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

n be such that hn (x) > 1, and take k > n so large that 1 + k < hn (x), and let

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

178 Chapter III. Partitions of unity 16.19

We ¬rst claim that u is well-de¬ned and continuous. Every coordinate x ’ u(x)γ

is continuous, so it remains to show that for every µ > 0 locally in x the set

of coordinates γ, where |u(x)γ | > µ is ¬nite. We do this for the ¬rst type of

coordinates. For this we may ¬x n, m and j (since the factors are bounded by 1).

Since Fn is a partition of unity, locally f (ˆ) = 0 for only ¬nitely many f ∈ Fn , so we

x

1

may also ¬x f ∈ Fn . For such an f the set ∆0 := {γ : |Tγ (x ’ xf )| ≥ π(x ’ xf ) + n }

is ¬nite by the proof of (13.17.3). Since x ’ xf = π(x ’ xf ) ¤ 1/n be have

ˆ

g(n Tγ (x ’ xf )) = 0 for γ ∈ ∆0 .

/

Thus only for those ∆ contained in the ¬nite set ∆0 , we have that the corresponding

coordinate does not vanish.

Next we show that u is injective. Let x = y ∈ E.

If x = y , then there is some n and a f ∈ Fn such that f (ˆ) = 0 = f (ˆ). Thus this

ˆˆ x y

is detected by the 4th row.

If x = y then S x = S y and since x ’ S x, y ’ S y ∈ c0 (“) there is a γ ∈ “ with

ˆˆ ˆ ˆ ˆ ˆ

Tγ (x ’ S x) = (x ’ S x)γ = (y ’ S y )γ = Tγ (y ’ S y ).

ˆ ˆ ˆ ˆ

We will make use of the following method repeatedly:

For every n there is a fn ∈ Fn with fn (ˆ) = 0 and hence x ’ xfn ¤ 1/n.

x ˆˆ

Since S is continuous we get xfn = S(ˆfn ) ’ S(ˆ) and thus limn Tγ (x ’ xfn ) =

x x

limn Tγ (x ’ S(ˆfn )) = Tγ (x ’ S(ˆ)).

x x

So we get

lim Tγ (x ’ xfn ) = Tγ (x ’ S(ˆ)) = Tγ (y ’ S(ˆ)) = lim Tγ (y ’ xfn ).

x y

n n

If all coordinates for u(x) and u(y) in the second row would be equal, then

g(n Tγ (x ’ xf )) = g(n Tγ (y ’ xf ))

since fγ (ˆ) = 0, and hence Tγ (x ’ xf ) ’ Tγ (y ’ xf ) ¤ 2/n, a contradiction.

x

Now let us show that u is a homeomorphism onto its image. We have to show

xk ’ x provided u(xk ) ’ u(x).

We consider ¬rst the case, where x = S x. As before we choose fn ∈ Fn with

ˆ

fn (ˆ) = 0 and get xfn = S(ˆfn ) ’ S(ˆ) = x. Let µ > 0 and j > 3/µ. Choose an n

x x x

such that xfn ’ x < 1/j. Then h(j (xfn ’ x)) = 0. From the coordinates in the

third and fourth row we conclude

f (ˆk ) h(j (xk ’ xfn )) ’ f (ˆ) h(j (x ’ xfn )) f (ˆk ) ’ f (ˆ) = 0.

x x and x x

Hence

h(j (xk ’ xfn )) ’ h(j (x ’ xfn )) = 0.

Thus xk ’ xfn < 2/j for all large k. But then

3

xk ’ x ¤ xk ’ xfn + xfn ’ x < < µ,

j

16.19

16.19 16. Smooth partitions of unity and smooth normality 179

i.e. xk ’ x.

Now the case, where x = S x. We show ¬rst that {xk : k ∈ N} is bounded. Pick

ˆ

n > x . From the coordinates in the last row we get that limk h(xk /n) = 0, i.e.

xk ¤ 2n for all large k.

We claim that for j ∈ N there is an n ∈ N and an f ∈ Fn with f (ˆ) = 0, a ¬nite

x

∆

set ∆ ⊆ “ with γ∈∆ g(n Tγ (x ’ xf )) = 0 and an m ∈ N with h(j πf,m (x)) = 0.

From 0 = (x ’ S x) ∈ c0 (“) we deduce that there is a ¬nite set ∆ ⊆ “ with

ˆ

Tγ (x ’ S x) = (x ’ S x)γ = 0 for all γ ∈ ∆ and dist(x ’ S x, eγ : γ ∈ ∆ ) < 1/(3j),

ˆ ˆ ˆ

i.e. |(x ’ S x)γ | ¤ 1/(3j) for all γ ∈ ∆. As before we choose fn ∈ Fn with fn (ˆ) = 0

ˆ / x

and get xfn = S(ˆfn ) ’ S(ˆ) and

x x

lim Tγ (x ’ xfn ) = (x ’ S x)γ = 0 for γ ∈ ∆.

ˆ

n

Thus g(n (Tγ (x ’ xfn ))) = 0 for all large n and γ ∈ ∆. Furthermore, dist(x, xfn +

∆

eγ : γ ∈ ∆ ) = dist(x ’ xfn , eγ : γ ∈ ∆ ) < 1/(2j). Since {yfn ,m : m ∈ N} is

∆

dense in xfn + eγ : γ ∈ ∆ there is an m such that x ’ yfn ,m < 1/(2j). Since

∆

πfn ,m ¤ 2 we get

πfn ,m (x) ¤ x ’ yfn ,m + |1 ’ ∆ ,m (x)| yfn ,m

∆ ∆ ∆

fn

1 1 1 1

+ ∆ ,m x ’ yfn ,m yfn ,m ¤

∆ ∆

¤ + =,

fn

2j 2j 2j j

∆

hence h(j πfn ,m (x)) = 0.

We claim that for every µ > 0 there is a ¬nite µ-net of {xk : k ∈ N}. Let µ > 0.

We choose j > 4/µ and we pick n ∈ N, f ∈ Fn , ∆ ⊆ “ ¬nite, and m ∈ N satisfying

the previous claim. From u(xk ) ’ u(x) we deduce from the coordinates in the ¬rst

row, that

∆

g(n Tγ (xk ’ xf )) ’

f (ˆk ) h(j πf,m (xk ))

x

γ∈∆

∆

’ f (ˆ) h(j πf,m (x)) g(n Tγ (x ’ xf )) for k ’ ∞

x

γ∈∆

and since by the coordinates in the fourth row f (ˆk ) ’ f (ˆ) = 0 we obtain from

x x

the coordinates in the second row, that

g(n Tγ (xk ’ xf )) ’ g(n Tγ (x ’ xf )) = 0 for γ ∈ ∆.

Hence

∆ ∆

h(j πf,m (xk )) ’ h(j πf,m (x)) = 0.

Therefore

1 µ

∆ ∆ ∆

xk ’ f,m (xk ) yf,n = πf,m (xk ) < < for all large k.

j 4

∆

Thus there is a ¬nite dimensional subspace in E spanned by yf,n and ¬nitely many

xk , such that all xk have distance ¤ µ/4 from it. Since {xk : k ∈ N} are bounded,

16.19

180 Chapter III. Partitions of unity 16.21

the compactness of the ¬nite dimensional balls implies that {xk : k ∈ N} has an

µ-net, hence {xk : k ∈ N} is relatively compact, and since u is injective we have

limk xk = x.

Now the result follows from (16.15).

Remark. In general, the existence of C ∞ -partitions of unity is not inherited by

the middle term of short exact sequences: Take a short exact sequence of Banach

spaces with Hilbert ends and non-Hilbertizable E in the middle, as in (13.18.6).

If both E and E — admitted C 2 -partitions of unity, then they would admit C 2 -

bump functions, hence E was isomorphic to a Hilbert space by [Meshkov, 1978], a

contradiction.

16.20. Results on C(K). Let K be compact. Then for the Banach space C(K)

we have:

(1) [Deville, Godefroy, Zizler, 1990]. If K (ω) = … then C(K) is C ∞ -paracom-

pact.

(2) [Vanderwer¬, 1992] If K (ω1 ) = … then C(K) is C 1 -paracompact.

(3) [Haydon, 1990] In contrast to (2) there exists a compact space K with

K (ω1 ) = {—}, but such that C(K) has no Gˆteaux-di¬erentiable norm. Nev-

a

ertheless C(K) is C 1 -regular by [Haydon, 1991]. Compare with (13.18.2).

(4) [Namioka, Phelps, 1975]. If there exists an ordinal number ± with K (±) = …

then the Banach space C(K) is Asplund (and conversely), hence it does not

admit a rough norm, by (13.8).

(5) [Ciesielski, Pol, 1984] There exists a compact K with K (3) = …. Conse-

quently, there is a short exact sequence c0 (“1 ) ’ C(K) ’ c0 (“2 ), and the

space C(K) is Lipschitz homeomorphic to some c0 (“). However, there is

no continuous linear injection of C(K) into some c0 (“).

Notes. (1) Applying theorem (16.19) recursively we get the result as in (13.17.5).

16.21. Some radial subsets are di¬eomorphic to the whole space. We are

now going to show that certain subsets of convenient vector spaces are di¬eomorphic

to the whole space. So if these subsets form a base of the c∞ -topology of the

modeling space of a manifold, then we may choose charts de¬ned on the whole

modeling space. The basic idea is to ˜blow up™ subsets U ⊆ E along all rays

starting at a common center. Without loss of generality assume that the center

is 0. In order for this technique to work, we need a positive function ρ : U ’ R,

1

which should give a di¬eomorphism f : U ’ E, de¬ned by f (x) := ρ(x) x. For

this we need that ρ is smooth, and since the restriction of f to U © R+ x ’ R+ x

has to be a di¬eomorphism as well, and since the image set is connected, we need

that the domain U © R+ x is connected as well, i.e., U has to be radial. Let Ux :=

t

{t > 0 : tx ∈ U }, and let fx : Ux ’ R be given by f (tx) = ρ(tx) x =: fx (t)x.

Since up to di¬eomorphisms this is just the restriction of the di¬eomorphism f , we

need that 0 < fx (t) = ‚t ρ(tx) = ρ(tx)’tρ (tx)(x) for all x ∈ U and 0 < t ¤ 1. This

‚ t

ρ(tx)2

means that ρ(y) > ρ (y)(y) for all y ∈ U , which is quite a restrictive condition,

16.21

16.21 16. Smooth partitions of unity and smooth normality 181

and so we want to construct out of an arbitrary smooth function ρ : U ’ R, which

tends to 0 towards the boundary, a new smooth function ρ satisfying the additional

assumption.

Theorem. Let U ⊆ E be c∞ -open with 0 ∈ U and let ρ : U ’ R+ be smooth, such

that for all x ∈ U with tx ∈ U for 0 ¤ t < 1 we have ρ(tx) ’ 0 for t

/ 1. Then

star U := {x ∈ U : tx ∈ U for all t ∈ [0, 1]} is di¬eomorphic to E.

Proof. First remark that star U is c∞ -open. In fact, let c : R ’ E be smooth

with c(0) ∈ star U . Then • : R2 ’ E, de¬ned by •(t, s) := t c(s) is smooth and

maps [0, 1] — {0} into U . Since U is c∞ -open and R2 carries the c∞ -topology there

exists a neighborhood of [0, 1] — {0}, which is mapped into U , and in particular

there exists some µ > 0 such that c(s) ∈ star U for all |s| < µ. Thus c’1 (star U )

is open, i.e., star U is c∞ -open. Note that ρ satis¬es on star U the same boundary

condition as on U . So we may assume without loss of generality that U is radial.

Furthermore, we may assume that ρ = 1 locally around 0 and 0 < ρ ¤ 1 everywhere,

by composing with some function which is constantly 1 locally around [ρ(0), +∞).

Now we are going to replace ρ by a new function ρ, and we consider ¬rst the

˜

case, where E = R. We want that ρ satis¬es ρ (t)t < ρ(t) (which says that the

˜ ˜ ˜

tangent to ρ at t intersects the ρ-axis in the positive part) and that ρ(t) ¤ ρ(t),

˜ ˜ ˜

i.e., log —¦˜ ¤ log —¦ρ, and since we will choose ρ(0) = 1 = ρ(0) it is su¬cient to have

ρ ˜

ρ (t)t

˜ ρ (t)t

ρ˜ ρ

ρ = (log —¦˜) ¤ (log —¦ρ) = ρ or equivalently ρ(t) ¤ ρ(t) for t > 0. In order

ρ

˜ ˜

to obtain this we choose a smooth function h : R ’ R which satis¬es h(t) < 1,

and h(t) ¤ t for all t, and h(t) = t for t near 0, and we take ρ as solution of the

˜

following ordinary di¬erential equation

ρ(t)

˜ ρ (t)t

·h

ρ (t) =

˜ with ρ(0) = 1.

˜

t ρ(t)

Note that for t near 0, we have 1 h ρρ(t) = ρ (t) , and hence locally a unique

(t)t

t ρ(t)

smooth solution ρ exists. In fact, we can solve the equation explicitly, since

˜

s

(log —¦˜) (t) = ρ (t) = 1 · h ρρ(t) , and hence ρ(s) = exp( 0 1 · h( ρρ(t) ) dt), which

˜ (t)t (t)t

ρ ˜

ρ(t)

˜ t t

is smooth on the same interval as ρ is.

Note that if ρ is replaced by ρs : t ’ ρ(ts), then the corresponding solution ρs

satis¬es ρs = ρs . In fact,

˜

(˜s ) (t)

ρ s˜ (st)

ρ 1 st˜ (st)

ρ 1 stρ (st) 1 t(ρs ) (t)

(log —¦˜s ) (t) = =·

ρ = =h =h .

ρs (t)

˜ ρ(st)

˜ t ρ(st)

˜ t ρ(st) t ρs (t)

For arbitrary E and x ∈ E let ρx : Ux ’ R+ be given by ρx (t) := ρ(tx), and let

ρ : U ’ R+ be given by ρ(x) := ρx (1), where ρx is the solution of the di¬erential

˜ ˜

equation above with ρx in place of ρ.

Let us now show that ρ is smooth. Since U is c∞ -open, it is enough to consider

˜

a smooth curve x : R ’ U and show that t ’ ρ(x(t)) = ρ(x(t)) (1) is smooth.

˜ ˜

ρx(t) (s)s ρ (s x(t))(s x(t))

1 1

This is the case, since (t, s) ’ sh = sh is smooth,

ρx(t) (s) ρ(s x(t))

16.21

182 Chapter III. Partitions of unity 16.21

ρ (s x(t))(s x(t)) •(t,s)

satis¬es •(t, 0) = 0, and hence 1 h(•(t, s)) =

since •(t, s) := =

ρ(s x(t)) s s

ρ (s x(t))(x(t))

locally.

ρ(s x(t))

From ρsx (t) = ρ(tsx) = ρx (ts) we conclude that ρsx (t) = ρx (ts), and hence ρ(sx) =

˜

‚ ‚

ρx (s). Thus, ρ (x)(x) = ‚t |t=1 ρ(tx) = ‚t |t=1 ρx (t) = ρx (1) < ρx (1) = ρ(x). This

˜ ˜ ˜ ˜ ˜ ˜

shows that we may assume without loss of generality that ρ : U ’ (0, 1] satis¬es

the additional assumption ρ (x)(x) < ρ(x).

t

Note that fx : t ’ ρ(tx) is bijective from Ux := {t > 0 : tx ∈ U } to R+ , since 0

t

is mapped to 0, the derivative is positive, and ρ(tx) ’ ∞ if either ρ(tx) ’ 0 or

t ’ ∞ since ρ(tx) ¤ 1.

1

It remains to show that the bijection x ’ ρ(x) x is a di¬eomorphism. Obviously,

its inverse is of the form y ’ σ(y)y for some σ : E ’ R+ . They are inverse

1

to each other so ρ(σ(y)y) σ(y)y = y, i.e., σ(y) = ρ(σ(y)y) for y = 0. This is

an implicit equation for σ. Note that σ(y) = 1 for y near 0, since ρ has this

property. In order to show smoothness, let t ’ y(t) be a smooth curve in E.

Then it su¬ces to show that the implicit equation (σ —¦ y)(t) = ρ((σ —¦ y)(t) · y(t))

satis¬es the assumptions of the 2-dimensional implicit function theorem, i.e., 0 =

‚

‚σ (σ ’ ρ(σ · y(t))) = 1 ’ ρ (σ · y(t))(y(t)), which is true, since multiplied with σ > 0

it equals σ ’ ρ (σ · y(t))(σ · y(t)) < σ ’ ρ(σ · y(t)) = 0.

16.21

183

Chapter IV

Smoothly Realcompact Spaces

17. Basic Concepts and Topological Realcompactness . . . . . . . . . . 184

18. Evaluation Properties of Homomorphisms . . . . . . . . . . . . . 188

19. Stability of Smoothly Realcompact Spaces . . . . . . . . . . . . . 203

20. Sets on which all Functions are Bounded . . . . . . . . . . . . . . 217

As motivation for the developments in this chapter let us tell a mathematical short

story which was posed as an exercise in [Milnor, Stashe¬, 1974, p.11]. For a ¬nite

dimensional Hausdor¬ second countable manifold M , one can prove that the space

of algebra homomorphisms Hom(C ∞ (M, R), R) equals M as follows. The kernel of

a homomorphism • : C ∞ (M, R) ’ R is an ideal of codimension 1 in C ∞ (M, R).

The zero sets Zf := f ’1 (0) for f ∈ ker • form a ¬lter of closed sets, since Zf © Zg =

Zf 2 +g2 , which contains a compact set Zf for a function f which is proper (i.e.,

compact sets have compact inverse images). Thus f ∈ker • Zf is not empty, it

contains at least one point x0 ∈ M . But then for any f ∈ C ∞ (M, R) the function

f ’ •(f )1 belongs to the kernel of •, so vanishes on x0 and we have f (x0 ) = •(f ).

This question has many rather complicated (partial) answers in any in¬nite dimen-

sional setting which are given in this chapter. One is able to prove that the answer

is positive surprisingly often, but the proofs are involved and tied intimately to the

class of spaces under consideration. The existing counter-examples are based on

rather trivial reasons. We start with setting up notation and listing some interesting

algebras of functions on certain spaces.

First we recall the topological theory of realcompact spaces from the literature and

discuss the connections to the concept of smooth realcompactness. For an algebra

homomorphism • : A ’ R on some algebra of functions on a space X we investigate

when •(f ) = f (x) for some x ∈ X for one function f , later for countably many, and

¬nally for all f ∈ A. We study stability of smooth realcompactness under pullback

along injective mappings, and also under (left) exact sequences. Finally we discuss

the relation between smooth realcompactness and bounding sets, i.e. sets on which

every function of the algebra is bounded. In this chapter, the ordering principle for

sections and results is based on the amount of evaluating properties obtained and

we do not aim for linearly ordered proofs. So we will often use results presented

later in the text. We believe that this is here a more transparent presentation than

the usual one. Most of the material in this chapter can also be found in the theses™

[Bistr¨m, 1993] and [Adam, 1993].

o

184 Chapter IV. Smoothly realcompact spaces 17.1

17. Basic Concepts and Topological Realcompactness

17.1. The setting. In [Hewitt, 1948, p.85] those completely regular topological

spaces were considered under the name Q-spaces, for which each real valued alge-

bra homomorphism on the algebra of all continuous functions is the evaluation at

some point of the space. Later on these spaces where called realcompact spaces.

Accordingly, we call a ˜space™ smoothly realcompact if this is true for ˜the™ algebra

of smooth functions. There are other algebras for which this question is interest-

ing, like polynomials, real analytic functions, C k -functions. So we will treat the

question in the following setting. Let

X be a set;

A ⊆ RX a point-separating subalgebra with unit; If X is a topological space we

also require that A ⊆ C(X, R); If X = E is a locally convex vector space we

also assume that A is invariant under all translations and contains the dual

E — of all continuous linear functionals;

XA the set X equipped with the initial topology with respect to A;

• : A ’ R an algebra homomorphism preserving the unit;

Zf := {x ∈ X : f (x) = •(f )} for f ∈ A;

Hom A be the set of all real valued algebra homomorphisms A ’ R preserving

the unit.

Moreover,

• is called F-evaluating for some subset F ⊆ A if there exists an x ∈ X with

•(f ) = f (x) for all f ∈ F; equivalently f ∈F Zf = …;

• is called m-evaluating for a cardinal number m if • is F-evaluating for all

F ⊆ A with cardinality of F at most m; This is most important for m = 1

and for m = ω, the ¬rst in¬nite cardinal number;

• is said to be ¯

1-evaluating if •(f ) ∈ f (X) for all f ∈ A.

• is said to be evaluating if • is A-evaluating, i.e., • = evx for some x ∈ X;

Homω A is the set of all ω-evaluating homomorphisms in Hom A;

A is called m-evaluating if • is m-evaluating for each algebra homomorphism

• ∈ Hom A;

A is called evaluating if • is evaluating for algebra homomorphism • ∈ Hom A;

X is called A-realcompact if A is evaluating; i.e., each algebra homomorphism

• ∈ Hom A is the evaluation at some point in X.

The algebra A is called

inversion closed if 1/f ∈ A for all f ∈ A with f (x) > 0 for every x ∈ X;

equivalently, if 1/f ∈ A for all f ∈ A with f nowhere 0 (use f 2 > 0).

bounded inversion closed if 1/f ∈ A for f ∈ A with f (x) > µ for some µ > 0 and

all x ∈ X;

C (∞) -algebra if h —¦ f ∈ A for all f ∈ A and h ∈ C ∞ (R, R);

C ∞ -algebra if h —¦ (f1 , . . . , fn ) ∈ A for all fj ∈ A and h ∈ C ∞ (Rn , R);

Clfs -algebra if it is a C ∞ -algebra which is closed under locally ¬nite sums, with

∞

respect to a speci¬ed topology on X. This holds if A is local, i.e., it contains

17.1

17.1 17. Basic concepts and topological realcompactness 185

any function f such that for each x ∈ X there is some fx ∈ A with f = fx

near x.

Clfcs -algebra if it is a C ∞ -algebra which is closed under locally ¬nite countable

∞

sums.

Interesting algebras are the following, where in this chapter in the notation we shall

generally omit the range space R.

y wC

“

‘

‘‘

Cb

‘‘

™

y wC y wC © Ce

se

∞ ∞ ∞

g

e

¢

Clfcs

ee

j

h

c

h ee

ee g

h

¨

h

p e

s

Py w P 99

y wC y wC © C&

4&

G

x

x C∞

ω ω

&&

f

99

c

( x xx

A

9 &

y wC

ω ω

Cconv

C(X) = C(X, R), the algebra of continuous functions on a topological space X.

It has all the properties from above.

Cb (X) = Cb (X, R), the algebra of bounded continuous functions on a topological

space X. It is only bounded inversion closed and a C ∞ -algebra, in general.

C ∞ (X) = C ∞ (X, R), the algebra of smooth functions on a Fr¨licher space X,

o

see (23.1), or on a smooth manifold X, see section (27). It has all properties

from above, where we may use the c∞ -topology.

C ∞ (E) © C(E), the algebra of smooth and continuous functions on a locally

convex space E. It has all properties from above, where we use the locally

convex topology on E.

∞ ∞

Cc (E) = Cc (E, R), the algebra of smooth functions, all of whose derivatives

are continuous on a locally convex space E. It has all properties from above,

again for the locally convex topology on E.

C ω (X) = C ω (X, R), the algebra of real analytic functions on a real analytic

manifold X. It is only inversion closed.

C ω (E) © C(E), the algebra of real analytic and continuous functions on a locally

convex space E. It is only inversion closed.

ω ω

Cc (E) = Cc (E, R), the algebra of real analytic functions, all of whose derivatives

are continuous on a locally convex space E. It is only inversion closed.

ω ω

Cconv (E) = Cconv (E, R), the algebra of globally convergent power series on a

locally convex space E.

Pf (E) = Polyf (E, R), the algebra of ¬nite type polynomials on a locally convex

space E, i.e. the algebra E Alg generated by E . This is the free commu-

tative algebra generated by the vector space E , see (18.12). It has none of

the properties from above.

17.1

186 Chapter IV. Smoothly realcompact spaces 17.3

P (E) = Poly(E, R), the algebra of polynomials on a locally convex space E, see

(5.15), (5.17), i.e. the homogeneous parts are given by bounded symmetric

multilinear mappings. No property from above holds.

∞ ∞ ∞

Clfcs (E) = Clfcs (E, R), the Clfcs -algebra (see below) generated by E , and hence

also called (E )∞ . Only the Clfs -property does not hold.

∞

lfcs

17.2. Results. For completely regular topological spaces X and A = C(X) the

following holds:

(1) Due to [Hewitt, 1948, p.85 + p.60] & [Shirota, 1952, p.24], see also [Engel-

king, 1989, 3.12.22.g & 3.11.3]. The space X is called realcompact if all

algebra homomorphisms in Hom C(X) are evaluations at points of X, equiv-

alently, if X is a closed subspace of a product of R™s.

(2) Due to [Hewitt, 1948, p.61] & [Katˇtov, 1951, p.82], see also [Engelking,

e

1989, 3.11.4 & 3.11.5]. Hence every closed subspace of a product of realcom-

pact spaces is realcompact.

(3) Due to [Hewitt, 1948, p.85], see also [Engelking, 1989, 3.11.12]. Each Lin-

del¨f space is realcompact.

o

(4) Due to [Katˇtov, 1951, p.82], see also [Engelking, 1989, 5.5.10]. Paracom-

e

pact spaces are realcompact if and only if all closed discrete subspaces are

realcompact.

(5) Due to [Hewitt, 1950, p.170, p.175] & [Mackey, 1944], see also [Engelk-

ing, 1989, 3.11.D.a]. Discrete spaces are realcompact if and only if their

cardinality is non-measurable.

(6) Hence Banach spaces are realcompact if and only if their density (i.e., the

cardinality of a maximal discrete or of a minimal dense subset) or their

cardinality is non-measurable.

(7) [Shirota, 1952], see also [Engelking, 1989, 5.5.10 & 8.5.13.h]. A space of

non-measurable cardinality is realcompact if and only if it admits a complete

uniformity.

(8) Due to [Dieudonn´, 1939,] see also [Engelking, 1989, 8.5.13.a]. A space

e

admits a complete uniformity, i.e. is Dieudonn´ complete, if and only if it

e

is a closed subspace of a product of metrizable spaces

Realcompact spaces where introduced by [Hewitt, 1948, p.85] under the name Q-

compact spaces. The equivalence in (1) is due to [Shirota, 1952, p.24]. The results

(1) and (2) are proved in [Engelking, 1989] for a di¬erent notion of realcompactness,

which was shown to be equivalent to the original one by [Katˇtov, 1951], see also

e

[Engelking, 1989, 3.12.22.g].

17.3. Lemma. [Kriegl, Michor, Schachermayer, 1989, 2.2, 2.3]. Let A be ¯

1-eval-

uating. Then we have a topological embedding

δ : XA ’ prf —¦δ := f,

R,

A

with dense image in the closed subset Hom A ⊆ R. Hence X is A-realcompact

A

if and only if δ has closed image.

17.3

17.6 17. Basic concepts and topological realcompactness 187

Proof. The topology of XA is by de¬nition initial with respect to all f = prf —¦δ,

hence δ is an embedding. Obviously Hom A ⊆ A R is closed. Let • : A ’ R be an

algebra-homomorphism. For f ∈ A consider Zf . If A is 1-evaluating then by (18.8)

for any ¬nite subset F ⊆ A there exists an xF ∈ f ∈F Zf . Thus δ(xF )f = •(f )

for all f ∈ F. If A is only ¯1-evaluating, then we get as in the proof of (18.3) for

every µ > 0 a point xF ∈ X such that |f (xF ) ’ •(f )| < µ for all f ∈ F. Thus δ(xF )

lies in the corresponding neighborhood of (•(f ))f . Thus δ(X) is dense in Hom A.

Now X is A-realcompact if and only if δ has Hom A as image, and hence if and

only if the image of δ is closed.

17.4. Theorem. [Kriegl, Michor, Schachermayer, 1989, 2.4] & [Adam, Bistr¨m, o

Kriegl, 1995, 3.1]. The topology of pointwise convergence on Homω A is realcom-

pact. If XA is not realcompact then there exists an ω-evaluating homomorphism •

which is not evaluating.

Proof. We ¬rst show the weaker statement, that: If XA is not realcompact then

there exists a non-evaluating •, i.e., X is not A-realcompact.

Assume that X is A-realcompact, then A is 1-evaluating and hence by lemma

(17.3) δ : XA ’ A R is a closed embedding. Thus by (17.2.1) the space XA is

realcompact.

Now we give a proof of the stronger statement that Homω (A, R) is realcompact:

Assume that all sets of homomorphisms are endowed with the pointwise topology.

Let M ⊆ 2A be the family of all countable subsets of A containing the unit.

For M ∈ M, consider the topological space Homω M , where M denotes the

subalgebra generated by M . Obviously the family (δf )f ∈M , where δf (•) = •(f ), is

a countable subset of C(Homω M ) that separates the points in Homω M . Hence

Homω M = Hom C(Homω M ) by (18.25), since C(Homω M ) is ω-evaluating by

(18.11), i.e. Homω M is realcompact. Now Homω A is an inverse limit of the spaces

Homω M for M ∈ M. Since Homω M is Hausdor¬, we obtain that Homω A as

a closed subset of a product of realcompact spaces is realcompact by (17.2.2).

Since X is not realcompact in the topology XA , which is that induced from the em-

bedding into Homω A, we have that X = Homω A and the statement is proved.

17.5. Counter-example. [Kriegl, Michor, 1993, 3.6.2]. The locally convex space

∞

R“

count of all points in the product with countable carrier is not C -realcompact, if

“ is uncountable and none-measurable.

Proof. By [Engelking, 1989, 3.10.17 & 3.11.2] the space R“ count is not realcompact,

in fact every c∞ -continuous function on it extends to a continuous function on R“ ,

see the proof of (4.27). Since the projections are smooth, XC ∞ is the product

topology. So the result follows from (17.4).

17.6. Theorem. [Kriegl, Michor, Schachermayer, 1989, 3.2] & [Garrido, G´mez,

o

Jaramillo, 1994, 1.8]. Let X be a realcompact and completely regular topological

space, let A be uniformly dense in C(X) and ¯

1-evaluating.

17.6

188 Chapter IV. Smoothly realcompact spaces 18.1

Then X is A-realcompact. Moreover, if X is A-paracompact then A is uniformly

dense in C(X).

∞

In [Kriegl, Michor, Schachermayer, 1989] it is shown that Clfcs -algebra A is uni-

formly dense in C(X) if and only if A © Cb (X) is uniformly dense in Cb (X). One

may ¬nd also other equivalent conditions there.

Proof. Since A ⊆ C(X) we have that the identity XA ’ X is continuous, and

hence A ⊆ C(XA ) ⊆ C(X). For each of these point-separating algebras we consider

the natural inclusion δ of X into the product of factors R over the algebra, given

by prf —¦δ = f . It is a uniform embedding for the uniformity induced on X by this

algebra and the complete product uniformity on R with basis formed by the sets

Uf,µ := {(u, v) : | prf (u) ’ prf (v)| < µ} with µ > 0.

The condition that A ⊆ C is dense implies that the uniformities generated by

C(X), by C(XA ) and by A coincide and hence we will consider X as a uniform

space endowed with this uniform structure in the sequel. In fact for an arbitrarily

given continuous map f and µ > 0 choose a g ∈ A such that |g ’ f | < µ. Then

{(x, y) : |f (x) ’ f (y)| < µ} ⊆ {(x, y) : |g(x) ’ g(y)| < 3µ}

⊆ {(x, y) : |f (x) ’ f (y)| < 5µ}.

Since XA is realcompact, δC (XA ) = Hom(C(XA )) and hence is closed and so the

uniform structure on X is complete. And similarly also if X is realcompact. Thus,

the image δA (X) is a complete uniform subspace of A R and so it is closed with

respect to the product topology, i.e. X is A-realcompact by (17.3).

17.7. In the case of a locally convex vector space the last result (17.6) can be

slightly generalized to:

Result. [Bistr¨m, Lindstr¨m, 1993b, Thm.6]. For E a realcompact locally convex

o o

vector space, let E ⊆ A ⊆ C(E) be a ω-evaluating C (∞) -algebra which is invari-

ant under translations and homotheties. Moreover, we assume that there exists

a 0-neighborhood U in E such that for each f ∈ C(E) there exists g ∈ A with

supx∈U |f (x) ’ g(x)| < ∞.

Then E is A-realcompact.

18. Evaluation Properties of Homomorphisms

In this section we consider ¬rst properties near the evaluation property at single

functions, then evaluation properties for homomorphisms on countable many func-

tions, and ¬nally direct situations where all homomorphisms are point evaluations.

18.1. Remark. If • in Hom A is 1-evaluating (i.e., •(f ) ∈ f (X) for all f in A),

then • is ¯

1-evaluating.

18.1

18.3 18. Evaluation properties of homomorphisms 189

18.2. Lemma. [Bistr¨m, Bjon, Lindstr¨m, 1991, p.181]. For a topological space

o o

X the following assertions are equivalent:

(1) • is ¯

1-evaluating;

˜x

ˇ

(2) There exists x in the Stone-Cech compacti¬cation βX with •(f ) = f (˜) for

˜

all f ∈ A.

˜ ˇ

Here f denotes the extension of f : X ’ R ’ R∞ to the Stone-Cech-compacti¬-

cation βX with values in the 1-point compacti¬cation R∞ of R.

In [Garrido, G´mez, Jaramillo, 1994, 1.3] it is shown for a subalgebra of Cb (R) that

o

x need not be unique.

˜

Proof. For f ∈ A and µ > 0 let U (f, µ) := {x ∈ X : |•(f ) ’ f (x)| < µ}. Then U :=

{U (f, µ) : f ∈ A, µ > 0} is a ¬lter basis on X. Consider X as embedded into βX and

˜

take an ultra¬lter U on βX that is ¬ner than U. For f := (f1 ’•(f1 ))2 +(f2 ’•(f2 ))2

we have in fact

U (f1 , µ1 ) © U (f2 , µ2 ) ⊇ U (f, min{µ1 , µ2 }2 ).

˜

Let x ∈ βX be the point to which U converges. For an arbitrary function f in A

˜

˜˜ ˜

the ¬lter f (U) converges to •(f ) by construction. But f (U) ≥ f (U) = f (U), so

˜x ˜x ˜

•(f ) = f (˜). The converse is obvious since •(f ) = f (˜) ∈ f (βX) ⊆ f (X) ⊆ R∞ ,

and •(f ) ∈ R.

18.3. Lemma. [Adam, Bistr¨m, Kriegl, 1995, 4.1]. An algebra homomorphism •

o

is ¯

1-evaluating if and only if • extends (uniquely) to an algebra homomorphism on

A∞ , the C ∞ -algebra generated by A.

Proof. For C ∞ -algebras A, we have that

•(h —¦ (f1 , . . . , fn )) = h(•(f1 ), . . . , •(fn ))

for all h ∈ C ∞ (Rn , R) and f1 , . . . , fn in A.

In fact set a := (•(f1 ), . . . , •(fn )) ∈ Rn . Then

1

ha (x) · (xj ’ aj ),

h(x) ’ h(a) = ‚j h(a + t(x ’ a)) dt · (xj ’ aj ) = j

0 j¤n j¤n

1

where ha (x) := 0 ‚j h(a + t(x ’ a))dt. Applying • to this equation composed with

j

the fi one obtains

•(h —¦ (f1 , . . . , fn )) ’ h(•(f1 ), . . . , •(fn )) =

•(ha —¦ (f1 , . . . , fn )) · (•(fj ) ’ •(fj )) = 0.

= j

j¤n

(’) We de¬ne •(h —¦ (f1 , . . . , fn )) := h(•(f1 ), . . . , •(fn )). By what we have shown

˜

above (1-preserving) algebra homomorphisms are C ∞ -algebra homomorphisms and

hence this is the only candidate for an extension. This map is well de¬ned. Indeed,

18.3

190 Chapter IV. Smoothly realcompact spaces 18.5

let h —¦ (f1 , . . . , fn ) = k —¦ (g1 , . . . , gm ). For each µ > 0 there is a point x ∈ E such

that |•(fi ) ’ fi (x)| < µ for i = 1, ..., n, and |•(gj ) ’ gj (x)| < µ for j = 1, ..., m. In

˜x

fact by (18.2) there is a point x ∈ βX with •(f ) = f (˜) for

˜

n m

(fi ’ •(fi ))2 + (gj ’ •(gj ))2 ,

f :=

i=1 j=1

˜x

and hence •(fi ) = fi (˜) and •(gj ) = gj (˜). Now approximate x by x ∈ X.

˜x ˜

By continuity of h and k we obtain that

h(•(f1 ), . . . , •(fn )) = k(•(f1 ), . . . , •(fm )),

and we therefore have a well de¬ned extension of •. This extension is a homo-

morphism, since for every polynomial θ on Rm (or even for θ ∈ C ∞ (Rm )) and

gi := hi —¦ (f1 , . . . , fni ) ∈ A∞ we have

i i

1 m

•(θ —¦ (g1 , . . . , gm )) = •(θ —¦ (h1 — . . . — hm ) —¦ (f1 , . . . , fnm ))

˜ ˜

1 m

= (θ —¦ (h1 — . . . — hm ))(•(f1 ), . . . , •(fnm ))

1 1 m m

= θ(h1 (•(f1 ), . . . , •(fn1 )), . . . , hm (•(f1 ), . . . , •(fnm ))

= θ(•(g1 ), . . . , •(gm )).

˜ ˜

(⇐) Suppose there is some f ∈ A with •(f ) ∈ f (X). Then we may ¬nd an

/

h ∈ C (R) with h(•(f )) = 1 and carr h © f (X) = …. Since A∞ is a C ∞ -algebra,

∞

we conclude from what we said above that •(h —¦ f ) = h(•(f )) = 1. But since

˜

h —¦ f = 0 we arrive at a contradiction.

18.4. Proposition. [Garrido, G´mez, Jaramillo, 1994, 1.2]. If A is bounded

o

inversion closed and • ∈ Hom A then • is ¯

1-evaluating.

Proof. We assume indirectly that there is a function f ∈ A with •(f ) ∈ f (X).

Let µ := inf x∈X |•(f ) ’ f (x)| and g(x) := 1 (•(f ) ’ f (x)). Then g ∈ A, •(g) = 0

µ

1

and |g(x)| = µ |•(f ) ’ f (x)| ≥ 1 for each x ∈ X. Thus 1/g ∈ A. But then

1 = •(g · 1/g) = •(g)•(1/g) = 0 gives a contradiction.

18.5. Lemma. Any C (∞) -algebra is bounded inversion closed.

Moreover, it is stable under composition with smooth locally de¬ned functions, which

contain the closure of the image in its domain of de¬nition.

Proof. Let A be a C ∞ -algebra (resp. C (∞) -algebra), n a natural number (resp.

n = 1), U ⊆ Rn open, h ∈ C ∞ (U, R), f := (f1 , . . . , fn ), with fi ∈ A such that

f (X) ⊆ U , then h —¦ f ∈ A . Indeed, choose ρ ∈ C ∞ (R) with ρ|f (X) = 1 and

supp ρ ⊆ U . Then k := ρ · h is a globally smooth function and h —¦ f = k —¦ f ∈ A.

18.5

18.9 18. Evaluation properties of homomorphisms 191

18.6. Lemma. Any inverse closed algebra A is 1-evaluating.

By (18.10) the converse is wrong.

Proof. Let f ∈ A and assume indirectly that Zf = …. Let g := f ’ •(f ). Then

g ∈ A and g(x) = 0 for all x ∈ X, by which 1/g ∈ A since A is inverse-closed. But

then 1 = •(g · 1/g) = •(g)•(1/g) = 0, which is a contradiction.

18.7. Proposition. [Bistr¨m, Jaramillo, Lindstr¨m, 1995, Lem.14] & [Adam, Bi-

o o

str¨m, Kriegl, 1995, 4.2]. For • in Hom A the following statements are equivalent:

o

(1) • is 1-evaluating.

(2) • extends to a unique (1-evaluating) homomorphism on the algebra RA :=

{f /g : f, g ∈ A, 0 ∈ g(X)}.

/

(3) • extends to a unique (1-evaluating) homomorphism on the following C ∞ -

algebra A ∞ constructed from A:

∞

A := {h —¦ (f1 , . . . , fn ) :fi ∈ A, (f1 , . . . , fn )(X) ⊆ U,

U open in some Rn , h ∈ C ∞ (U )}.

Proof. (1) ’ (3) We de¬ne •(h —¦ (f1 , . . . , fn )) := h(•(f1 ), . . . , •(fn )). Since there

exists by (18.8) an x with •(fi ) = fi (x), we have (•(f1 ), . . . , •(fn )) ∈ U , hence the

right side makes sense. The rest follows in the same way as in the proof of (18.3).

∞

(3) ’ (2) Existence is obvious, since RA ⊆ A , and uniqueness follows from the

de¬nition of RA.

(2) ’ (1) Since RA is inverse-closed, the extension of • to this algebra is 1-

evaluating by (18.6), hence the same is true for • on A.

18.8. Lemma. Every 1-evaluating homomorphism is ¬nitely evaluating.

Proof. Let F be a ¬nite subset of A. De¬ne a function f : X ’ R by

(g ’ •(g))2 .

f :=

g∈F

Then f ∈ A and •(f ) = 0. By assumption there is a point x ∈ X with •(f ) = f (x).

Hence g(x) = •(g) for all g ∈ F.

18.9. Theorem. Automatic boundedness. [Kriegl, Michor, 1993] & [Arias-

de-Reyna, 1988] Every 1-evaluating homomorphism • ∈ Hom A is positive, i.e.,

0 ¤ •(f ) for all 0 ¤ f ∈ A. Moreover we even have •(f ) > 0 for f ∈ A with

f (x) > 0 for all x ∈ X.

Every positive homomorphism • ∈ Hom A is bounded for any convenient algebra

structure on A.

A convenient algebra structure on A is a locally convex topology, which turns A

into a convenient vector space and such that the multiplication A — A ’ A is

bounded, compare (5.21).

18.9

192 Chapter IV. Smoothly realcompact spaces 18.11

Proof. Positivity: Let f1 ¤ f2 . By (17) and (18.8) there exists an x ∈ X such

that •(fi ) = fi (x) for i = 1, 2. Thus •(f1 ) = f1 (x) ¤ f2 (x) = •(f2 ). Note that if

f (x) > 0 for all x, then •(f ) > 0.

Boundedness: Suppose fn is a bounded sequence, but |•(fn )| is unbounded. By

2

replacing fn by fn we may assume that fn ≥ 0 and hence also •(fn ) ≥ 0. Choosing

a subsequence we may even assume that •(fn ) ≥ 2n . Now consider n 21 fn . This

n

series converges Mackey, and since the bornology on A is by assumption complete

the limit is an element f ∈ A. Applying • yields

N N

1 1 1 1

≥

•(f ) = • fn + fn = •(fn ) + • fn

2n 2n 2n 2n

n=0 n=0

n>N n>N

N N

1 1

≥ •(fn ) + 0 = •(fn ),

2n 2n

n=0 n=0

where we used the monotonicity of • applied to n>N 21 fn ≥ 0. Thus the series

n

N

N ’ n=0 21 •(fn ) is bounded and increasing, hence converges, but its summands

n

are bounded by 1 from below. This is a contradiction.

18.10. Lemma. For a locally convex vector space E the algebra Pf (E) is 1-

evaluating.

More on the algebra Pf (E) can be found in (18.27), (18.28), and (18.12).

Proof. Every ¬nite type polynomial p is a polynomial in a ¬nite number of linearly

independent functionals 1 , . . . , n in E . So there is for each i = 1, . . . , n some point

ai ∈ E such that i (ai ) = •( i ) and j (ai ) = 0 for all j = i. Let a = a1 +· · ·+an ∈ E.

Then i (a) = i (ai ) = •( i ) for i = 1, . . . , n hence •(p) = p(a).

Countably Evaluating Homomorphisms

18.11. Theorem. Idea of [Arias-de-Reyna, 1988, proof of thm.8], [Adam, Bis-

∞

tr¨m, Kriegl, 1995, 2.5]. For a topological space X any Clfcs -algebra A ⊆ C(X) is

o

closed under composition with local smooth functions and is ω-evaluating.