(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