) < µ(x)}, where g : X ’ F , µ : X ’ R+ are continuous

and p is a continuous seminorm on F .

Proof. For every x ∈ X let yx be chosen such that (x, yx ) ∈ U . Next choose open

neighborhoods Ux of x such that Ux — {yx } ⊆ U . Since X is S-paracompact there

exists a S-partition of unity F subordinated to the covering {Ux : x ∈ X}. In

particular, for every • ∈ F there exists an x• ∈ X with carr • ⊆ Ux• . Now de¬ne

16.2

168 Chapter III. Partitions of unity 16.4

yx• •. Then f ∈ S and for every x ∈ X we have

f := •∈F

yx• •(x) ∈ ι’1 (U ),

f (x) = yx• •(x) = x

x∈carr •

•∈F

since ι’1 (U ) is convex, contains yx• for x ∈ carr(•) ⊆ Ux• , and •(x) ≥ 0 with

x

1 = • •(x) = x∈carr • •(x).

16.3. Lemma. Lip2 -functions on Rn . [Wells, 1973]. Let B ∈ N and A := {x ∈

RN : xi ¤ 0 for all i and x ¤ 1}. Suppose that f ∈ CB (RN , R) with f |A = 0 and

3

f (x) ≥ 1 for all x with dist(x, A) ≥ 1. Then N < B 2 + 36 B 4 .

Proof. Suppose N ≥ B 2 +36 B 4 . We may assume that f is symmetric by replacing

f with x ’ N ! σ f (σ — x), where σ runs through all permutations, and σ — just

1

permutes the coordinates. Consider points xj ∈ RN for j = 0, . . . , B 2 of the form

xj = 1 1 1 1

B , . . . , B , ’ B , . . . , ’ B , 0, . . . , 0 .

>36 B 4

B 2 ’j

j

2

Then xj = 1, x0 ∈ A and d(xB , A) ≥ 1. Since f is symmetric and y j :=

1 j j+1

) has vanishing j, B 2 + 1, . . . , N coordinates, we have for the partial

2 (x + x

derivatives ‚j f (y j ) = ‚k f (y j ) for k = B 2 + 1, . . . , N . Thus

N

f (y j ) 2

f (y j ) 2

1 1

2

|‚j f (y j )|2 = j 2

|‚k f (y )| ¤ ¤

= ,

N ’ B2 36 B 4 36 B 4 36 B 2

k=B 2 +1

since from f |A = 0 we conclude that f (0) = f (0) = f (0) = f (0) and hence

f (j) (h) ¤ B h 3’j for j ¤ 3, see (15.6).

From |f (x + h) ’ f (x) ’ f (x)(h) ’ 1 f (x)(h2 )| ¤ B 3! h

1 3

we conclude that

2

|f (x + h) ’ f (x ’ h)| ¤ |f (x + h) ’ f (x) ’ f (x)(h) ’ 1 f (x)(h2 )|

2

+ |f (x ’ h) ’ f (x) + f (x)(h) ’ 1 f (x)(h2 )|

2

+ 2|f (x)(h)|

3

2

¤ Bh + 2|f (x)(h)|.

3!

1

If we apply this to x = y j and h = B ej , where ej denotes the j-th unit vector, then

we obtain

2 1 1 2

|f (xj+1 ) ’ f (xj )| ¤ B 3 + 2|‚j f (y j )| ¤ .

3B 2

3! B B

2 2

2

Summing up yields 1 ¤ |f (xB )| = |f (xB ) ’ f (x0 )| ¤ < 1, a contradiction.

3

16.4. Corollary. 2 is not Lip2 -normal. [Wells, 1973]. Let A0 := {x ∈ 2 :

glob

xj ¤ 0 for all j and x ¤ 1} and A1 := {x ∈ 2 : d(x, A) ≥ 1} and f ∈ C 3 ( 2 , R)

with f |Aj = j for j = 0, 1. Then f (3) is not bounded.

Proof. By the preceding lemma a bound B of f (3) must satisfy for f restricted to

RN , that N < B 2 + 36B 4 . This is not for all N possible.

16.4

16.6 16. Smooth partitions of unity and smooth normality 169

16.5. Corollary. Whitney™s extension theorem is false on 2 . [Wells, 1973].

Let E := R — 2 ∼ 2 and π : E ’ R be the projection onto the ¬rst factor.

=

For subsets A ⊆ 2 consider the cone CA := {(t, ta) : t ≥ 0, a ∈ A} ⊆ E. Let

A := C(A0 ∪ A1 ) with A0 and A1 as in (16.4). Let a jet (f j ) on A be de¬ned by

f j = 0 on the cone CA1 and f j (x)(v 1 , . . . , v j ) = h(j) (π(x))(π(v 1 ), . . . , π(v j )) for

all x in the cone of CA0 , where h ∈ C ∞ (R, R) is in¬nite ¬‚at at 0 but with h(t) = 0

for all t = 0. This jet has no C 3 -prolongation to E.

Proof. Suppose that such a prolongation f exists. Then f (3) would be bounded

1

locally around 0, hence fa (x) := 1 ’ h(a) f (a, ax) would be a CB function on 2 for

3

small a, which is 1 on A1 and vanishes on A0 . This is a contradiction to (16.4).

So it remains to show that the following condition of Whitney (22.2) is satis¬ed:

k’j

1 j+i

f j (y) ’ f (x)(y ’ x)j = o( x ’ y k’j

x, y ’ a.

) for A

i!

i=0

j j

Let f1 := 0 and f0 (x) := h(j) (π(x)) —¦ (π — . . . — π). Then both are smooth on R • 2 ,

and thus Whitney™s condition is satis¬ed on each cone separately. It remains to

show this when x is in one cone and y in the other and both tend to 0. Thus,

we have to replace f at some places by f1 and at others by f0 . Since h is in¬nite

j

¬‚at at 0 we have f0 (z) = o( z n ) for every n. Furthermore for xi ∈ CAi for

i = 0, 1 we have that x1 ’ x0 ≥ sin(arctan 2 ’ arctan 1) max{ x0 , x1 }. Thus,

j j

we may replace f0 (y) by f1 (y) and vice versa. So the condition is reduced to the

case, where y and z are in the same cone CAi .

16.6. Lemma. Smoothly regular strict inductive limits. Let E be the strict

inductive limit of a sequence of C ∞ -normal convenient vector spaces En such that

En ’ En+1 is closed and has the extension property for smooth functions. Then

E is C ∞ -regular.

Proof. Let U be open in E and 0 ∈ U . Then Un := U © En is open in En . We

choose inductively a sequence of functions fn ∈ C ∞ (En , R) such that supp(fn ) ⊆

Un , fn (0) = 1, and fn |En’1 = fn’1 . If fn is already constructed, we may choose by

C ∞ -normality a smooth g : En+1 ’ R with supp(g) ⊆ Un+1 and g|supp(fn ) = 1. By

assumption, fn extends to a function fn ∈ C ∞ (En+1 , R). The function fn+1 := g·fn

has the required properties.

Now we de¬ne f : E ’ R by f |En := fn for all n. It is smooth since any

c ∈ C ∞ (R, E) locally factors to a smooth curve into some En by (1.8) since a

strict inductive limit is regular by (52.8), so f —¦ c is smooth. Finally, f (0) = 1,

and if f (x) = 0 then x ∈ En for some n, and we have fn (x) = f (x) = 0, thus

x ∈ Un ⊆ U .

For counter-examples for the extension property see (21.7) and (21.11). However,

for complemented subspaces the extension property obviously holds.

16.6

170 Chapter III. Partitions of unity 16.9

16.7. Proposition. Cc is C ∞ -regular. The space Cc (Rm , R) of smooth func-

∞ ∞

tions on Rm with compact support satis¬es the assumptions of (16.6).

∞

Let Kn := {x ∈ Rm : |x| ¤ n}. Then Cc (Rm , R) is the strict inductive limit of the

∞

closed subspaces CKn (Rm , R) := {f : supp(f ) ⊆ Kn }, which carry the topology of

uniform convergence in all partial derivatives separately. They are nuclear Fr´chet

e

spaces and hence separable, see (52.27). Thus they are C ∞ -normal by (16.10)

below.

In order to show the extension property for smooth functions we proof more general

that for certain sets A the subspace {f ∈ C ∞ (E, R) : f |A = 0} is a complemented

subspace of C ∞ (E, R). The ¬rst result in this direction is:

16.8. Lemma. [Seeley, 1964] The subspace {f ∈ C ∞ (R, R) : f (t) = 0 for t ¤ 0}

of the Fr´chet space C ∞ (R, R) is a direct summand.

e

Proof. We claim that the following map is a bounded linear mapping being left

inverse to the inclusion: s(g)(t) := g(t) ’ k∈N ak h(’t2k )g(’t2k ) for t > 0 and

s(g)(t) = 0 for t ¤ 0. Where h : R ’ R is a smooth function with compact support

satisfying h(t) = 1 for t ∈ [’1, 1] and (ak ) is a solution of the in¬nite system of

linear equations k∈N ak (’2k )n = 1 (n ∈ N) (the series is assumed to converge

absolutely). The existence of such a solution is shown in [Seeley, 1964] by taking

the limit of solutions of the ¬nite subsystems. Let us ¬rst show that s(g) is smooth.

For t > 0 the series is locally around t ¬nite, since ’t2k lies outside the support of

h for k su¬ciently large. Its derivative (sg)(n) (t) is

n

(n) kn

h(j) (’t2k )g (n’j) (’t2k )

(t) ’

g ak (’2 )

j=0

k∈N

and this converges for t ’ 0 towards g (n) (0)’ k∈N ak (’2k )n g (n) (0) = 0. Thus s(g)

is in¬nitely ¬‚at at 0 and hence smooth on R. It remains to show that g ’ s(g) is a

bounded linear mapping. By the uniform boundedness principle (5.26) it is enough

to show that g ’ (sg)(t) is bounded. For t ¤ 0 this map is 0 and hence bounded.

For t > 0 it is a ¬nite linear combination of evaluations and thus bounded.

Now the general result:

16.9. Proposition. Let E be a convenient vector space, and let p be a smooth

seminorm on E. Let A := {x : p(x) ≥ 1}. Then the closed subspace {f : f |A = 0}

in C ∞ (E, R) is complemented.

Proof. Let g ∈ C ∞ (E, R) be a smooth reparameterization of p with support in

E \ A equal to 1 near p’1 (0). By lemma (16.8), there is a bounded projection

P : C ∞ (R, R) ’ C(’∞,0] (R, R). The following mappings are smooth in turn by the

∞

16.9

16.10 16. Smooth partitions of unity and smooth normality 171

properties of the cartesian closed smooth calculus, see (3.12):

(x, t) ’ f (et , x) ∈ R

E—R

x) ∈ C ∞ (R, R)

x ’ f (e( )

E

∞

x ’ P (f (e( )

x)) ∈ C(’∞,0] (R, R)

E

(x, r) ’ P (f (e( )

E—R x))(r) ∈ R

x

’ P f (e( )x

x’ (ln(p(x))) ∈ R.

carr p , ln(p(x)) p(x) )

p(x)

So we get the desired bounded linear projection

¯

P : C ∞ (E, R) ’ {f ∈ C ∞ (E, R) : f |A = 0},

¯

(P (f ))(x) := g(x) f (x) + (1 ’ g(x)) P (f (e( )x

p(x) ))(ln(p(x))).

16.10. Theorem. Smoothly paracompact Lindel¨f. [Wells, 1973]. If X is

o

Lindel¨f and S-regular, then X is S-paracompact. In particular, all nuclear Fr´chet

o e

spaces and strict inductive limits of sequences of such spaces are C ∞ -paracompact.

Furthermore, nuclear Silva spaces, see (52.37), are C ∞ -paracompact.

The ¬rst part was proved by [Bonic, Frampton, 1966] under stronger assumptions.

The importance of the proof presented here lies in the fact that we need not assume

1

that S is local and that f ∈ S for f ∈ S. The only things used are that S is an

algebra and for each g ∈ S there exists an h : R ’ [0, 1] with h —¦ g ∈ S and h(t) = 0

for t ¤ 0 and h(t) = 1 for t ≥ 1. In particular, this applies to S = Lippglobal and X

a separable Banach space.

Proof. Let U be an open covering of X.

Claim. There exists a sequence of functions gn ∈ S(X, [0, 1]) such that {carr gn :

’1

n ∈ N} is a locally ¬nite family subordinated to U and {gn (1) : n ∈ N} is a

covering of X.

For every x ∈ X there exists a neighborhood U ∈ U (since U is a covering) and

hence an hx ∈ S(X, [0, 2]) with hx (x) = 2 and carr(hx ) ⊆ U (since X is S-regular).

Since X is Lindel¨f we ¬nd a sequence xn such that {x : hn (x) > 1 : n ∈ N} is

o

a covering of X (we denote hn := hxn ). Now choose an h ∈ C ∞ (R, [0, 1]) with

h(t) = 0 for t ¤ 0 and h(t) = 1 for t ≥ 1. Set

gn (x) := h(n (hn (x) ’ 1) + 1) h(n (1 ’ hj (x)) + 1).

j<n

Note that

1

0 for hn (x) ¤ 1 ’ n

h(n (hn (x) ’ 1) + 1) =

1 for hn (x) ≥ 1

1

0 for hj (x) ≥ 1 + n

h(n (1 ’ hj (x)) + 1) =

for hj (x) ¤ 1

1

16.10

172 Chapter III. Partitions of unity 16.10

Then gn ∈ S(X, [0, 1]) and carr gn ⊆ carr hn . Thus, the family {carr gn : n ∈ N} is

subordinated to U.

’1

The family {gn (1) : n ∈ N} covers X since for each x ∈ X there exists a minimal

n with hn (x) ≥ 1, and thus gn (x) = 1.

If we could divide in S, then fn := gn / j gj would be the required partition of

unity (and we do not need the last claim in this strong from).

Instead we proceed as follows. The family {carr gn : n ∈ N} is locally ¬nite: Let

1