1

Ux := {y : hn (y) > 1 + k }, which is a neighborhood of x. For m ≥ k and y ∈ Ux

1 1

we have that hn (y) > 1 + k ≥ 1 + m , hence the (n + 1)-st factor of gm vanishes at

y, i.e. {j : carr gj © Ux = …} ⊆ {1, . . . , m ’ 1}.

Now de¬ne fn := gn j<n (1 ’ gj ) ∈ S. Then carr fn ⊆ carr gn , hence {carr fn :

n ∈ N} is a locally ¬nite family subordinated to U. By induction, one shows that

j¤n fj = 1 ’ j¤n (1 ’ gj ). In fact j<n (1 ’

j¤n fj = fn + j<n fj = gn

gj ) + 1 ’ j<n (1 ’ gj ) = 1 + (gn ’ 1) j<n (1 ’ gj ). For every x ∈ U there exists

∞

an n with gn (x) = 1, hence fk (x) = 0 for k > n and j=0 fj (x) = j¤n fj (x) =

1 ’ j¤n (1 ’ gj (x)) = 1.

Let us consider a nuclear Silva space. By (52.37) its dual is a nuclear Fr´chet space.

e

By (4.11.2) on the strong dual of a nuclear Fr´chet space the c∞ -topology coincides

e

∞

with the locally convex one. Hence, it is C -regular since it is nuclear, so it has

a base of (smooth) Hilbert seminorms. A Silva space is an inductive limit of a

sequence of Banach spaces with compact connecting mappings (see (52.37)), and

we may assume that the Banach spaces are separable by replacing them by the

closures of the images of the connecting mappings, so the topology of the inductive

limit is Lindel¨f. Therefore, by the ¬rst assertion we conclude that the space is

o

C ∞ -paracompact.

In order to obtain the statement on nuclear Fr´chet spaces we note that these are

e

separable, see (52.27), and thus Lindel¨f. A strict inductive limit of a sequence of

o

nuclear Fr´chet spaces is C ∞ -regular by (16.6), and it is also Lindel¨f for its c∞ -

e o

topology, since this is the inductive limit of topological spaces (not locally convex

spaces).

Remark. In particular, every separable Hilbert space has Lip2 global -partitions of

unity, thus there is such a Lip2 2

\ A0

global -partition of functions • subordinated to

and 2 \ A1 , with A0 and A1 mentioned in (16.4). Hence, f := carr •©A0 =… • ∈ C 2

satis¬es f |Aj = j for j = 0, 1. However, f ∈ Lip2

/ global . The reason behind this is

that Lip2global is not a sheaf.

Open problem. Classically, one proves the existence of continuous partitions of

unity from the paracompactness of the space. So the question arises whether theorem

(16.10) can be strengthened to: If the initial topology with respect to S is paracom-

pact, do there exist S-partitions of unity? Or equivalently: Is every paracompact

S-regular space S-paracompact?

16.10

16.14 16. Smooth partitions of unity and smooth normality 173

16.11. Theorem. Smoothness of separable Banach spaces. Let E be a

separable Banach space. Then the following conditions are equivalent.

E has a C 1 -norm;

(1)

E has C 1 -bump functions, i.e., E is C 1 -regular;

(2)

The C 1 -functions separate closed sets, i.e., E is C 1 -normal;

(3)

E has C 1 -partitions of unity, i.e., E is C 1 -paracompact;

(4)

(5) E has no rough norm, i.e. E is Asplund;

(6) E is separable.

Proof. The implications (1) ’ (2) and (4) ’ (3) ’ (2) are obviously true. The

implication (2) ’ (4) is (16.10). (2) ’ (5) holds by (14.9). (5) ’ (6) follows from

(14.10) since E is separable. (6) ’ (1) is (13.22) together with (13.20).

A more general result is:

16.12. Result. [John, Zizler, 1976] Let E be a WCG Banach space. Then the

following statements are equivalent:

E is C 1 -normable;

(1)

E is C 1 -regular;

(2)

E is C 1 -paracompact;

(3)

(4) E has norm, whose dual norm is LUR;

E has shrinking Markuˇeviˇ basis, i.e. vectors xi ∈ E and x— ∈ E with

(5) sc i

xi (xj ) = δi,j and the span of the xi is dense in E and the span of x— is

—

i

dense in E .

16.13. Results.

(1) [Godefroy, Pelant, et. al., 1983] ( [Vanderwer¬, 1992]) Let E is WCG Ba-

nach space (or even WCD, see (53.8)). Then E is C 1 -regular.

(2) [Vanderwer¬, 1992] Let K be compact with K (ω1 ) = …. Then C(K) is C 1 -

paracompact. Compare with (13.18.2) and (13.17.5).

(3) [Godefroy, Troyanski, et. al., 1983] Let E be a subspace of a WCG Banach

space. If E is C k -regular then it is C k -paracompact. This will be proved in

(16.18).

(4) [MacLaughlin, 1992] Let E be a WCG Banach space. If E is C k -regular

then it is C k -paracompact.

16.14. Lemma. Smooth functions on c0 (“). [Toru´czyk, 1973]. The norm-

n

topology of c0 (“) has a basis which is a countable union of locally ¬nite families of

carriers of smooth functions, each of which depends locally only on ¬nitely many

coordinates.

Proof. The open balls Br := {x : x ∞ < r} are carriers of such functions: In

fact, similarly to (13.16) we choose a h ∈ C ∞ (R, R) with h = 1 locally around 0

and carr h = (’1, 1), and de¬ne f (x) := γ∈“ h(xγ ). Let

Un,r,q = {Br + q1 eγ1 + · · · + qn eγn : {γ1 , . . . , γn } ⊆ “}

16.14

174 Chapter III. Partitions of unity 16.15

where n ∈ N, r ∈ Q, q ∈ Qn with |qi | > 2r for 1 ¤ i ¤ n. This is the required

countable family.

Un,r,q is a basis for the topology.

Claim. The union n,r,q

µ

Let x ∈ c0 (“) and µ > 0. Choose 0 < r < 2 such that r = |xγ | for all γ (note that

|xγ | ≥ µ/4 only for ¬nitely many γ). Let {γ1 , . . . , γn } := {γ : |xγ | > r}. For qi with

|qi ’ xγi | < r and |qi | > 2r we have

x’ qi eγi ∈ Br ,

i

and hence

n

x ∈ Br + qi eγi ⊆ x + B2r ⊆ {y : y ’ x ¤ µ}.

∞

i=1

Claim. Each family Un,r,q is locally ¬nite.

r

For given x ∈ c0 (“), let {γ1 , . . . , γm } := {γ : |xγ | > 2 } and assume there exists a

n

y ∈ (x + B r ) © (Br + i=1 qi eβi ) = …. For y ∈ x + B r we have |ya | < r for all γ ∈ /

2 2

n

{γ1 , . . . , γm } and for y ∈ Br + i=1 qi eβi we have |yγ | > r for all γ ∈ {β1 , . . . , βn }.

Hence, {β1 , . . . , βn } ⊆ {γ1 , . . . , γm } and Un,r,q is locally ¬nite.

16.15. Theorem, Smoothly paracompact metrizable spaces. [Toru´czyk, n

1973]. Let X be a metrizable smooth space. Then the following are equivalent:

(1) X is S-paracompact, i.e. admits S-partitions of unity.

(2) X is S-normal.

(3) The topology of X has a basis which is a countable union of locally ¬nite

families of carriers of smooth functions.

(4) There is a homeomorphic embedding i : X ’ c0 (A) for some A (with image

in the unit ball) such that eva —¦ i is smooth for all a ∈ A.

Proof. (1) ’ (3) Let Un be the cover formed by all open balls of radius 1/n. By

(1) there exists a partition of unity subordinated to it. The carriers of these smooth

functions form a locally ¬nite re¬nement Vn . The union of all Vn is clearly a base

of the topology since that of all Un is one.

(3) ’ (2) Let A1 and A2 be two disjoint closed subsets of X. Let furthermore Un

be a locally ¬nite family of carriers of smooth functions such that n Un is a basis.

i

Let Wn := {U ∈ Un : U © Ai = …}. This is the carrier of the smooth locally

i

¬nite sum of the carrying functions of the U ™s. The family {Wn : i ∈ {0, 1}, n ∈ N}

forms a countable cover of X. By the argument used in the proof of (16.10) we

may shrink the Wn to a locally ¬nite cover of X. Then W 1 = n Wn is a carrier

i 1

containing A2 and avoiding A1 . Now use (16.2.2).

(2) ’ (1) is lemma (16.2), since metrizable spaces are paracompact.

(3) ’ (4) Let Un be a locally ¬nite family of carriers of smooth functions such that

1

U := n Un is a basis. For every U ∈ Un let fU : X ’ [0, n ] be a smooth function

with carrier U . We de¬ne a mapping i : X ’ c0 (U), by i(x) = (fU (x))U ∈U . It

16.15

16.18 16. Smooth partitions of unity and smooth normality 175

is continuous at x0 ∈ X, since for n ∈ N there exists a neighborhood V of x0

1

which meets only ¬nitely many sets U ∈ k¤2n Uk , and so i(x) ’ i(x0 ) ¤ n

1

for those x ∈ V with |fU (x) ’ fU (x0 )| < n for all U ∈ k¤n Uk meeting V .

The mapping i is even an embedding, since for x0 ∈ U ∈ U and x ∈ U we have

/

i(x) ’ i(x0 ) = fU (x0 ) > 0.

(4) ’ (3) By (16.14) the Banach space c0 (A) has a basis which is a countable union

of locally ¬nite families of carriers of smooth functions, all of which depend locally

only on ¬nitely many coordinates. The pullbacks of all these functions via i are

smooth on X, and their carriers furnish the required basis.

16.16. Corollary. Hilbert spaces are C ∞ -paracompact. [Toru´czyk, 1973].n

Every space c0 (“) (for arbitrary index set “) and every Hilbert space (not necessarily

separable) is C ∞ -paracompact.

Proof. The assertion for c0 (“) is immediate from (16.15). For a Hilbert space

2

(“) we use the embedding i : 2 (“) ’ c0 (“ ∪ {—}) given by

for γ ∈ “

xγ

i(x)γ = 2

for γ = —

x

This is an embedding: From xn ’ x ∞ ’ 0 we conclude by H¨lder™s inequality

o

that y, xn ’ x ’ 0 for all y ∈ 2 and hence xn ’ x 2 = xn 2 + x 2 ’ 2 x, xn ’

2 x 2 ’ 2 x 2 = 0.

16.17. Corollary. A countable product of S-paracompact metrizable spaces is

again S-paracompact.

Proof. By theorem (16.15) we have certain embeddings in : Xn ’ c0 (An ) with

images contained in the unit balls. We consider the embedding i : n Xn ’

1

c0 ( n An ) given by i(x)a = n in (xn ) for a ∈ An which has the required properties

for theorem (16.15). It is an embedding, since i(xn ) ’ i(x) if and only if xn ’ xk

k

for all k (all but ¬nitely many coordinates are small anyhow).

16.18. Corollary. [Godefroy, Troyanski, et. al., 1983]

Let E be a Banach space with a separable projective resolution of identity, see

(53.13). If E is C k -regular, then it is C k -paracompact.

Proof. By (53.20) there exists a linear, injective, norm 1 operator T : E ’ c0 (“1 )

for some “1 and by (53.13) projections P± for ω ¤ ± ¤ dens E. Let “2 := {∆ :

∆ ⊆ [ω, dens E), ¬nite}. For ∆ ∈ “2 choose a dense sequence (x∆ )n in the unit

n

∆ ∆

sphere of Pω (E) • ±∈∆ (P±+1 ’ P± )(E) and let yn ∈ E be such that yn = 1

and yn (x∆ ) = 1. For n ∈ N let πn : x ’ x ’ yn (x)x∆ . Choose a smooth function

∆ ∆ ∆

n n

∞

h ∈ C (E, [0, 1]) with h(x) = 0 for x ¤ 1 and h(x) = 1 for x ≥ 2. Let

R± := (P±+1 ’ P± )/ P±+1 ’ P± .

16.18

176 Chapter III. Partitions of unity 16.18

Now de¬ne an embedding as follows: Let “ := N3 — “2 N — [ω, dens E) “1

N

and let u : E ’ c0 (“) be given by

1 ∆

for γ = (m, n, l, ∆) ∈ N3 — “2 ,

h(mπn x) h(lR± x)

±

2n+m+l ±∈∆

1

for γ = (m, ±) ∈ N — [ω, dens E),

2m h(mR± x)

u(x)γ := 1 x

for γ = m ∈ N,

2 h( m )

for γ = ± ∈ “1 .

T (x)±

Let us ¬rst show that u is well-de¬ned and continuous. We do this only for the

coordinates in the ¬rst row (for the others it is easier, the third has locally even

¬nite support).

Let x0 ∈ E and 0 < µ < 1. Choose n0 with 1/2n0 < µ. Then |u(x)γ | < µ for all

x ∈ X and all ± = (m, n, l, ∆) with m + n + l ≥ n0 .

For the remaining coordinates we proceed as follows: We ¬rst choose δ < 1/n0 . By

(53.13.8) there is a ¬nite set ∆0 ∈ “2 such that R± x0 < δ/2 for all ± ∈ ∆0 . For

/

those ± and x ’ x0 < δ/2 we get

δ δ

R± (x) ¤ R± (x0 ) + R± (x ’ x0 ) < + = δ,

22

hence u(x)γ = 0 for all γ = (m, n, l, ∆) with m + n + l < n0 and ∆ © ([ω, dens E \

∆0 ) = ….

For the remaining ¬nitely many coordinates γ = (m, n, l, ∆) with m+n+l < n0 and

∆ ⊆ ∆0 we may choose a δ1 > 0 such that |u(x)γ ’u(x0 )γ | < µ for all x’x0 < δ1 .

Thus for x ’ x0 < min{δ/2, δ1 } we have |u(x)γ ’ u(x0 )γ | < 2µ for all γ ∈ N3 — “2

and |u(x0 )γ | ≥ µ only for ± = (m, n, l, ∆) with m + n + l < n0 and ∆ ⊆ ∆0 .

Since T is injective, so is u. In order to show that u is an embedding let x∞ , xp ∈ E

with u(xp ) ’ u(x∞ ). Then xp is bounded, since for n0 > x∞ implies that

h(x∞ /n0 ) = 0 and from h(xp /n0 ) ’ h(x∞ /n0 ) we conclude that xp /n0 ¤ 2 for

large p.

Now we show that for any µ > 0 there is a ¬nite µ-net for {xp : p ∈ N}: For this

we choose m0 > 2/µ. By (53.13.8) there is a ¬nite set ∆0 ⊆ Λ(x∞ ) := µ>0 {± <

∆

dens E : R± (x∞ ) ≥ µ} and an n0 := n ∈ N such that m0 πn 0 (x∞ ) ¤ 1 and

∆

hence h(m0 πn 0 (x∞ )) = 0. In fact by (53.13.9) there is a ¬nite linear combination

of vectors R± (x∞ ), which has distance less than µ from x∞ , let δ := min{ R± (x) :

∆

for those ±} > 0. Since the yn 0 are dense in the unit sphere of Pω • ±∈∆0 R± E

1

we may choose an n such that x∞ ’ x∞ x∆0 < 2m0 and hence

n

πn 0 (x∞ ) = x∞ ’ yn 0 (x∞ )x∆0

∆ ∆

n

¤ x∞ ’ x∞ x∆0 + x∞ x∆0 ’ yn 0 (x∆0 )x∆0

∆

n n n n

∆

x∞ x∆0 ’ x∞ ) x∆0

+ yn 0 n n

1 1 1

¤ +0+ =

2m0 2m0 m0

16.18

16.19 16. Smooth partitions of unity and smooth normality 177

Next choose l0 := l ∈ N such that l0 δ0 ≥ 2 and hence l0 R± x∞ ≥ 2 for all ± ∈ ∆0 .

Then

∆ ∆

h(l0 R± xp ) ’ h(m0 πn00 x∞ )

h(m0 πn00 xp ) h(l0 R± x∞ )

±∈∆0 ±∈∆0

h(l0 R± xp ) ’ h(l0 R± x∞ ) = 1 for ± ∈ ∆0

and

Hence

∆ ∆

h(m0 πn00 xp ) ’ h(m0 πn00 x∞ ) = 0,

and so πn00 xp ¤ 2/m0 < µ for all large p. Thus d(xp , R x∆00 ) ¤ µ, hence {xp : p ∈

∆

n

N} has a ¬nite µ-net, since its projection onto the one dimensional subspace Rx∆00n

is bounded.

Thus {x∞ , xp : p ∈ N} is relatively compact, and hence u restricted to its closure

is a homeomorphism onto the image. So xp ’ x∞ .

Now the result follows from (16.15).

16.19. Corollary. [Deville, Godefroy, Zizler, 1990]. Let c0 (“) ’ E ’ F be a

short exact sequence of Banach spaces and assume F admits C p -partitions of unity.

Then E admits C p -partitions of unity.

Proof. Without loss of generality we may assume that the norm of E restricted

to c0 (“) is the supremum norm. Furthermore there is a linear continuous splitting

T : 1 (“) ’ E by (13.17.3) and a continuous splitting S : F ’ E by (53.22) with

S(0) = 0. We put Tγ := T (eγ ) for all γ ∈ “. For n ∈ N let Fn be a C p -partition

of unity on F with diam(carr(f )) ¤ 1/n for all f ∈ Fn . Let F := n Fn and let

“2 := {∆ ⊆ “ : ∆ is ¬nite}. For any f ∈ F choose xf ∈ S(carr(f )) and for any

∆

∆ ∈ “2 choose a dense sequence {yf,m : m ∈ N} 0 in the linear subspace generated

by {xf + eγ : γ ∈ ∆}. Let ∆ ∈ E be such that ∆ (yf,m ) = ∆ · yf,m = 1.

∆ ∆

f,m f,m f,m

Let πf,m : E ’ E be given by πf,m (x) := x ’ ∆ (x) yf,m . Let h : E ’ R be

∆ ∆ ∆

f,m

C p with h(x) = 0 for x ¤ 1 and h(x) = 1 for x ≥ 2. Let g : R ’ [’1, 1] be

C p with g(t) = 0 for |t| ¤ 1 and injective on {t : |t| > 1}. Now de¬ne a mapping

˜

u : E ’ c0 (“), where

˜

“ := (F — “2 — N2 ) (F — “) (F — N) F N N

by

1 ∆

g(n Tγ (x ’ xf ))

u(x)γ := f (ˆ) h(j πf,m (x))

x

˜

2n+m+j

γ∈∆

for γ = (f, ∆, j, m) ∈ Fn — “2 — N2 , and by

˜

±1

2n f (ˆ) g(n Tγ (x ’ xf )) for γ = (f, γ) ∈ Fn — “

x ˜

1

n+j f (ˆ) h(j (x ’ xf )) for γ = (f, j) ∈ Fn — N

x ˜

2

1

= f ∈ Fn ⊆ F

u(x)γ := 2n f (ˆ)

x for γ

˜

˜

1

=n∈N

n h(n x) for γ

˜

2

1

= n ∈ N.

2n h(x/n) for γ

˜

16.19