•(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.

Proof. (1) This is (19.18).

∞

(2) follows directly from (19.15) applied to the algebra A = Clfs .

∞

(3) By (19.13.2) the space E has ω-small Clfs -zerosets. By (19.14.1) we have as-

∞

sumption (iv) in (19.13), and then by (19.13.4) we have E = Homω (Clfs (E)). Thus

E belongs to RZ.

(4) Since every space E in c0 -ext is obtained by applying ¬nitely many constructions

as in (2) and a last one as in (3) we get it for E.

19.25. Remark. [Adam, Bistr¨m, Kriegl, 1995]. The class RZ is ˜quite big™. By

o

(19.24.4) we have that c0 -ext is a subclass of RZ. Also the following spaces are in

RZ:

The space C(K) where K is the one-point compacti¬cation of the topological

(ω)

disjoint union of a sequence of compact spaces Kn with Kn = …. In fact we

have a continuous injection given by the countable product of the restriction maps

C(K) ’ C(Kn ). Hence the result follows from (19.24.4) using also the remark in

(19.16) for the C(Kn ), followed by (19.20) for the product and by (19.21) for C(K).

Remark that in such a situation we might have K (ω) = {∞} = ….

The space D[0, 1] of all functions f : [0, 1] ’ R which are right continuous and have

left limits and endowed with the sup norm is in RZ. Indeed it contains C[0, 1] as a

subspace and D[0, 1]/C[0, 1] ∼ c0 [0, 1] according to [Corson, 1961]. By (18.27) we

=

have that C[0, 1] is weakly Lindel¨f and Pf is ω-isolating, since {evt : t ∈ Q © [0, 1]}

o

are point-separating. Now we use (19.24.3).