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);