Proof. We ¬rst show closedness under local smooth functions (and hence in partic-

ular under inversion), i.e. if h ∈ C ∞ (U ), where U ⊆ Rn is open and f := (f1 , . . . , fn )

with fi ∈ A has values in U , then h —¦ f ∈ A .

Consider a smooth partition of unity {hj : j ∈ N} of U , such that supp hj ⊆ U .

Then hj ·h is a smooth function on Rn vanishing outside supp hj . Hence (hj ·h)—¦f ∈

A . Since we have

carr (hj · h) —¦ f ⊆ f ’1 (carr hj ),

18.11

18.12 18. Evaluation properties of homomorphisms 193

the family {carr((hj · h) —¦ f ) : j ∈ N} is locally ¬nite, f is continuous, and since

1 = j∈N hj on U we obtain that h —¦ f = j∈N (hj · h) —¦ f ∈ A .

By (18.6) we have that • is 1-evaluating, hence ¬nitely evaluating by (18.8). We

now show that • is countably evaluating:

For this take a sequence (fn )n in A. Then hn : x ’ (fn (x) ’ •(fn ))2 belongs to

A and •(hn ) = 0. We have to show that there exists an x ∈ X with hn (x) = 0

for all n. Assume that this were not true, i.e. for all x ∈ X there exists an n with

hn (x) > 0. Take h ∈ C ∞ (R, [0, 1]) with carr h = {t : t > 0} and let gn : x ’

h(hn (x)) · h( n ’ h1 (x)) · . . . · h( n ’ hn’1 (x)). Then gn ∈ A and the sum n 21 gn

1 1

n

is locally ¬nite, hence de¬nes a function g ∈ A. Since • is 1-evaluating there exists

for any n an xn ∈ X with hn (xn ) = •(hn ) = 0 and •(gn ) = gn (xn ). Hence

1 1

•(gn ) = gn (xn ) = h(hn (xn )) · h( n ’ h1 (xn )) · . . . · h( n ’ hn’1 (xn )) = 0.

By assumption on the hn and h we have that g > 0. Hence by (18.9) •(g) > 0,

since • is 1-evaluating. Let N be so large that 1/2N < •(g). Again since A is

1-evaluating, there is some a ∈ X such that •(g) = g(a) and •(gj ) = gj (a) for

j ¤ N . Then

1 1 1 1 1

gn (a) ¤ 0 + N

< •(g) = g(a) = gn (a) = •(gn ) +

2N 2n 2n 2n 2

n n>N

n¤N

gives a contradiction.

18.12. Counter-example. [Bistr¨m, Jaramillo, Lindstr¨m, 1995, Prop.17]. For

o o

any non-re¬‚exive weakly realcompact locally convex space (and any non-re¬‚exive

Banach space) E the algebra Pf (E) of ¬nite type polynomials is not ω-evaluating.

Moreover, EA is realcompact, but E is not A-realcompact, for A = Pf (E), so that

the converse of the assertion in (17.4) holds only under the additional assumptions

of (17.6).

1

As example we may take E = , which is non-re¬‚exive, but by (18.27) weakly

realcompact.

By (18.10) the algebra Pf (E) is 1-evaluating and hence by (18.7) it has the same

homomorphisms as RPf (E), Pf (E)∞ or even Pf (E) ∞ . So these algebras are not

ω-evaluating for spaces E as above.

Proof. By the universal property (5.10) of Pf (E) we get Hom Pf (E) ∼ (E )— , the

=

space of (not necessarily bounded) linear functionals on E . For weakly realcompact

E by (18.27) we have Homω Pf (E) = E. So if Pf (E) were ω-evaluating then even

E = Hom Pf (E). Any bounded subset of E is obviously Pf -bounding and hence

by (20.2) relatively compact in the weak topology, since EPf (E) = (E, σ(E, E )).

Since E is not semi-re¬‚exive, this is a contradiction, see [Jarchow, 1981, 11.4.1].

If we have a (not necessarily weakly compact) Banach space, we can replace in the

argument above (20.2) by the following version given in [Bistr¨m, 1993, 5.10]: If

o

Homω Pf (E) = Hom Pf (E) then every A-bounding set with complete closed convex

hull is relatively compact in the weak topology.

18.12

194 Chapter IV. Smoothly realcompact spaces 18.15

18.13. Lemma. The Clfs -algebra A∞ generated by an algebra A can be obtained

∞

lfs

in two steps as (A )lfs . Also the Clfcs -algebra A∞ can be obtained in two steps as

∞ ∞

lfcs

∞

(A )lfcs .

Proof. We prove the result only for countable sums, the general case is easier. We

have to show that (A∞ )lfcs is closed under composition with smooth mappings. So

take h ∈ C ∞ (Rn ) and j≥1 fi,j ∈ (A∞ )lfcs for i = 1, . . . , n. We put h0 := 0 and

k k

fn,j ) ∈ A∞ and obtain

hk := h —¦ ( f1,j , . . . ,

j=1 j=1

h—¦( (hk ’ hk’1 ),

f1,j , . . . , fn,j ) =

j≥1 j≥1 k≥1

where the right member is locally ¬nite and hence an element of (A∞ )lfcs .

18.14. Theorem. [Adam, Bistr¨m, Kriegl, 1995, 4.3]. A homomorphism • in

o

Hom A is ω-evaluating if and only if • extends (uniquely) to an algebra homomor-

phism on the Clfcs -algebra A∞ generated by A, which can be obtained in two steps

∞

lfcs

∞

as (A )lfcs (and this extension is ω-evaluating by (18.11)).

Proof. (’) The algebra A∞ is the union of the algebras obtained by a ¬nite

lfcs

iteration of passing to Alf cs and A∞ , where Alfcs := {f : f = n fn , fn ∈

A, the sum is locally ¬nite}. To A∞ it extends by (18.3). It is countably eval-

uating there, since in any f ∈ A∞ only ¬nitely many elements of A are involved.

Remains to show that • can be extended to Alf cs and that this extension is also

countably evaluating.

For a locally ¬nite sum f = k fk we de¬ne •(f ) := k •(fk ). This makes sense,

since there exists an x ∈ X with •(fn ) = fn (x), and since n fn is point ¬nite, we

have that the sum n •(fn ) = n fn (x) is in fact ¬nite. It is well de¬ned, since for

n gn we can choose an x ∈ X with •(fn ) = fn (x) and •(gn ) = gn (x) for

n fn =

all n, and hence n •(fn ) = n fn (x) = n gn (x) = n •(gn ). The extension is

a homomorphism, since for the product for example we have

• fn gk =• fn gk = •(fn gk ) =

n k n,k n,k

= •(fn ) •(gk ) = •(fn ) •(gk ) .

n

n,k k

Remains to show that the extension is countably evaluating. So let f k = n fn be

k

k k

given. By assumption there exists an x such that •(fn ) = fn (x) for all n and all

k. Thus •(f k ) = n •(fn ) = n fn (x) = f k (x) for all k.

k k

(⇐) Since A∞cs is a Clfcs -algebra we conclude from (18.11) that the extension of •

∞

lf

is countably evaluating.

18.15. Proposition. [Garrido, G´mez, Jaramillo, 1994, 1.10]. Let • in Hom A

o

j 1

be 1-evaluating, and let fn ∈ A be such that n »n fn ∈ A for all » ∈ and

j ∈ {1, 2}.

18.15

18.17 18. Evaluation properties of homomorphisms 195

Then • is {fn : n ∈ N}-evaluating.

For a convenient algebra structure on A and {fn : n ∈ N} bounded in A the second

condition holds, as used in (18.26).

It would be interesting to know if the assumption for j = 2 can be removed, since

then the application in (18.26) to ¬nite type polynomials would work.

Proof. Choose a positive absolutely summable sequence (»n )n∈N such that the

sequences (»n •(fn ))n∈N and (»n •(fn )2 )n∈N are summable. Then the sum

∞

»j (fj ’ •(fj ))2 ∈ A.

g :=

j=1

If there exists x ∈ X with g(x) = 0, we are done. If not, then consider the (positive)

function

∞

1

»j (fj ’ •(fj ))2 ∈ A.

h :=

2j

j=1

For every n ∈ N there exists xn ∈ X such that •(fj ) = fj (xn ) for all j ¤ n,

•(g) = g(xn ) and •(h) = h(xn ). But then for all n ∈ N we have by (18.9) that

0 < 2n •(h) = 2n’j »j •(fj ’ •(fj ))2 ¤ »j •(fj ’ •(fj ))2 = •(g),

j>n j>n

a contradiction.

18.16. Corollary. [Bistr¨m, Jaramillo, Lindstr¨m, 1995, Prop.9]. Let E be a

o o

Banach space and A a 1-evaluating algebra containing P (E). Then for each • ∈

Hom A, each f ∈ A, and each sequence (pn )n∈N in P (E) with uniformly bounded

degree, there exists a ∈ E with •(f ) = f (a) and •(pn ) = pn (a) for all n ∈ N.

Proof. Let (»n )n∈N be a sequence of positive reals such that {»n pn : n ∈ N} is

bounded. Then by (18.15) the set {f, pn } is evaluated.

18.17. Theorem. [Adam, Bistr¨m, Kriegl, 1995, 3.3]. Let (fγ )γ∈“ be a family in

o

A such that γ∈“ zγ fγ is a pointwise convergent sum in A for all z = (zγ ) ∈ ∞ (“)

j

and j = 1, 2. Let |“| be non-measurable, and let • be ω-evaluating.

Then • is {fγ : γ ∈ “}-evaluating.

We will apply this in particular if {fγ : γ ∈ “} is locally ¬nite, and A stable

under locally ¬nite sums. Note that we can always add ¬nitely many f ∈ A to

{fγ : γ ∈ “}.

Again it would be nice to get rid of the assumption for j = 2.

Proof. Let x ∈ X and set zγ := sign(fγ (x)) for all γ ∈ “. Then z = (zγ ) ∈ ∞ (“)

1

and γ∈“ |fγ (x)| = γ∈“ zγ fγ (x) < ∞, i.e. (fγ (x))γ∈“ ∈ (“). Next observe

that (•(fγ ))γ∈“ ∈ c0 (“), since otherwise there exists some µ > 0 and a countable

18.17

196 Chapter IV. Smoothly realcompact spaces 18.18

set Λ ⊆ “ with |•(fγ )| ≥ µ for each γ ∈ Λ. By the countably evaluating property

of • there is a point x ∈ X with |fγ (x)| = |•(f» | ≥ µ for each γ ∈ Λ, violating the

condition (fγ (x))γ∈“ ∈ 1 (“). Since as a vector in c0 (“) it has countable support

and since • is countably evaluating we get even (•(fγ ))γ∈“ ∈ 1 (“). Therefore we

may consider g, de¬ned by

x ’ g(x) := (fγ (x) ’ •(fγ ))2 1

∈

X (“).

γ∈“

This gives a map g — : ∞ 1

(“) ’ A, by

(“) =

g — (z) : x ’ z, g(x) = zγ · (fγ (x) ’ •(fγ ))2 ,

γ∈“

since (•(fγ )γ∈“ ∈ 1 (“). Let ¦ : ∞ (“) ’ R be the linear map ¦ := • —¦ g — :

∞

(“) ’ A ’ R. By the countably evaluating property of •, for any sequence

(zn ) in ∞ (“) there exists an x ∈ X such that ¦(zn ) = •(g — (zn )) = g — (zn )(x) =

zn , g(x) for all n. For non-measurable |“| the weak topology on 1 (“) is realcom-

pact by [Edgar, 1979, p.575]. By (18.19) there exists a point c ∈ 1 (“) such that

¦(z) = z, c for all z ∈ ∞ (“). For each standard unit vector eγ ∈ ∞ (“) we have

0 = ¦(eγ ) = eγ , c = cγ . Hence c = 0 and therefore ¦ = 0. For the constant

vector 1 in ∞ (“), we get 0 = ¦(1) = •(g — (1)). Since • is 1-evaluating there exists

an a ∈ X with •(g — (1)) = g — (1)(a) = 1, g(a) = γ∈“ (fγ (a) ’ •(fγ ))2 , hence

•(fγ ) = fγ (a) for each γ ∈ “.

18.18. Valdivia gives in [Valdivia, 1982] a characterization of the locally convex

spaces which are realcompact in their weak topologies. Let us mention some classes

of spaces that are weakly realcompact:

Result.

(1) All locally convex spaces E with σ(E , E)-separable E .

(2) All weakly Lindel¨f locally convex spaces, and hence in particular all weakly

o

countably determined Banach spaces, see [Vaˇ´k, 1981]. In particular this

sa

applies to c0 (X) for locally compact metrizable X by [Corson, 1961, p.5].

(3) The Banach spaces E with angelic weak— dual unit ball [Edgar, 1979, p.564].

Note that (E — , weak— ) is angelic :” for B ⊆ E — bounded the weak— -closure

is obtained by weak— -convergent sequences in B, i.e. sequentially for the

weak— -topology.

(4) 1 (“) for |“| non-measurable. Furthermore the spaces C[0, 1], ∞ , L∞ [0, 1],

the space JL of [Johnson, Lindenstrauss, 1974] (a short exact sequence

c0 ’ JL ’ 2 (“) exists), the space D[0, 1] or right-continuous functions

having left sided limits, by [Edgar, 1979, p.575] and [Edgar, 1977]. All these

spaces are not weakly Lindel¨f.o

(5) All closed subspaces of products of the spaces listed above.

(6) Not weakly realcompact are C[0, ω1 ] and ∞ [0, 1], the space of bounded

count

functions on [0, 1] with countable support, by [Edgar, 1979].

18.18

18.20 18. Evaluation properties of homomorphisms 197

18.19. Lemma. [Corson, 1961]. If E is a weakly realcompact locally convex space,

then every linear countably evaluating ¦ : E ’ R is given by a point-evaluation

evx on E with x ∈ E.

Proof. Since ¦ : E ’ R is countably evaluating it is linear and F := {ZK : K ⊆

E countable} does not contain the empty set and generates a ¬lter. We claim that

this ¬lter is Cauchy with respect to the uniformity de¬ned by the weakly continuous

real functions on E:

To see this, let f : E ’ R be weakly continuous. For each r ∈ R, let Lr := {x ∈

E : f (x) < r} and similarly Ur := {x ∈ E : f (x) > r}. By [Jarchow, 1981, 8.1.4]

— —

we have that E is σ(E , E )-dense in E . Thus there are open disjoint subsets

—

˜ ˜

Lr and Ur on E having trace Lr and Ur on E (take the complements of the

closures of the complements). Let B ⊆ E be an algebraic basis of E . Then the

— —

map χ : E ’ RB , l ’ (l(x ))x ∈B is a topological isomorphism for σ(E , E ).

By [Bockstein, 1948] there exists a countable subset Kr ⊆ B ⊆ E , such that the

˜ ˜

images under prKr : RB ’ RKr of the open sets Lr and Ur are disjoint. Let

K = r∈Q Kr . For µ > 0 we have that ZK — ZK ⊆ {(x1 , x2 ) : f (x1 ) = f (x2 )} ⊆

{(x1 , x2 ) : |f (x1 ) ’ f (x2 )| < µ}, i.e. the ¬lter generated by F is Cauchy. In fact, let

x1 , x2 ∈ ZK . Then x (x1 ) = •(x ) = x (x2 ) for all x ∈ K. Suppose f (x1 ) = f (x2 ).

Without loss of generality we ¬nd a r ∈ Q with f (x1 ) < r < f (x2 ), i.e. x1 ∈ Lr

and x2 ∈ Ur . But then x (x1 ) = x (x2 ) for all x ∈ Kr ⊆ K gives a contradiction.

By realcompactness of (E, σ(E, E )) the uniform structure generated by the weakly

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

AY -realcompact. Then we have:

(1) [Jaramillo, 1992, 5]. If AX is 1-evaluating and AY is 1-isolating on Y , then

X is AX -realcompact and AX is 1-isolating on X.

(2) [Bistr¨m, Lindstr¨m, 1993a, Thm.2]. If AX is ω-evaluating and AY is

o o

ω-isolating on Y , then X is AX -realcompact and AX is ω-isolating on X.

We say that AX is 1-isolating on X if for every x ∈ X there is an f ∈ AX with

{x} = f ’1 (f (x)).

Similarly AX is called ω-isolating on X if for every x ∈ X there exists a sequence

’1

(fn )n in AX such that {x} = n fn (fn (x)). This was called A-countably sepa-

rated in [Bistr¨m, Lindstr¨m, 1993a].

o o

Proof. There is a point y ∈ Y with ψ = evy . Let G ⊆ AY be such that

{y} = g∈G g ’1 (g(y)), where G is either a single function or countably many

functions. Let f ∈ AX be arbitrary. By assumption there exists xf ∈ X with

19.1

204 Chapter IV. Smoothly realcompact spaces 19.4

•(f ) = f (xf ) and •(T — (g)) = T — (g)(xf ) for all g ∈ G. Since g(y) = ψ(g) =

•(T — (g) = T — (g)(xf ) = g(T (xf )) for all g ∈ G, we obtain that y = T (xf ). Since T

is injective, we get that xf does not depend on f , and hence • is evaluating.

19.2. Lemma. If E is a convenient vector space which admits a bounded point-

separating sequence in the dual E then the algebra P (E) of polynomials is 1-

isolating on E.

Proof. Let {xn : n ∈ N} ⊆ E be such a sequence and let a ∈ E be arbitrary. Then

∞

the series x ’ n=1 2’n xn (x ’ a)2 converges in P (E), since xn ( ’a)2 is bounded

∞

and n=1 2’n < ∞. It gives a polynomial which vanishes exactly at a.

19.3. Examples. [Garrido, G´mez, Jaramillo, 1994, 2.4 and 2.5.2]. Any super-

o

re¬‚exive Banach space X of non-measurable cardinality is AX -realcompact, for each

1-isolating and 1-evaluating algebra AX as in (17.1) which contains the algebra of

rational functions RP (X), see (18.7.2).

A Banach-space E is called super-re¬‚exive, if all Banach-spaces F which are ¬nitely

representable in E (i.e. for any ¬nite dimensional subspace F1 and µ > 0 there exists

a isomorphism T : F1 ∼ E1 ⊆ E onto a subspace E1 of E with T · T ’1 ¤ 1 + µ)

=

are re¬‚exive (see [En¬‚o, Lindenstrauss, Pisier, 1975]). This is by [En¬‚o, 1972]

equivalent to the existence of an equivalent uniformly convex norm, i.e. inf{2’ x+

y : x = y = 1, x ’ y ≥ µ} > 0 for all 0 < µ < 2. In [En¬‚o, Lindenstrauss,

Pisier, 1975] it is shown that superre¬‚exivity has the 3-space property.

Proof. A super-re¬‚exive Banach space injects continuously and linearly into p (“)

for some p > 1 and some “ by [John, Torunczyk, Zizler, 1981, p.133] and hence into

some 2n (“). We apply (19.1.1) to the situation X := E ’ 2n (“) =: Y , which is

possible because the algebra P (Y ) is 1-isolating on Y , since the 2n-th power of the

norm is a polynomial and can be used as isolating function. By (18.6) the algebra

RP (Y ) is 1-evaluating, and by (18.29) it is thus evaluating on Y .

19.4. Lemma.

(1) Every 1-isolating algebra is ω-isolating.

(2) If X is A-regular and XA has ¬rst countable topology then A is ω-isolating.

(3) If for a convenient vector space the dual (E , σ(E , E)) is separable then the

algebra Pf (E) of ¬nite type polynomials is ω-isolating on E.

Proof. (1) is trivial.

(2) Let x ∈ X be given and consider a countable neighborhood base (Un )n of x.

Since X is assumed to be A-regular, there exist fn ∈ A with fn (y) = 0 for y ∈ Un

’1

and fn (x) = 1. Thus n fn (fn (x)) = {x}.

(3) Let {xn : n ∈ N} be dense in (E , σ(E , E)) and 0 = x ∈ E. Then there

is some x ∈ E with x (x) = 1. By the denseness there is some n such that

|xn (x) ’ x (x)| < 1 and hence xn (x) > 0. So {0} = n (xn )’1 (0).

19.4

19.8 19. Stability of smoothly realcompact spaces 205

19.5. Example. For “ of non-measurable cardinality, the Banach space E :=

∞

c0 (“) is Clfs (E)-paracompact by (16.15), and hence any ω-evaluating algebra A ⊇

∞

Clfs (E) is ω-isolating and evaluating.

∞

Proof. The Banach space E is Clfs (E)-paracompact by (16.16). By (17.6) the

∞

space E is A-realcompact for any A ⊇ Clfs (E) and is ω-isolating by (19.4.2).

19.6. Example. Let K be a compact space of non-measurable cardinality with

K (ω0 ) = ….

Then the Banach space C(K) is C ∞ -paracompact by (16.20.1), hence C ∞ (C(K))

is ω-isolating and C(K) is C ∞ -realcompact.

Proof. We use the exact sequence

c0 (K \ K ) ∼ {f ∈ C(K) : F |K = 0} ’ C(K) ’ C(K )

=

to obtain that C(K) is C ∞ -paracompact, see (16.19). By (17.6) the space E is

C ∞ -realcompact, is ω-isolating by (19.4.2).

19.7. Example. [Bistr¨m, Lindstr¨m, 1993a, Corr.3bac]. The following locally

o o

∞

convex space are A-realcompact for each ω-evaluating algebra A ⊇ Clfs , if their

cardinality is non-measurable.

(1) Weakly compactly generated (WCG) Banach spaces, in particular separable

Banach spaces and re¬‚exive ones. More generally weakly compactly deter-

mined (WCD) Banach spaces.

(2) C(K) for Valdivia-compact spaces K, i.e. compact subsets K ⊆ R“ with

K © {x ∈ R“ : supp x countable} being dense in K.

(3) The dual of any realcompact Asplund Banach space.

Proof. All three classes of spaces inject continuous and linearly into some c0 (“)

∞

with non-measurable “ by (53.21). Now we apply (19.5) for the algebra Clfs on

c0 (“) to see that the conditions of (19.1.2) for the range space Y = c0 (“) are

satis¬ed. So (19.1.2) implies the result.

19.8. Proposition. Let T : X ’ Y be a closed embedding between topological

spaces equipped with algebras of continuous functions such that T — (AY ) ⊆ AX . Let

• ∈ Hom AX such that ψ := • —¦ T — is AY -evaluating.

(1) [Kriegl, Michor, 1993, 8]. If • is 1-evaluating on AX and AY has 1-small

zerosets on Y then • is AX -evaluating, and AX has 1-small zerosets on X.

(2) [Bistr¨m, Lindstr¨m, 1993b, p.178]. If • is ω-evaluating on AX and AY

o o

has ω-small zerosets on Y then • is AX -evaluating, and AX has ω-small

zerosets on X.

Let m be a cardinal number (often 1 or ω). We say that there are m-small AY -

zerosets on Y or AY has m-small zerosets on Y if for every y ∈ Y and neighborhood

U of y there exists a subset G ⊆ AY with g∈G g ’1 (g(y)) ⊆ U and |G| ¤ m.

19.8

206 Chapter IV. Smoothly realcompact spaces 19.10

For m = 1 this was called large A-carriers in [Kriegl, Michor, 1993], and for m = ω

it was called weakly A-countably separated in [Bistr¨m, Lindstr¨m, 1993b, p.178].

o o

Proof. Let y ∈ Y be a point with ψ = evy . Since Y admits m-small AY -

zerosets there exists for every neighborhood U of y a set G ⊆ AY of functions

with g∈G g ’1 (g(y)) ⊆ U with |G| ¤ m. Let now f ∈ AX be arbitrary. Since AX

is assumed to be m-evaluating, there exists a point xf,U such that f (xf,U ) = •(f )

and g(T (xf,U )) = T — (g)(xf,U ) = •(T — g) = ψ(g) = g(y) for all g ∈ G, hence

T (xf,U ) ∈ U . Thus the net T (xf,U ) converges to y for U ’ y and since T is

a closed embedding there exists a unique x with T (x) = y and x = limU xf,U .

Consequently f (x) = f (limU xf,U ) = limU f (xf,U ) = limU •(f ) = •(f ) since f is

continuous.

The additional assertions are obvious.

19.9. Corollary. [Adam, Bistr¨m, Kriegl, 1995, 5.6]. Let E be a locally convex

o

space, A ⊇ E , and let • ∈ Hom A be ω-evaluating. Assume • is E -evaluating

(this holds if (E, σ(E, E )) is realcompact by (18.27), e.g.). Let E admit ω-small

((E )∞ © A)lfs © A-zerosets. Then • is evaluating on A.

∞

In particular, if E is realcompact in the weak topology and admits ω-small Clfs -

∞

zerosets then E = Homω Clfs (E).

Proof. We may apply (19.8.2) to X = Y := E, AX = A and AY := ((E )∞ ©

A)lfs ©A . Note that • is evaluating on AY by (18.17) and that Clfs (E) = ((E )∞ )lfs

∞

by (18.13).

19.10. Lemma. [Adam, Bistr¨m, Kriegl, 1995, 5.5].

o

(1) If a space is A-regular then it admits 1-small A-zerosets (and in turn also

ω-small A-zerosets).

(2) For any cardinality m, any m-isolating algebra A has m-small A-zerosets.

(3) If a topological space X is ¬rst countable and admits ω-small A-zerosets

then A is ω-isolating.

(4) Any Lindel¨f locally convex space admits ω-small Pf -zerosets.

o

The converse to (1) is false for Pf (E), where E is an in¬nite dimensional separable

Banach space E, see [Adam, Bistr¨m, Kriegl, 1995, 5.5].

o

The converse to (2) is false for Pf (R“ ) with uncountable “, see [Adam, Bistr¨m,

o

Kriegl, 1995, 5.5].

Proof. (1) and (2) are obvious.

(3) Let x ∈ X and U a countable neighborhood basis of x. For every U ∈ U there

is a countable set GU ⊆ A with g∈GU g ’1 (g(y)) ⊆ U . Then G := U ∈U GU is

countable and

g ’1 (g(y)) ⊆ g ’1 (g(y)) ⊆ U = {y}

U ∈U g∈GU U ∈U

g∈G

19.10

19.12 19. Stability of smoothly realcompact spaces 207

(4) Take a point x and an open set U with x ∈ U ⊆ E. For each y ∈ E \ U let

py ∈ E ⊆ Pf (E) with py (x) = 0 and py (y) = 1. Set Vy := {z ∈ E : py (z) > 0}. By

the Lindel¨f property, there is a sequence (yn ) in E \ U such that {U } ∪ {Vyn }n∈N

o

is a cover of E. Hence for each y ∈ E \ U there is some n ∈ N such that y ∈ Vyn ,

i.e. pyn (y) > 0 = pyn (x).

19.11. Theorem. [Kriegl, Michor, 1993] & [Bistr¨m, Lindstr¨m, 1993b, Prop.4].

o o

Let m be 1 or an in¬nite cardinal and let X be a closed subspace of i∈I Xi , let

A be an algebra of functions on X and let Ai be algebras on Xi , respectively, such

that pr— (Ai ) ⊆ A for all i.

i

If each Xi admits m-small Ai -zerosets then X admits m-small A-zerosets.

If in addition • ∈ Hom A is m-eval on A and •i := • —¦ pr— ∈ Hom Ai is evaluating

i

on Ai for all i, then • is evaluating A on X.

Proof. We consider Y := i Xi and the algebra AY generated by i {fi —¦ pri :

fi ∈ AXi }, where prj : i Xi ’ Xj denotes the canonical projection.

Now we prove the ¬rst statement for AY . Let x ∈ Y and U a neighborhood of

x = (xi )i in Y . Thus there exists a neighborhood in i Xi contained in U , which

we may assume to be of the form i Ui with Ui = Xi for all but ¬nitely many

i. Let F be the ¬nite set of those exceptional i. For each i ∈ F we choose a set

Gi ⊆ A with g∈Gi g ’1 (g(xi )) ⊆ Ui . Without loss of generality we may assume

g(xi ) = 0 and g ≥ 0 (replace g by x ’ (g(x) ’ g(xi ))2 ). For any g ∈ i∈F Gi we

de¬ne g ∈ AY by g := i∈F gi —¦ pri ∈ AY . Then g (x) = i∈F gi (x) = 0

˜ ˜ ˜

g ’1 (0) ⊆ U,

˜

Gi

g∈ i∈F

since for z ∈ U we have zi ∈ Ui for at least one i ∈ F. Note that | Gi | ¤ m,

/ / i∈F

since m is either 1 or in¬nite.

That AY is evaluating follows trivially since •i := • —¦ pri — : AXi ’ AX ’ R is an

algebra homomorphism and AXi is evaluating, so there exists a point ai ∈ Xi such

that •(fi —¦ pri ) = (• —¦ pri — )(fi ) = fi (ai ) for all fi ∈ AXi . Let a := (ai )i . Then

obviously every f ∈ AY is evaluated at a.

If now X is a closed subspace of the product Y := Xi then we can apply (19.8.1)

i

and (19.8.2).

19.12. Theorem (19.11) is usually applied as follows. Let U be a zero-neighborhood

basis of a locally convex space E. Then E embeds into U ∈U E(U ) , where E(U )

denotes the completion of the Banach space E(U ) := E/ ker pU , where pU denotes

the Minkowski functional of U . If E is complete, then this is a closed embedding,

and in order to apply (19.11) we have to ¬nd an appropriate basis U and for each

U ∈ U an algebra AU on E(U ) , which pulls back to A along the canonical projections

πU : E ’ E(U ) ⊆ E(U ) , such that the Banach space E(U ) is AU -realcompact and

has m-small AU -zerosets.

19.12

208 Chapter IV. Smoothly realcompact spaces 19.12

Examples.

(1) [Kriegl, Michor, 1993]. A complete, trans-separable (i.e. contained in prod-

uct of separable normed spaces) locally convex space is A-realcompact for

—

every 1-evaluating algebra A ⊇ U πU (Pf ).

Note that for products of separable Banach spaces one has C ∞ = Cc , see

∞

[Adam, 1993, 9.18] & [Kriegl, Michor, 1993].

(2) [Bistr¨m, 1993, 4.5]. A complete, Hilbertizable (i.e. there exists a basis of

o

Hilbert seminorms, in particular nuclear spaces) locally convex space is A-

—

realcompact for every 1-evaluating A ⊇ U πU (P ).

(3) [Bistr¨m, Lindstr¨m, 1993b, Cor.3]. Every complete non-measurable WCG

o o

locally convex space is C ∞ -realcompact.

(4) [Bistr¨m, Lindstr¨m, 1993b, Cor.5]. Any re¬‚exive non-measurable Fr´chet

o o e

space is C ∞ = Cc -realcompact.

∞

(5) [Bistr¨m, Lindstr¨m, 1993b, Cor.4]. Any complete non-measurable infra-

o o

∞

Schwarz space is Cc -realcompact.

(6) [Bistr¨m, 1993, 4.16-4.18]. Every countable coproduct of locally convex

o

spaces, and every countable p -sum or c0 -sum of Banach-spaces injects con-

tinuously into the corresponding product. Thus from A being ω-isolating

and evaluating on each factor, we deduce the same for the total space by

(19.1.2) if A is ω-evaluating on it.

A locally convex space is usually called WCG if there exists a sequence of absolutely

convex, weakly-compact subsets, whose union is dense.

Proof. (1) We have for E(U ) that it is A-realcompact for every 1-evaluating A ⊇

P by (18.26) and Pf is 1-isolating by (19.2) and hence has 1-small zero sets by

(19.10.2).

For a product E of metrizable spaces the two algebras C ∞ (E) and Cc (E) coin-

∞

cide: For every countable subset A of the index set, the corresponding product is

separable and metrizable, hence C ∞ -realcompact. Thus there exists a point xA

in this countable product such that •(f ) = f (xA ) for all f which factor over the

projection to that countable subproduct. Since for A1 ⊆ A2 the projection of xA2

to the product over A1 is just xA1 (use the coordinate projections for f ), there is

a point x in the product, whose projection to the subproduct with index set A is

just xA . Every Mackey continuous function, and in particular every C ∞ -function,

depends only on countable many coordinates, thus factors over the projection to

some subproduct with countable index set A, hence •(f ) = f (xA ) = f (x). This

can be shown by the same proof as for a product of factors R in (4.27), since the

result of [Mazur, 1952] is valid for a product of separable metrizable spaces.

2

(“) is A-realcompact for every 1-evaluating A ⊇ P

(2) By (19.3) we have that

and P is 1-isolating.

(3) For every U the Banach space E(U ) is then WCG, hence as in (19.7.1) there is a

SPRI, and by (53.20) a continuous linear injection into some c0 (“). By (19.5) any

∞

ω-evaluating algebra A on c0 (“) which contains Clfs is evaluating and ω-isolating.

19.12

19.13 19. Stability of smoothly realcompact spaces 209

By (19.1.2) this is true for such stable algebras on E(U ) , and hence by (19.11) for

E.

(4) Here E(U ) embeds into C(K), where K := (U o , σ(E , E )) is Talagrand compact

[Cascales, Orihuela, 1987, theorem 12] and hence Corson compact [Negrepontis,

1984, 6.23]. Thus by (19.7.2) we have PRI. Now we proceed as in (3).

(5) Any complete infra-Schwarz space is a closed subspace of a product of re¬‚exive

and hence WCG Banach spaces, since weakly compact mappings factor over such

spaces by [Jarchow, 1981, p.374]. Hence we may proceed as in (3).

Short Exact Sequences

In the following we will consider exact sequences of locally convex spaces

ι π

0 ’ H ’ E ’ F,

’ ’

i.e. ι : H ’ E is a embedding of a closed subspace and π has ι(H) as kernel. Let

algebras AH , AE and AF on H, E and F be given, which satisfy π — (AF ) ⊆ AE and

ι— (AE ) ⊇ AH , the latter one telling us that AH functions on H can be extended

to AE functions on E. This is a very strong requirement, since by (21.11) not even

polynomials of degree 2 on a closed subspace of a Banach space can be extended

to a smooth function. The only algebra, where we have the extension property in

general is that of ¬nite type polynomials. So we will apply the following theorem

in (19.14) and (19.15) to situations, where AH is of quite di¬erent type then AE

and AF .

ι π

19.13. Theorem. [Adam, Bistr¨m, Kriegl, 1995, 6.1]. Let 0 ’ H ’ E ’ F be

’ ’

o

an exact sequence of locally convex spaces equipped with algebras satisfying

(i) π — (AF ) ⊆ AE and ι— (AE ) ⊇ AH .

(ii) AF is ω-isolating on F .

(iii) AE is translation invariant.

Then we have:

(1) If AH is ω-isolating on H then AE is ω-isolating on E.

(2) If H has ω-small AH -zerosets then E has ω-small AE -zerosets.

If in addition

(iv) Homω AF = F and Homω AH = H,

then we have:

(3) If • ∈ Hom AE is ω-evaluating on AE then • is evaluating on A0 := {f ∈

AE : ι— (f ) ∈ AH }.

(4) If • ∈ Hom AE is ω-evaluating on AE and if AH is ω-isolating on H then

• is evaluating on AE ; i.e., E = Homω AE .

Proof. Let x ∈ E. Since AE is translation invariant, we may assume x = 0. By

(ii) there is a sequence (gn ) in AF which isolates π(x) in F , i.e. gn (π(x)) = 0 and

’1

gn (0) = {π(x)}.

19.13

210 Chapter IV. Smoothly realcompact spaces 19.14

(1) By the special assumption in (19.13.1) there exist countable many hn ∈ AH

˜

which isolate 0 in H. According to (i) π — (gn ) ∈ AE and there exist hn ∈ AE with

˜ ˜

hn —¦ ι = hn . By (iii) we have that fn := hn ( ’x) ∈ AE . Now the functions

π — (gn ) together with the sequence (fn ) isolate x. Indeed, if x ∈ E is such that

(gn —¦ π)(x ) = (gn —¦ π)(x) for all n, then π(x ) = π(x), i.e. x ’ x ∈ H. From

˜

fn (x ) = fn (x) we conclude that hn (x ’ x) = hn (x ’ x) = fn (x ) = fn (x) = hn (0),

and hence x = x.

(2) Let U be a 0-neighborhood in E. By the special assumption there are countably

many hn ∈ AH with 0 ∈ n Z(hn ) ⊆ U © H. As before consider the sequence of

˜

functions fn := hn ( ’x). The common kernel of the functions in the sequences

(fn ) and (π — (gn )) contains x and is contained in π ’1 (π(x)) = x + H and hence in

(x + U ) © (x + H) ⊆ x + U .

Now the remaining two statements:

Let • ∈ Homω AE . Then • —¦ π — : AF ’ R is a ω-evaluating homomorphism, and

hence by (iv) given by the evaluation at a point y ∈ F . By (ii) there is a sequence

(gn ) in AF which isolates y. Since • is ω-evaluating there exists a point x ∈ E,

such that gn (y) = •(π — (gn )) = π — (gn )(x) = gn (π(x)) for all n. Hence y = π(x).

Since • obviously evaluates each countable set in AE at a point in π ’1 (y) ∼ K, •

=

—

induces a ω-evaluating homomorphism •H : AH ’ R by •H (ι (f )) := •(f ( ’x))

¯ ¯

for f ∈ A0 . In fact let f , f ∈ A0 with ι— (f ) = ι— (f ). Then • evaluates f ( ’x),

¯

f ( ’x) and all π — (gn ) at some common point x. So π(¯) = y = π(x), hence

¯ x

¯x

x ’ x ∈ H and f (¯ ’ x) = f (¯ ’ x).

¯ x

By (iv), •H is given by the evaluation at a point z ∈ H.

(3) Here we have that AH = ι— (A0 ), and hence

•(f ) = •H (ι— (f ( +x)) = ι— (f ( +x)(z)) = f (ι(z) + x)

for all f ∈ A0 . So • is evaluating on A0 .

(4) We show that • = δι(z)+x on AE . Indeed, by the special assumption there is

a sequence (hn ) in AH which isolates z. By (i) and (iii), we may ¬nd fn ∈ AE

such that hn = ι— (fn ( +x)). The sequences (π — (gn )) and (fn ) isolate z + x. So

let f ∈ AE be arbitrary. Then there exists a point z ∈ E, such that • = δz for all

these functions, hence z = ι(z) + x.

ι π

19.14. Corollary. [Adam, Bistr¨m, Kriegl, 1995, 6.3]. Let 0 ’ H ’ E ’ F be

’ ’

o

a left exact sequence of locally convex spaces and let AF and AE ⊇ E be algebras on

F and E, respectively, that satisfy all the assumptions (i-iv) of (19.13) not involving

AH . Let furthermore • : AE ’ R be ω-evaluating and • —¦ π — be evaluating on AF .

Then we have

(1) The homomorphism • is AE -evaluating if (H, σ(H, H )) is realcompact and

admits ω-small Pf -zerosets.

(2) The homomorphism • is A0 -evaluating if (H, σ(H, ι— (A0 ))) is Lindel¨f and

o

A0 ⊆ AE is some subalgebra.

(3) The homomorphism • is E -evaluating if (H, σ(H, H )) is realcompact.

19.14

19.15 19. Stability of smoothly realcompact spaces 211

Proof. We will apply (19.13.3). For this we choose appropriate subalgebras A0 ⊆

AE and put AH := ι— (A0 ). Then (i-iii) of (19.13) is satis¬ed. Remains to show for

(iv) that Homω (AH ) = H in the three cases:

(1) Let A0 := AE . Then we have Homω (AH ) = H by (19.9) using (18.27).

(2) If HAH = (H, σ(H, AH )) is Lindel¨f, then H = Homω (AH ), by (18.24).

o

(3) Let A0 := Pf (E). Then AH := ι— (A0 ) = Pf (H) by Hahn-Banach. If H is

σ(H, H )-realcompact, then H = Homω (AH ), by (18.27).

ι

19.15. Theorem. [Adam, Bistr¨m, Kriegl, 1995, 6.4 and 6.5]. Let c0 (“) ’ ’

o

π

E ’ F be a short exact sequence of locally convex spaces where AE is translation

’

invariant and contains (π — (AF ) ∪ E )∞ , and where F is AF -regular.

lfs

Then ι— (AE ) contains the algebra Ac0 (“) which is generated by all functions x ’

∞

γ∈“ ·(xγ ), where · ∈ C (R, R) is 1 near 0.

If AF is ω-isolating on F then AE is ω-isolating on E. If in addition F = Homω AF

and “ is non-measurable then E = Homω AE .

Proof. Let us show that the function x ’ γ∈“ ·(xγ ) can be extended to a

function in AE .

Remark that this product is locally ¬nite, since x ∈ c0 (“) and · = 1 locally around

0. Let p be an extension of the supremum norm ∞ on c0 (“) to a continuous

seminorm on E, and let p be the corresponding quotient seminorm on F de¬ned by

˜

p(y) := inf{p(x) : π(x) = y}. Let furthermore γ be a continuous linear extensions

˜

of prγ : c0 (“) ’ R which satisfy | γ (x)| ¤ p(x) for all x ∈ E. Finally let µ > 0 be

such that ·(t) = 1 for |t| ¤ µ.

We show ¬rst, that for the open subset {x ∈ E : p(π(x)) < µ} the product

˜

γ∈“ ·( γ (x)) is locally ¬nite as well. So let p(π(x)) < µ and 3 δ := µ ’ p(π(x)).

˜ ˜

We claim that

“x := {γ : | γ (x)| ≥ p(π(x)) + 2δ}

˜

is ¬nite. In fact by de¬nition of the quotient seminorm p(π(x)) := inf{p(x + y) :

˜

y ∈ c0 (“)} there is a y ∈ c0 (“) such that p(x + y) ¤ p(π(x)) + δ. Since y ∈ c0 (“)

˜

the set “0 := {γ : |yγ | ≥ δ} is ¬nite. For all γ ∈ “0 we have

/

| γ (x)| ¤ | γ (x + y)| + | γ (y)| ¤ p(x + y) + |yγ | < p(π(x)) + 2 δ,

˜

hence “x ⊆ “0 is ¬nite.

Now take z ∈ E with p(z ’ x) ¤ δ. Then for γ ∈ “x we have

/

| γ (z)| ¤ | γ (x)| + | γ (z ’ x)| < p(π(x)) + 2 δ + p(z ’ x) ¤ p(π(x)) + 3 δ = µ,

˜ ˜

hence ·( γ (z)) = 1 and the product is a locally ¬nite.

In order to obtain the required extension to all of E, we choose 0 < µ < µ and a

function g ∈ AF with carrier contained inside {z : p(z) ¤ µ } and with g(0) = 1.

˜

Then f : E ’ R de¬ned by

f (x) := g(π(x)) ·( γ (x))

γ∈“

19.15

212 Chapter IV. Smoothly realcompact spaces 19.16

is an extension belonging to π — (AF ) ∪ (E )∞ ⊆ (π — (AF ) ∪ E )∞ ⊆ AE .

Alg

lfs lfs

Let us now show that we can ¬nd such an extension with arbitrary small carrier,

and hence that E is AE -regular.

So let an arbitrary seminorm p on E be given. Then there exists a δ > 0 such

that δ p|c0 (“) ¤ ∞ . Let q be an extension of ∞ to a continuous seminorm

on E. By replacing p with max{q, δ p} we may assume that p|c0 (“) = ∞ and

the unit ball of the original p contains the δ-ball of the new p. Let again p be the

˜

corresponding quotient norm on F .

Then the construction above with some 0 < µ < µ < µ ¤ δ/3, for · ∈ C ∞ (R, R)

with ·(t) = 1 for |t| ¤ µ and ·(t) = 0 for |t| > µ > µ and g ∈ C ∞ (F, R) with

carr(g) ⊆ {y ∈ F : p(y) ¤ µ < µ} gives us a function f ∈ AE and it remains

˜

to show that the carrier of f is contained in the δ-ball of p. So let x ∈ E be

such that f (x) = 0. Then on one hand g(π(x)) = 0 and hence p(π(x)) ¤ µ and

˜

on the other hand ·( γ (x)) = 0 for all γ ∈ “ and hence | γ (x)| ¤ µ . We have

a unique continuous linear mapping T : 1 (“) ’ E , which maps prγ to γ , and

satis¬es |T (y — )(z)| ¤ y — p(z) for all z ∈ E since the unit ball of 1 (“) is the closed

absolutely convex hull of {prγ : γ ∈ “}. By Hahn-Banach there is some ∈ E

be such that | (z)| ¤ p(z) for all z and (x) = p(x). Hence ι— ( ) = |c0 (“) is in

the unit ball of 1 (“), and hence |T (ι— ( ))(x)| ¤ µ , since | γ (x)| ¤ µ . Moreover

|T (ι— ( ))(z)| ¤ p(z). Then 0 := (T —¦ ι— ’ 1)( ) = T ( |c0 (“) ) ’ ∈ E vanishes

on c0 (“) and | 0 (z)| ¤ 2 p(z) for all z. Hence | 0 (x)| ¤ 2 p(π(x)) ¤ 2 µ . So

˜

p(x) = | (x)| ¤ |T (ι— ( ))(x)| + | 0 (x)| ¤ µ + 2 µ < δ.

Because of the extension property Ac0 (“) ⊆ ι— (AE ) and since c0 (“) is Ac0 (“) -regular

and hence by (19.10.1) ω-isolated, we can apply (19.13.1) to obtain the statement

on ω-isolatedness. The evaluating property now follows from (19.13.4) using that

Homω Ac0 (“) = c0 (“) by (18.30.1).

19.16. The class c0 -ext. We shall show in (19.18) that in the short exact sequence

of (19.15) we can in fact replace c0 (“) by spaces from a huge class which we now

de¬ne.

De¬nition. Let c0 -ext be the class of spaces H, for which there are short exact

sequences c0 (“j ) ’ Hj ’ Hj+1 for j = 1, ..., n, with |“j | non-measurable, Hn+1 =

{0} and T : H ’ H1 an operator whose kernel is weakly realcompact and has

ω-small Pf -zerosets (By (18.18.1) and (19.2) these two conditions are satis¬ed, if it

has for example a weak— -separable dual).

Of course all spaces which admit a continuous linear injection into some c0 (“) with

non-measurable “ belong to c0 -ext. Besides these there are other natural spaces

in c0 -ext. For example let K be a compact space with |K| non-measurable and

K (ω0 ) = …, where ω0 is the ¬rst in¬nite ordinal and K (ω0 ) the corresponding ω0 -th

derived set. Then the Banach space C(K) belongs to c0 -ext, but is in general not

even injectable into some c0 (“), see [Godefroy, Pelant, et. al., 1988]. In fact, from

K (ω) = … and the compactness of K, we conclude that K (n) = … for some integer

19.16

19.17 19. Stability of smoothly realcompact spaces 213

n. We have the short exact sequence

ι π

c0 (K \ K (1) ) ∼ E ’ C(K) ’ C(K)/E ∼ C(K (1) ),

=’ ’ =

where E := {f ∈ C(K) : f |K (1) = 0}. By using (19.15) inductively the space

∞

C(K) is Clfs -regular. Also it is again an example of a Banach space E with E =

Hom C ∞ (E) that we are able to obtain without using the quite complicated result

(16.20.1) that it admits C ∞ -partition of unity.

19.17. Lemma. Pushout. [Adam, Bistr¨m, Kriegl, 1995, 6.6]. Let a closed

o

subspace ι : H ’ E and a continuous linear mapping T : H ’ H1 of locally convex

spaces be given.

Then the pushout of ι and T is the locally convex space E1 := H1 — E/{(T z, ’z) :

z ∈ H}. The natural mapping ι1 : H ’ E1 , given by u ’ [(u, 0)] is a closed

embedding and the natural mapping T1 : E ’ E1 given by T1 (x) := [(0, x)] is

continuous and linear. Moreover, if T is a quotient mapping then so is T1 .

ι π

Given a short exact sequence H ’ E ’ F of locally convex vector spaces and a

’ ’

continuous linear map T : H ’ H1 then we obtain by this construction a short

ι π

exact sequence H1 ’1 E1 ’ 1 F and a (unique) extension T1 : E ’ E1 of T , with

’ ’

ker T = ker T1 , such that the following diagram commutes

z ww 0z

z

ker T ker T1

u u u

y wE ww

ι π

H F

u u

T

T 1

y wE ww

ι1 π1

H1 F

1

Proof. Since H is closed in E the space E1 is a Hausdor¬ locally convex space.

The mappings ι1 and T1 are clearly continuous and linear. And ι1 is injective, since

(u, 0) ∈ {(T (z), ’z) : z ∈ H} implies 0 = z and u = T (z) = T (0) = 0. In order to

see that ι1 is a topological embedding let U be an absolutely convex 0-neighborhood

in H1 . Since ι is a topological embedding there is a 0-neighborhood W in E with

W © H = T ’1 (U ). Now consider the image of U — W ⊆ H1 — E under the quotient

map H1 — E ’ E1 . This is a 0-neighborhood in E1 and its inverse image under ι1

is contained in 2U . Indeed, if [(u, 0)] = [(x, z)] with u ∈ H1 , x ∈ U and z ∈ W ,

then x ’ u = T (z) and z ∈ H © W , by which u = x ’ T (z) ∈ U ’ U = 2U . Hence

ι1 embeds H1 topologically into E1 .

We have the universal property of a pushout, since for any two continuous linear

mappings ± : E ’ G and β : H1 ’ G with β —¦ T = ± —¦ ι, there exists a unique

linear mapping γ : E1 ’ G, given by [(u, x)] ’ ±(x) ’ β(u) with γ —¦ T1 = ± and

γ —¦ ι1 = β. Since H1 • E ’ E1 is a quotient mapping γ is continuous as well.

19.17

214 Chapter IV. Smoothly realcompact spaces 19.18

Let now π : E ’ F be a continuous linear mapping with kernel H, e.g. π the

natural quotient mapping E ’ F := E/H. Then by the universal property we get

a unique continuous linear π1 : E1 ’ F with π1 —¦ T1 = π and π1 —¦ ι1 = 0. We have

ι1 (H1 ) = ker(π1 ), since 0 = π1 [(u, z)] = π(z) if and only if z ∈ H, i.e. if and only if

[(u, z)] = [(u + T z, 0)] lies in the image of ι1 . If π is a quotient map then clearly so

is π1 . In particular the image of ι1 is closed.

Since T (x) = 0 if and only if [(0, x)] = [(0, 0)], we have that ker T = ker T1 . Assume

now, in addition, that T is a quotient map. Given any [(y, x)] ∈ E1 , there is then

some z ∈ H with T (z) = y. Thus T1 (x + z) = [(0, x + z)] = [(T (z), x)] = [(y, x)]

and T1 is onto. Remains to prove that T1 is ¬nal, which follows by categorical

reasoning. In fact let g : E1 ’ G be a mapping with g —¦ T1 continuous and linear.

Then g —¦ ι1 : H1 ’ G is a mapping with (g —¦ ι1 ) —¦ T = g —¦ T1 —¦ ι continuous and linear

and since T is ¬nal also g —¦ ι1 is continuous. Thus g composed with the quotient

mapping H1 • E ’ E1 is continuous and linear and thus also g itself.

ι π

19.18. Theorem. [Adam, Bistr¨m, Kriegl, 1995, 6.7]. Let H ’ E ’ F be a

’ ’

o

∞

short exact sequence of locally convex spaces, let F be Clfs -regular and let H be of

class c0 -ext, see (19.16).

∞ ∞

If Clfs (F ) is ω-isolating on F then Clfs (E) is ω-isolating on E. If, in addition,

∞ ∞

F = Homω Clfs (F ) then E = Homω Clfs (E).

Proof. Since H is of class c0 -ext there are short exact sequences c0 (“j ) ’ Hj ’

Hj+1 for j = 1, ..., n such that |“j | is non-measurable, Hn+1 = {0}, and T : H ’ H1

is an operator whose kernel is weakly realcompact and has ω-small Pf -zerosets. We

proceed by induction on the length of the resolution

H0 := H ’ H1 ··· Hn+1 = {0}.

According to (19.17) we have for every continuous linear T : Hj ’ Hj+1 the

following diagram

z ww z

z

ker T ker T1 0

u u u

y wE ww

ιj pj

Hj F

j

u u

T1

T

y wE ww

ιj+1 πj+1

Hj+1 F

j+1

For j > 0 we have that ker T = c0 (“) for some none-measurable “, and T and T1

are quotient mappings. So let as assume that we have already shown for the bottom

∞

row, that Ej+1 has the required properties and is in addition Clfs -regular. Then by

the exactness of the middle column we get the same properties for Ej using (19.15).

If j = 0, then the kernel is by assumption weakly paracompact and admits ω-small

Pf -zerosets. Thus applying (19.14.1) and (19.13.1) to the left exact middle column

we get the required properties for E = E0 .

19.18

19.23 19. Stability of smoothly realcompact spaces 215

∞

A Class of Clfs -Realcompact Locally Convex Spaces

19.19. De¬nition. Following [Adam, Bistr¨m, Kriegl, 1995] let RZ denote the

o

∞

class of all locally convex spaces E which admit ω-small Clfs -zerosets and have the

∞

property that E = Homω A for each translation invariant algebra A with Clfs (E) ⊆

A ⊆ C(E). In particular this applies to the algebras C, Cc and C ∞ © C.

∞

Note that for every continuous linear T : E ’ F we have T — : Clfs (F ) ’ Clfs (E).

∞ ∞

In fact we have T — (F ) ⊆ E , hence T — : (F )∞ ’ (E )∞ and T — ( i fi ) is again

locally ¬nite, if T is continuous and i fi is it.

∞

A locally convex space E with ω-small Clfs -zerosets belongs to RZ if and only if

∞ ∞ ∞

E = Homω Clfs (E) = Hom Clfs (E). In fact by (18.11) we have Homω Clfs (E) =

∞ ∞

Hom Clfs (E). Now let A ⊇ Clfs (E) and let • ∈ Homω A be countably evaluating.

∞

Then by (19.8.2) applied to X = Y = E, AX := A and AY := Clfs (E) the

homomorphism • is evaluating on A.

∞

Note that by (19.10.3) for metrizable E the condition of having ω-small Clfs -zerosets

∞

can be replaced by Clfs being ω-isolating. Moreover, by (19.10.1) it is enough to

∞

assume that E is Clfs -regular in order that E belongs to RZ.

19.20. Proposition. The class RZ is closed under formation of arbitrary products

and closed subspaces.

Proof. This is a direct corollary of (19.11).

19.21. Proposition. [Adam, Bistr¨m, Kriegl, 1995]. Every locally convex space

o

that admits a linear continuous injection into a metrizable space of class RZ is

itself of class RZ.

Proof. Use (19.1.2) and (19.10.3).

19.22. Corollary. [Adam, Bistr¨m, Kriegl, 1995]. The countable locally convex

o

direct sum of a sequence of metrizable spaces in RZ belongs to RZ.

The class of Banach spaces in RZ is closed under forming countable c0 -sums and

p -sums with 1 ¤ p ¤ ∞.

Proof. By (19.20) the class RZ is stable under (countable) products. And (19.21)

applies since a countable product of metrizable is again metrizable.

19.23. Corollary. [Adam, Bistr¨m, Kriegl, 1995]. Among the complete locally

o

convex spaces the following belong to the class RZ:

(1) All trans-separable (i.e. subspaces of products of separable Banach spaces)

locally convex spaces;

(2) All Hilbertizable locally convex spaces;

(3) All non-measurable WCG locally convex spaces;

(4) All non-measurable re¬‚exive Fr´chet spaces;

e

(5) All non-measurable infra-Schwarz locally convex spaces.

19.23

216 Chapter IV. Smoothly realcompact spaces 19.25

Proof. By (19.20), (19.5), and (19.21) we see that every complete locally convex

space E belongs to RZ, if it admits a zero-neighborhood basis U such that each

Banach space E(U ) for U ∈ U injects into some c0 (“U ) with non-measurable “U .

Apply this to the examples (19.12.1)-(19.12.5).

19.24. Proposition. [Adam, Bistr¨m, Kriegl, 1995]. Let 0 ’ H ’ E ’ F be

o

∞

an exact sequence. Let F be in RZ and let Clfs be ω-isolating on F .

Then E is in RZ under any of the following assumptions.

(1) The sequence 0 ’ H ’ E ’ F ’ 0 is exact, H is in c0 -ext and F is

∞ ∞

Clfs -regular; Here it follows also that Clfs is ω-isolating on E.

(2) The sequence 0 ’ H ’ E ’ F ’ 0 is exact, H = c0 (“) for some

∞

none-measurable “ and F is Clfs -regular; Here it follows also that E is

∞

Clfs -regular.

(3) The weak topology on H is realcompact and H admits ω-small Pf -zerosets.

4 The class c0 -ext is a subclass of RZ.