continuous functions E ’ R is complete (see [Gillman, Jerison, 1960, p.226]) and

hence the ¬lter F converges to a point a ∈ E. Thus a ∈ ZK for all countable

K ⊆ E , and in particular ¦(x ) = x (a) for all x ∈ E .

18.20. Proposition. [Bistr¨m, Jaramillo, Lindstr¨m, 1995, Thm.10]. Let E be a

o o

ω

Banach space, let A ⊇ Cconv (E) be 1-evaluating, let f ∈ A, and let F be a countable

ω

subset of Cconv (E).

ω

Then {f } ∪ F is evaluating. In particular, RCconv (E) (see (18.7.2)) is ω-evaluating

for every Banach space E.

Proof. Let (pn )n∈N be a sequence in P (E) and (kn )n∈N a sequence of odd natural

numbers with k1 = 1 and kn+1 > 2kn (1 + deg pn ) for n ∈ N. Then |pkn (x)| ¤

n

kn kn deg pn

·x for every x ∈ E. Set

pn

∞

1 1 1

· n · 2kn deg pn (pkn ’ •(pkn ))2 ,

g := n n

»n 2 n

n=1

where (»n )n∈N is a sequence of reals with

2kn

+ 2|•(pkn )| · pn kn

+ (•(p2kn ))2 for all n ∈ N.

» n > pn n n

Then

∞

1 1 kn 2 deg pn kn deg pn

g(x) ¤ · 2kn deg pn x +x +1

2n n

n=1

∞

1 x x

2kn deg pn kn deg pn

¤ + 1 < ∞ for all x ∈ E

+

2n n n

n=1

18.20

198 Chapter IV. Smoothly realcompact spaces 18.23

ω

Since g is pointwise convergent, it is a function in Cconv (E). By the technique used

in (18.15) we obtain that there exists x ∈ E with •(f ) = f (x) and •(pkn ) = pkn (x)

n n

for all n ∈ N. As for each n ∈ N the number kn is odd, it follows that •(pn ) = pn (x)

for all n ∈ N. Since each g ∈ F is a sum n∈N pn,g of homogeneous polynomials

pn,g ∈ P (E) of degree n for n ∈ N, there exists x ∈ E with •(g) = g(x) for all

g ∈ F, and •(pn,g ) = pn,g (x) for all n ∈ N, whence •(g) = n∈N •(pn,g ) for all

g ∈ F. Let a ∈ E with •(f ) = f (a) and •(pn,g ) = pn,g (a) for all n ∈ N and all

g ∈ F. Then

pn,g (a) = g(a) for all g ∈ F.

•(g) = •(pn,g ) =

n∈N n∈N

18.21. Result. [Adam, Bistr¨m, Kriegl, 1995, 2.1]. Given two in¬nite cardinals

o

m < n, let E := {x ∈ Rn : | supp x| ¤ m} Then for any algebra A ⊆ C(E),

containing the natural projections (prγ )γ∈n , there is a homomorphism • on A that

is m-evaluating but not n-evaluating.

Evaluating Homomorphisms

18.22. Proposition. [Garrido, G´mez, Jaramillo, 1994, 1.7]. Let X be a closed

o

subspace of a product R“ . Let A ⊆ C(X) be a subalgebra containing the projections

prγ |X : X ⊆ R“ ’ R, and let • ∈ Hom A be ¯ 1-evaluating.

Then • is A-evaluating.

Proof. Set aγ = •(prγ |X ). Then the point a = (aγ )γ∈“ is an element in X.

Otherwise, since X is closed there exists a ¬nite set J ⊆ “ and µ > 0 such that

no point y with |yγ ’ aγ | < µ for all γ ∈ J is contained in X. Set p(x) :=

2

γ∈J (prγ (x) ’ aγ ) for x ∈ X. Then p ∈ A and •(p) = 0. By assumption there is

an x ∈ X, such that |•(p) ’ p(x)| < µ2 , but then | prγ (x) ’ aγ | < µ for all γ ∈ J, a

contradiction. Thus a ∈ X and •(g) = g(a) for all g in the algebra A0 generated

by all functions prγ |X .

ˇ

By the assumption and by (18.2) there exists a point x in the Stone-Cech compact-

˜

˜x ˜

i¬cation βX such that •(f ) = f (˜) for all f ∈ A, where f is the unique continuous

extension βX ’ R∞ of f . We claim that x = a. This holds if x ∈ X since the prγ

˜ ˜

separate points on X. So let x ∈ βX \ X. Then x is the limit of an ultra¬lter U

˜ ˜

in X. Since U does not converge to a, there is a neighborhood of a in X, without

loss of generality of the form U = {x ∈ X : f (x) > 0} for some f ∈ A0 . But then

˜x

the complement of U is in the ultra¬lter U, thus f (˜) ¤ 0. But this contradicts

˜x

f (˜) = •(f ) = f (a) for all f ∈ A0 .

18.23. Corollary. [Kriegl, Michor, 1993, 1]. If A is ¬nitely generated then each

1-evaluating • ∈ Hom A is evaluating.

Finitely generated can even be meant in the sense of C ∞ -algebra, see the proof.

This applies to the algebras RP , C ω , Cconv and C ∞ on Rn (or a closed submanifold

ω

of Rn ).

18.23

18.26 18. Evaluation properties of homomorphisms 199

Proof. Let F ⊆ A be a ¬nite subset which generates A in the sense that A ⊆

F ∞ := ( F Alg ) ∞ , compare (18.7.3). By (18.7) again we have that • restricted

to F Alg extends to • ∈ Hom F ∞ by •(h —¦ (f1 , . . . , fn )) = h(•(f1 ), . . . , •(fn ))

˜

for fi ∈ F, h ∈ C ∞ (U, R) where (f1 , . . . , fn )(X) ⊆ U and U is open in Rn . For

f ∈ A there exists x ∈ X such that • = evx on f and on F, which implies that

•(f ) = f (x) = •(f ). Finally note that if • = evx on F then • = evx on F ∞ ,

˜ ˜

thus • = evx on A.

18.24. Proposition. [Bistr¨m, Bjon, Lindstr¨m, 1992, Prop.4]. Let • ∈ Hom A

o o

be ω-evaluating and X be Lindel¨f (for some topology ¬ner than XA ).

o

Then • is evaluating.

This applies to any ω-evaluating algebra on a separable Fr´chet space, [Arias-de-

e

Reyna, 1988, 8].

∞

It applies also to A = Clfcs (E) for any weakly Lindel¨f space by (18.27). In par-

o

ticular, for 1 < p ¤ ∞ the space p (“) is weakly Lindel¨f by (18.18.1) as weak— -

o

1

dual of the normed space q with q := 1/(1 ’ p ) and the same holds for the

spaces ( 1 (“), σ( 1 (“), c0 (“))). Furthermore it is true for ( 1 (“), σ( 1 (“), ∞ (“)))

by [Edgar, 1979], and for (c0 (“), σ(c0 (“), 1 (“))) by [Corson, 1961, p.5].

Proof. By the sequentially evaluating property of A the family (Zf )f ∈A of closed

sets Zf = {x ∈ X : f (x) = •(f )} has the countable intersection property. Since X

is Lindel¨f, the intersection of all sets in this collection is non-empty. Thus • is a

o

point evaluation with a point in this intersection.

18.25. Proposition. Let A be an algebra which contains a countable point-

separating subset.

Then every ω-evaluating • in Hom A is A is evaluating.

If a Banach space E has weak— -separable dual and D ⊆ E is countable and weak— -

dense, then D is point-separating, since for x = 0 there is some ∈ E with (x) = 1

and since {x ∈ E : x (x) > 0} is open in the weak— -topology also an ∈ D with

(x) > 0. The converse is true as well, see [Bistr¨m, 1993, p.28].

o

Thus (18.25) applies to all Banach-spaces with weak— -separable dual and the alge-

bras RP , C ω , RCconv , C ∞ .

ω

Proof. Let {fn }n be a countable subset of A separating the points of X. Let

f ∈ A. Since A is ω-evaluating there exists a point xf ∈ X with f (xf ) = •(f )

and fn (xf ) = •(fn ). Since the fn are point-separating this point xf is uniquely

determined and hence independent on f ∈ A.

18.26. Proposition. [Arias-de-Reyna,1988, Thm.8] for C m on separable Banach

spaces; [G´mez, Llavona, 1988, Thm.1] for ω-evaluating algebras on locally convex

o

spaces with w— -separable dual; [Adam, 1993, 6.40]. Let E be a convenient vector

space, let A ⊇ P be an algebra containing a point separating bounded sequence of

homogeneous polynomials of ¬xed degree.

18.26

200 Chapter IV. Smoothly realcompact spaces 18.29

Then each 1-evaluating homomorphism is evaluating.

In particular this applies to c0 and p for 1 ¤ p ¤ ∞. It also applies to a dual of a

separable Fr´chet space, since then any dense countable subset of E can be made

e

equicontinuous on E by [Bistr¨m, 1993, 4.13].

o

Proof. Let {pn : n ∈ N} be a point-separating bounded sequence. By the polar-

ization formulas given in (7.13) this is equivalent to boundedness of the associated

multilinear symmetric mappings, hence {pn : n ∈ N} satis¬es the assumptions of

(18.15) and thus {pn : n ∈ N} is evaluated. Now the result follows as in (18.25).

18.27. Theorem. [Adam, Bistr¨m, Kriegl, 1995, 5.1]. A locally convex space E

o

∞

is weakly realcompact if and only if E = Homω Pf (E)(= Hom Clfcs (E)).

∞

Proof. By (18.14) we have Homω Pf (E) = Hom Clfcs (E).

— —

(’) Let E be weakly realcompact. Since E is σ(E , E )-dense in E (see [Jarchow,

∞

1981, 8.1.4]), it follows from (18.19) that any • ∈ Homω Pf (E) = Hom Clfcs (E) is

E -evaluating and hence also evaluating on the algebra Pf (E) generated by E .

(⇐) By (17.4) the space Homω (Pf (E)) is realcompact in the topology of pointwise

convergence. Since E = Homω Pf (E) and σ(E, E ) equals the topology of pointwise

convergence on Homω (Pf (E)), we have that (E, σ(E, E )) is realcompact.

18.28. Proposition. [Bistr¨m, Jaramillo, Lindstr¨m, 1995, Thm.13]. Let E be a

o o

Banach space with the Dunford-Pettis property that does not contain a copy of 1 .

Then Pf (E) is dense in P (E) for the topology of uniform convergence on bounded

sets.

A Banach space E is said to have the Dunford-Pettis property [Diestel, 1984, p.113]

if x— ’ 0 in σ(E , E )) and xn ’ 0 in σ(E, E ) implies x— (xn ) ’ 0. Well known

n n

1

Banach spaces with the Dunford-Pettis property are L (µ), C(K) for any compact

K, and ∞ (“) for any “. Furthermore c0 (“) and 1 (“) belong to this class since

if E has the Dunford-Pettis property then also E has. According to [Aron, 1976,

p.215], the space 1 is not contained in C(K) if and only if K is dispersed, i.e.

K (±) = … for some ±, or equivalently whenever its closed subsets admit isolated

points.

Proof. According to [Carne, Cole, Gamelin, 1989, theorem 7.1], the restriction of

any p ∈ P (E) to a weakly compact set is weakly continuous if E has the Dunford-

Pettis property and, consequently, sequentially weakly continuous. By [Llavona,

1986, theorems 4.4.7 and 4.5.9], such a polynomial p is weakly uniformly continuous

on bounded sets if E, in addition, does not contain a copy of 1 . The assertion

therefore follows from [Llavona, 1986, theorem 4.3.7].

18.29. Theorem. [Garrido, G´mez, Jaramillo, 1994, 2.4] & [Adam, Bistr¨m,

o o

Kriegl, 1995, 3.4]. Let E be 2n (“) for some n and some “ of non-measurable

cardinality. Let P (E) ⊆ A ⊆ C(E).

18.29

18.30 18. Evaluation properties of homomorphisms 201

Then every 1-evaluating homomorphism • is evaluating.

Proof. For f ∈ A let Af be the algebra generated by f and all i-homogeneous

polynomials in P (E) with degree i ¤ 4n + 2. Take a sequence (pn ) of continuous

polynomials with degree i ¤ 2n + 1. Then there is a sequence (tn ) in R+ such

that {tn pn : n ∈ N} is bounded, hence • is by (18.15) evaluating on it, i.e. • is

ω-evaluating on Af .

∞

Given z = (zγ ) ∈ (“) and x ∈ E, set

fz,j (x) := f (x)j + zγ prγ (x)(2n+1)j ,

γ∈“

where j = 1, 2. Then fz,j ∈ Af and we can apply (18.17). Thus there is a point

xf ∈ E with •(f ) = f (xf ) and •(prγ )2n+1 = prγ (xf )2n+1 for all γ ∈ “. Hence

•(prγ ) = prγ (xf ), and since (prγ )γ∈“ is point separating, xf is uniquely determined

and thus not depending on f and we are ¬nished.

18.30. Proposition. Let E = c0 (“) with “ non-measurable. If one of the follow-

ing conditions is satis¬ed, then • is evaluating:

∞

(1) [Bistr¨m, 1993, 2.22] & [Adam, Bistr¨m, Kriegl, 1995, 5.4]. Clfs (E) ⊆ A

o o

and • is ω-evaluating.

(2) [Garrido, G´mez, Jaramillo, 1994, 2.7]. P (E) ⊆ A, every f ∈ A depends

o

only on countably many coordinates and • is 1-evaluating.

Proof. (1) Since • is ω-eval, it follows that (•(prγ ))γ∈“ ∈ c0 (“), where prγ :

c0 (“) ’ R are the natural coordinate projections (see the proof of (18.17)). Fix n

and consider the function fn : c0 (“) ’ R de¬ned by the locally ¬nite product

h n · (prγ (x) ’ •(prγ )) ,

fn (x) :=

γ∈“

where h ∈ C ∞ (R, [0, 1]) is chosen such that h(t) = 1 for |t| ¤ 1/2 and h(t) = 0 for

|t| ≥ 1. Note that a locally ¬nite product f := i∈I fi (i.e. locally only ¬nitely

many factors fi are unequal to 1) can be written as locally ¬nite sum f = J gJ ,

where gi := fi ’ 1 and for ¬nite subsets J ⊆ I let gJ := j∈J gj ∈ A and the index

J runs through all ¬nite subsets of I. Since I is at least countable, the set of these

indices has the same cardinality as I has.

By means of (18.17) •(fn ) = γ∈“ h(0) = 1 for all n. Now let f ∈ A. Then there

exists a xf ∈ E with •(f ) = f (xf ) and 1 = •(fn ) = fn (xf ). Hence |n · (prγ (xf ) ’

•(prγ ))| ¤ 1 for all n, i.e. prγ (xf ) = •(prγ ) for all γ ∈ “. Since (prγ )γ∈“ is point

separating, the point xf ∈ E is unique and thus does not depend on f .

(2) By (18.15) or (18.16) the restriction of • to the algebra generated by {prγ :

γ ∈ “} is ω-evaluating. Since c0 (K) is weakly-realcompact by [Corson, 1961] for

locally compact metrizable K and hence in particular for discrete K, we have by

(18.19) that • is evaluating on this algebra, i.e. there exists a = (aγ )γ∈“ ∈ E with

aγ = prγ (a) = •(prγ ) for all γ ∈ “.

18.30

202 Chapter IV. Smoothly realcompact spaces 18.32

Every f ∈ A(E) depends only on countably many coordinates, i.e. there exists a

˜ ˜

countable “f ⊆ “ and a function f : c0 (“f ) ’ R with f —¦ pr“f = f . Let

Af := {g ∈ Rc0 (“f ) : g —¦ pr“f ∈ A}

and let • : Af ’ R be given by • = • —¦ pr“f . Since “f is countable there is by

˜ ˜

˜˜ ˜

(18.15) an xf ∈ c0 (“f ) with •(f ) = f (xf ) and aγ = •(prγ ) = •(prγ ) = prγ (xf ) =

˜

xf for all γ ∈ “f . Thus pr“f (a) = x and

γ

˜ ˜˜ ˜

•(f ) = •(f —¦ pr“f ) = •(f ) = f (pr“f (a)) = f (a).

18.31. Proposition. [Garrido, G´mez, Jaramillo, 1994, 2.7]. Each f ∈ C ω (c0 (“))

o

depends only on countably many coordinates.

Proof. Let f : c0 (“) ’ R be real analytic. So there is a ball Bµ (0) ⊆ c0 (“) such

∞

that f (x) = n=1 pn (x) for all x ∈ Br (0), where pn ∈ Ln (c0 (“); R) for all n ∈ N.

sym

By (18.28) the space Pf (c0 (“)) is dense in P (c0 (“)) for the topology of uniform

convergence on bounded sets, since c0 (“) has the Dunford-Pettis property and does

not contain 1 as topological linear subspace. Thus we have that for any n, k ∈ N

there is some qnk ∈ Pf (c0 (“)) with

1

sup{|pn (x) ’ qnk (x)| : x ∈ Bµ (0)} < .

k

Since each q ∈ Pf (c0 (“)) is a polynomial form in elements of 1 (“), there is a count-

able set Λnk ⊆ “ such that qnk only depends on the coordinates with index in Λnk ,

whence pn on Bµ (0) only depends on coordinates with index in Λn := k∈N Λnk .

Set Λ := n∈N Λn and let ιΛ : c0 (Λ) ’ c0 (“) denote the natural injection given by

(ιΛ (x))γ = xγ if γ ∈ Λ and (ιΛ (x))γ = 0 otherwise. By construction f = f —¦ ιΛ —¦ prΛ

on Bµ (0). Since both functions are real analytic and agree on Bµ (0), they also agree

on c0 (“).

18.32. Example. [Garrido, G´mez, Jaramillo, 1994, 2.6]. For uncountable “ the

o

space c0 (“) \ {0} is not C ω -realcompact.

But for non-measurable “ the whole space c0 (“) is C ω -evaluating by (18.30) and

(18.31).

Proof. Let „¦ := c0 (“) \ {0}, let f : „¦ ’ R be real analytic and consider any

sequence (um )m∈N in „¦ with um ’ 0. For each m ∈ N there exists µm > 0 and

n

homogeneous Pm in P (c0 (“)) of degree n for all n, such that, for h < µm

∞

f (um + h) = f (um ) + n

Pm (h).

n=1

n

As carried out in (18.31), each Pm only depends on coordinates with index in some

countable set Λn ⊆ “. The set Λ := ( n,m∈N Λn ) ∪ ( m∈N supp um ) is countable.

m m

18.32

19.1 19. Stability of smoothly realcompact spaces 203

Let γ ∈ “ \ Λ. Then, since Pm (eγ ) = 0 and um + teγ = 0 for all m, n ∈ N and

n

t ∈ R, we get f (um + teγ ) = f (um ) for all |t| < µm . Thus f (um + teγ ) = f (um ) for

every t ∈ R, since the function t ’ f (um + teγ ) is real analytic on R. In particular,

f (um + eγ ) = f (um ) and, since um + eγ ’ eγ , there exists

•(f ) := lim f (um ) = lim f (um + eγ ) = f (eγ ).

m∈N m∈N

Then • is an algebra homomorphism, since a common γ can be found for ¬nitely

many f . And since 1 (“) ⊆ C ω („¦) is point separating the homomorphism • cannot

be an evaluation at some point of „¦.

18.33. Example. [Bistr¨m, Jaramillo, Lindstr¨m, 1995, Prop.16]. The algebra

o o

∞

ω

Cconv ( ) is not 1-evaluating.

Proof. Suppose that Cconv ( ∞ ) is 1-evaluating. By (20.3) the unit ball Bc0 of c0

ω

is Cconv -bounding in ∞ . By (18.20) the algebra Cconv ( ∞ ) is ω-evaluating and,

ω ω

since ( ∞ ) admits a point separating sequence, we have ∞ = Hom(Cconv ( ∞ )) by

ω

(18.25). Hence by (20.2), every Cconv -bounding set in ∞ is relatively compact in

ω

the initial topology induced by Cconv ( ∞ ) and in particular relatively σ( ∞ , ( ∞ ) )-

ω

compact. Therefore, since the topologies σ(c0 , 1 ) and σ( ∞ , ( ∞ ) ) coincide on c0 ,

we have that Bc0 is σ(c0 , 1 )-compact, which contradicts the non-re¬‚exivity of c0

by by [Jarchow, 1981, 11.4.4].

19. Stability of Smoothly Realcompact Spaces

In this section stability of evaluation properties along certain mappings are studied

which furnish some large classes of smoothly realcompact spaces.

19.1. Proposition. Let AX and AY be algebras of functions on sets X and Y as

in (17.1), let T : X ’ Y be injective with T — (AY ) ⊆ AX , and suppose that Y is