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choose inductively a sequence of functions fn ∈ C ∞ (En , R) such that supp(fn ) ⊆
Un , fn (0) = 1, and fn |En’1 = fn’1 . If fn is already constructed, we may choose by
C ∞ -normality a smooth g : En+1 ’ R with supp(g) ⊆ Un+1 and g|supp(fn ) = 1. By
assumption, fn extends to a function fn ∈ C ∞ (En+1 , R). The function fn+1 := g·fn
has the required properties.
Now we de¬ne f : E ’ R by f |En := fn for all n. It is smooth since any
c ∈ C ∞ (R, E) locally factors to a smooth curve into some En by (1.8) since a
strict inductive limit is regular by (52.8), so f —¦ c is smooth. Finally, f (0) = 1,
and if f (x) = 0 then x ∈ En for some n, and we have fn (x) = f (x) = 0, thus
x ∈ Un ⊆ U .

For counter-examples for the extension property see (21.7) and (21.11). However,
for complemented subspaces the extension property obviously holds.

16.6
170 Chapter III. Partitions of unity 16.9

16.7. Proposition. Cc is C ∞ -regular. The space Cc (Rm , R) of smooth func-
∞ ∞

tions on Rm with compact support satis¬es the assumptions of (16.6).


Let Kn := {x ∈ Rm : |x| ¤ n}. Then Cc (Rm , R) is the strict inductive limit of the

closed subspaces CKn (Rm , R) := {f : supp(f ) ⊆ Kn }, which carry the topology of
uniform convergence in all partial derivatives separately. They are nuclear Fr´chet
e
spaces and hence separable, see (52.27). Thus they are C ∞ -normal by (16.10)
below.
In order to show the extension property for smooth functions we proof more general
that for certain sets A the subspace {f ∈ C ∞ (E, R) : f |A = 0} is a complemented
subspace of C ∞ (E, R). The ¬rst result in this direction is:

16.8. Lemma. [Seeley, 1964] The subspace {f ∈ C ∞ (R, R) : f (t) = 0 for t ¤ 0}
of the Fr´chet space C ∞ (R, R) is a direct summand.
e

Proof. We claim that the following map is a bounded linear mapping being left
inverse to the inclusion: s(g)(t) := g(t) ’ k∈N ak h(’t2k )g(’t2k ) for t > 0 and
s(g)(t) = 0 for t ¤ 0. Where h : R ’ R is a smooth function with compact support
satisfying h(t) = 1 for t ∈ [’1, 1] and (ak ) is a solution of the in¬nite system of
linear equations k∈N ak (’2k )n = 1 (n ∈ N) (the series is assumed to converge
absolutely). The existence of such a solution is shown in [Seeley, 1964] by taking
the limit of solutions of the ¬nite subsystems. Let us ¬rst show that s(g) is smooth.
For t > 0 the series is locally around t ¬nite, since ’t2k lies outside the support of
h for k su¬ciently large. Its derivative (sg)(n) (t) is

n
(n) kn
h(j) (’t2k )g (n’j) (’t2k )
(t) ’
g ak (’2 )
j=0
k∈N


and this converges for t ’ 0 towards g (n) (0)’ k∈N ak (’2k )n g (n) (0) = 0. Thus s(g)
is in¬nitely ¬‚at at 0 and hence smooth on R. It remains to show that g ’ s(g) is a
bounded linear mapping. By the uniform boundedness principle (5.26) it is enough
to show that g ’ (sg)(t) is bounded. For t ¤ 0 this map is 0 and hence bounded.
For t > 0 it is a ¬nite linear combination of evaluations and thus bounded.

Now the general result:

16.9. Proposition. Let E be a convenient vector space, and let p be a smooth
seminorm on E. Let A := {x : p(x) ≥ 1}. Then the closed subspace {f : f |A = 0}
in C ∞ (E, R) is complemented.

Proof. Let g ∈ C ∞ (E, R) be a smooth reparameterization of p with support in
E \ A equal to 1 near p’1 (0). By lemma (16.8), there is a bounded projection
P : C ∞ (R, R) ’ C(’∞,0] (R, R). The following mappings are smooth in turn by the




16.9
16.10 16. Smooth partitions of unity and smooth normality 171

properties of the cartesian closed smooth calculus, see (3.12):

(x, t) ’ f (et , x) ∈ R
E—R
x) ∈ C ∞ (R, R)
x ’ f (e( )
E

x ’ P (f (e( )
x)) ∈ C(’∞,0] (R, R)
E
(x, r) ’ P (f (e( )
E—R x))(r) ∈ R
x
’ P f (e( )x
x’ (ln(p(x))) ∈ R.
carr p , ln(p(x)) p(x) )
p(x)
So we get the desired bounded linear projection

¯
P : C ∞ (E, R) ’ {f ∈ C ∞ (E, R) : f |A = 0},
¯
(P (f ))(x) := g(x) f (x) + (1 ’ g(x)) P (f (e( )x
p(x) ))(ln(p(x))).



16.10. Theorem. Smoothly paracompact Lindel¨f. [Wells, 1973]. If X is
o
Lindel¨f and S-regular, then X is S-paracompact. In particular, all nuclear Fr´chet
o e
spaces and strict inductive limits of sequences of such spaces are C ∞ -paracompact.
Furthermore, nuclear Silva spaces, see (52.37), are C ∞ -paracompact.

The ¬rst part was proved by [Bonic, Frampton, 1966] under stronger assumptions.
The importance of the proof presented here lies in the fact that we need not assume
1
that S is local and that f ∈ S for f ∈ S. The only things used are that S is an
algebra and for each g ∈ S there exists an h : R ’ [0, 1] with h —¦ g ∈ S and h(t) = 0
for t ¤ 0 and h(t) = 1 for t ≥ 1. In particular, this applies to S = Lippglobal and X
a separable Banach space.

Proof. Let U be an open covering of X.
Claim. There exists a sequence of functions gn ∈ S(X, [0, 1]) such that {carr gn :
’1
n ∈ N} is a locally ¬nite family subordinated to U and {gn (1) : n ∈ N} is a
covering of X.
For every x ∈ X there exists a neighborhood U ∈ U (since U is a covering) and
hence an hx ∈ S(X, [0, 2]) with hx (x) = 2 and carr(hx ) ⊆ U (since X is S-regular).
Since X is Lindel¨f we ¬nd a sequence xn such that {x : hn (x) > 1 : n ∈ N} is
o
a covering of X (we denote hn := hxn ). Now choose an h ∈ C ∞ (R, [0, 1]) with
h(t) = 0 for t ¤ 0 and h(t) = 1 for t ≥ 1. Set

gn (x) := h(n (hn (x) ’ 1) + 1) h(n (1 ’ hj (x)) + 1).
j<n

Note that
1
0 for hn (x) ¤ 1 ’ n
h(n (hn (x) ’ 1) + 1) =
1 for hn (x) ≥ 1
1
0 for hj (x) ≥ 1 + n
h(n (1 ’ hj (x)) + 1) =
for hj (x) ¤ 1
1

16.10
172 Chapter III. Partitions of unity 16.10

Then gn ∈ S(X, [0, 1]) and carr gn ⊆ carr hn . Thus, the family {carr gn : n ∈ N} is
subordinated to U.
’1
The family {gn (1) : n ∈ N} covers X since for each x ∈ X there exists a minimal
n with hn (x) ≥ 1, and thus gn (x) = 1.
If we could divide in S, then fn := gn / j gj would be the required partition of
unity (and we do not need the last claim in this strong from).
Instead we proceed as follows. The family {carr gn : n ∈ N} is locally ¬nite: Let
1
n be such that hn (x) > 1, and take k > n so large that 1 + k < hn (x), and let
1
Ux := {y : hn (y) > 1 + k }, which is a neighborhood of x. For m ≥ k and y ∈ Ux
1 1
we have that hn (y) > 1 + k ≥ 1 + m , hence the (n + 1)-st factor of gm vanishes at
y, i.e. {j : carr gj © Ux = …} ⊆ {1, . . . , m ’ 1}.
Now de¬ne fn := gn j<n (1 ’ gj ) ∈ S. Then carr fn ⊆ carr gn , hence {carr fn :
n ∈ N} is a locally ¬nite family subordinated to U. By induction, one shows that
j¤n fj = 1 ’ j¤n (1 ’ gj ). In fact j<n (1 ’
j¤n fj = fn + j<n fj = gn
gj ) + 1 ’ j<n (1 ’ gj ) = 1 + (gn ’ 1) j<n (1 ’ gj ). For every x ∈ U there exists

an n with gn (x) = 1, hence fk (x) = 0 for k > n and j=0 fj (x) = j¤n fj (x) =
1 ’ j¤n (1 ’ gj (x)) = 1.
Let us consider a nuclear Silva space. By (52.37) its dual is a nuclear Fr´chet space.
e
By (4.11.2) on the strong dual of a nuclear Fr´chet space the c∞ -topology coincides
e

with the locally convex one. Hence, it is C -regular since it is nuclear, so it has
a base of (smooth) Hilbert seminorms. A Silva space is an inductive limit of a
sequence of Banach spaces with compact connecting mappings (see (52.37)), and
we may assume that the Banach spaces are separable by replacing them by the
closures of the images of the connecting mappings, so the topology of the inductive
limit is Lindel¨f. Therefore, by the ¬rst assertion we conclude that the space is
o
C ∞ -paracompact.
In order to obtain the statement on nuclear Fr´chet spaces we note that these are
e
separable, see (52.27), and thus Lindel¨f. A strict inductive limit of a sequence of
o
nuclear Fr´chet spaces is C ∞ -regular by (16.6), and it is also Lindel¨f for its c∞ -
e o
topology, since this is the inductive limit of topological spaces (not locally convex
spaces).

Remark. In particular, every separable Hilbert space has Lip2 global -partitions of
unity, thus there is such a Lip2 2
\ A0
global -partition of functions • subordinated to
and 2 \ A1 , with A0 and A1 mentioned in (16.4). Hence, f := carr •©A0 =… • ∈ C 2
satis¬es f |Aj = j for j = 0, 1. However, f ∈ Lip2
/ global . The reason behind this is
that Lip2global is not a sheaf.

Open problem. Classically, one proves the existence of continuous partitions of
unity from the paracompactness of the space. So the question arises whether theorem
(16.10) can be strengthened to: If the initial topology with respect to S is paracom-
pact, do there exist S-partitions of unity? Or equivalently: Is every paracompact
S-regular space S-paracompact?


16.10
16.14 16. Smooth partitions of unity and smooth normality 173

16.11. Theorem. Smoothness of separable Banach spaces. Let E be a
separable Banach space. Then the following conditions are equivalent.
E has a C 1 -norm;
(1)
E has C 1 -bump functions, i.e., E is C 1 -regular;
(2)
The C 1 -functions separate closed sets, i.e., E is C 1 -normal;
(3)
E has C 1 -partitions of unity, i.e., E is C 1 -paracompact;
(4)
(5) E has no rough norm, i.e. E is Asplund;
(6) E is separable.

Proof. The implications (1) ’ (2) and (4) ’ (3) ’ (2) are obviously true. The
implication (2) ’ (4) is (16.10). (2) ’ (5) holds by (14.9). (5) ’ (6) follows from
(14.10) since E is separable. (6) ’ (1) is (13.22) together with (13.20).

A more general result is:

16.12. Result. [John, Zizler, 1976] Let E be a WCG Banach space. Then the
following statements are equivalent:
E is C 1 -normable;
(1)
E is C 1 -regular;
(2)
E is C 1 -paracompact;
(3)
(4) E has norm, whose dual norm is LUR;
E has shrinking Markuˇeviˇ basis, i.e. vectors xi ∈ E and x— ∈ E with
(5) sc i
xi (xj ) = δi,j and the span of the xi is dense in E and the span of x— is

i
dense in E .

16.13. Results.
(1) [Godefroy, Pelant, et. al., 1983] ( [Vanderwer¬, 1992]) Let E is WCG Ba-
nach space (or even WCD, see (53.8)). Then E is C 1 -regular.
(2) [Vanderwer¬, 1992] Let K be compact with K (ω1 ) = …. Then C(K) is C 1 -
paracompact. Compare with (13.18.2) and (13.17.5).
(3) [Godefroy, Troyanski, et. al., 1983] Let E be a subspace of a WCG Banach
space. If E is C k -regular then it is C k -paracompact. This will be proved in
(16.18).
(4) [MacLaughlin, 1992] Let E be a WCG Banach space. If E is C k -regular
then it is C k -paracompact.

16.14. Lemma. Smooth functions on c0 (“). [Toru´czyk, 1973]. The norm-
n
topology of c0 (“) has a basis which is a countable union of locally ¬nite families of
carriers of smooth functions, each of which depends locally only on ¬nitely many
coordinates.

Proof. The open balls Br := {x : x ∞ < r} are carriers of such functions: In
fact, similarly to (13.16) we choose a h ∈ C ∞ (R, R) with h = 1 locally around 0
and carr h = (’1, 1), and de¬ne f (x) := γ∈“ h(xγ ). Let

Un,r,q = {Br + q1 eγ1 + · · · + qn eγn : {γ1 , . . . , γn } ⊆ “}

16.14
174 Chapter III. Partitions of unity 16.15

where n ∈ N, r ∈ Q, q ∈ Qn with |qi | > 2r for 1 ¤ i ¤ n. This is the required
countable family.
Un,r,q is a basis for the topology.
Claim. The union n,r,q
µ
Let x ∈ c0 (“) and µ > 0. Choose 0 < r < 2 such that r = |xγ | for all γ (note that
|xγ | ≥ µ/4 only for ¬nitely many γ). Let {γ1 , . . . , γn } := {γ : |xγ | > r}. For qi with
|qi ’ xγi | < r and |qi | > 2r we have

x’ qi eγi ∈ Br ,
i


and hence
n
x ∈ Br + qi eγi ⊆ x + B2r ⊆ {y : y ’ x ¤ µ}.

i=1

Claim. Each family Un,r,q is locally ¬nite.
r
For given x ∈ c0 (“), let {γ1 , . . . , γm } := {γ : |xγ | > 2 } and assume there exists a
n
y ∈ (x + B r ) © (Br + i=1 qi eβi ) = …. For y ∈ x + B r we have |ya | < r for all γ ∈ /
2 2
n
{γ1 , . . . , γm } and for y ∈ Br + i=1 qi eβi we have |yγ | > r for all γ ∈ {β1 , . . . , βn }.
Hence, {β1 , . . . , βn } ⊆ {γ1 , . . . , γm } and Un,r,q is locally ¬nite.

16.15. Theorem, Smoothly paracompact metrizable spaces. [Toru´czyk, n
1973]. Let X be a metrizable smooth space. Then the following are equivalent:
(1) X is S-paracompact, i.e. admits S-partitions of unity.
(2) X is S-normal.
(3) The topology of X has a basis which is a countable union of locally ¬nite
families of carriers of smooth functions.
(4) There is a homeomorphic embedding i : X ’ c0 (A) for some A (with image
in the unit ball) such that eva —¦ i is smooth for all a ∈ A.

Proof. (1) ’ (3) Let Un be the cover formed by all open balls of radius 1/n. By
(1) there exists a partition of unity subordinated to it. The carriers of these smooth
functions form a locally ¬nite re¬nement Vn . The union of all Vn is clearly a base
of the topology since that of all Un is one.
(3) ’ (2) Let A1 and A2 be two disjoint closed subsets of X. Let furthermore Un
be a locally ¬nite family of carriers of smooth functions such that n Un is a basis.
i
Let Wn := {U ∈ Un : U © Ai = …}. This is the carrier of the smooth locally
i
¬nite sum of the carrying functions of the U ™s. The family {Wn : i ∈ {0, 1}, n ∈ N}
forms a countable cover of X. By the argument used in the proof of (16.10) we
may shrink the Wn to a locally ¬nite cover of X. Then W 1 = n Wn is a carrier
i 1

containing A2 and avoiding A1 . Now use (16.2.2).
(2) ’ (1) is lemma (16.2), since metrizable spaces are paracompact.
(3) ’ (4) Let Un be a locally ¬nite family of carriers of smooth functions such that
1
U := n Un is a basis. For every U ∈ Un let fU : X ’ [0, n ] be a smooth function
with carrier U . We de¬ne a mapping i : X ’ c0 (U), by i(x) = (fU (x))U ∈U . It

16.15
16.18 16. Smooth partitions of unity and smooth normality 175

is continuous at x0 ∈ X, since for n ∈ N there exists a neighborhood V of x0
1
which meets only ¬nitely many sets U ∈ k¤2n Uk , and so i(x) ’ i(x0 ) ¤ n
1
for those x ∈ V with |fU (x) ’ fU (x0 )| < n for all U ∈ k¤n Uk meeting V .
The mapping i is even an embedding, since for x0 ∈ U ∈ U and x ∈ U we have
/
i(x) ’ i(x0 ) = fU (x0 ) > 0.
(4) ’ (3) By (16.14) the Banach space c0 (A) has a basis which is a countable union
of locally ¬nite families of carriers of smooth functions, all of which depend locally
only on ¬nitely many coordinates. The pullbacks of all these functions via i are
smooth on X, and their carriers furnish the required basis.

16.16. Corollary. Hilbert spaces are C ∞ -paracompact. [Toru´czyk, 1973].n
Every space c0 (“) (for arbitrary index set “) and every Hilbert space (not necessarily
separable) is C ∞ -paracompact.

Proof. The assertion for c0 (“) is immediate from (16.15). For a Hilbert space
2
(“) we use the embedding i : 2 (“) ’ c0 (“ ∪ {—}) given by

for γ ∈ “

i(x)γ = 2
for γ = —
x

This is an embedding: From xn ’ x ∞ ’ 0 we conclude by H¨lder™s inequality
o
that y, xn ’ x ’ 0 for all y ∈ 2 and hence xn ’ x 2 = xn 2 + x 2 ’ 2 x, xn ’
2 x 2 ’ 2 x 2 = 0.

16.17. Corollary. A countable product of S-paracompact metrizable spaces is
again S-paracompact.

Proof. By theorem (16.15) we have certain embeddings in : Xn ’ c0 (An ) with
images contained in the unit balls. We consider the embedding i : n Xn ’
1
c0 ( n An ) given by i(x)a = n in (xn ) for a ∈ An which has the required properties
for theorem (16.15). It is an embedding, since i(xn ) ’ i(x) if and only if xn ’ xk
k
for all k (all but ¬nitely many coordinates are small anyhow).

16.18. Corollary. [Godefroy, Troyanski, et. al., 1983]
Let E be a Banach space with a separable projective resolution of identity, see
(53.13). If E is C k -regular, then it is C k -paracompact.

Proof. By (53.20) there exists a linear, injective, norm 1 operator T : E ’ c0 (“1 )
for some “1 and by (53.13) projections P± for ω ¤ ± ¤ dens E. Let “2 := {∆ :
∆ ⊆ [ω, dens E), ¬nite}. For ∆ ∈ “2 choose a dense sequence (x∆ )n in the unit
n
∆ ∆
sphere of Pω (E) • ±∈∆ (P±+1 ’ P± )(E) and let yn ∈ E be such that yn = 1
and yn (x∆ ) = 1. For n ∈ N let πn : x ’ x ’ yn (x)x∆ . Choose a smooth function
∆ ∆ ∆
n n

h ∈ C (E, [0, 1]) with h(x) = 0 for x ¤ 1 and h(x) = 1 for x ≥ 2. Let
R± := (P±+1 ’ P± )/ P±+1 ’ P± .

16.18
176 Chapter III. Partitions of unity 16.18

Now de¬ne an embedding as follows: Let “ := N3 — “2 N — [ω, dens E) “1
N
and let u : E ’ c0 (“) be given by
1 ∆
for γ = (m, n, l, ∆) ∈ N3 — “2 ,
h(mπn x) h(lR± x)
±
2n+m+l ±∈∆


1
for γ = (m, ±) ∈ N — [ω, dens E),

2m h(mR± x)

u(x)γ := 1 x
for γ = m ∈ N,
2 h( m )




for γ = ± ∈ “1 .
T (x)±

Let us ¬rst show that u is well-de¬ned and continuous. We do this only for the
coordinates in the ¬rst row (for the others it is easier, the third has locally even
¬nite support).
Let x0 ∈ E and 0 < µ < 1. Choose n0 with 1/2n0 < µ. Then |u(x)γ | < µ for all
x ∈ X and all ± = (m, n, l, ∆) with m + n + l ≥ n0 .
For the remaining coordinates we proceed as follows: We ¬rst choose δ < 1/n0 . By
(53.13.8) there is a ¬nite set ∆0 ∈ “2 such that R± x0 < δ/2 for all ± ∈ ∆0 . For
/
those ± and x ’ x0 < δ/2 we get

δ δ
R± (x) ¤ R± (x0 ) + R± (x ’ x0 ) < + = δ,
22
hence u(x)γ = 0 for all γ = (m, n, l, ∆) with m + n + l < n0 and ∆ © ([ω, dens E \
∆0 ) = ….
For the remaining ¬nitely many coordinates γ = (m, n, l, ∆) with m+n+l < n0 and
∆ ⊆ ∆0 we may choose a δ1 > 0 such that |u(x)γ ’u(x0 )γ | < µ for all x’x0 < δ1 .
Thus for x ’ x0 < min{δ/2, δ1 } we have |u(x)γ ’ u(x0 )γ | < 2µ for all γ ∈ N3 — “2
and |u(x0 )γ | ≥ µ only for ± = (m, n, l, ∆) with m + n + l < n0 and ∆ ⊆ ∆0 .
Since T is injective, so is u. In order to show that u is an embedding let x∞ , xp ∈ E
with u(xp ) ’ u(x∞ ). Then xp is bounded, since for n0 > x∞ implies that
h(x∞ /n0 ) = 0 and from h(xp /n0 ) ’ h(x∞ /n0 ) we conclude that xp /n0 ¤ 2 for
large p.
Now we show that for any µ > 0 there is a ¬nite µ-net for {xp : p ∈ N}: For this
we choose m0 > 2/µ. By (53.13.8) there is a ¬nite set ∆0 ⊆ Λ(x∞ ) := µ>0 {± <

dens E : R± (x∞ ) ≥ µ} and an n0 := n ∈ N such that m0 πn 0 (x∞ ) ¤ 1 and

hence h(m0 πn 0 (x∞ )) = 0. In fact by (53.13.9) there is a ¬nite linear combination
of vectors R± (x∞ ), which has distance less than µ from x∞ , let δ := min{ R± (x) :

for those ±} > 0. Since the yn 0 are dense in the unit sphere of Pω • ±∈∆0 R± E
1
we may choose an n such that x∞ ’ x∞ x∆0 < 2m0 and hence
n


πn 0 (x∞ ) = x∞ ’ yn 0 (x∞ )x∆0
∆ ∆
n

¤ x∞ ’ x∞ x∆0 + x∞ x∆0 ’ yn 0 (x∆0 )x∆0

n n n n



x∞ x∆0 ’ x∞ ) x∆0
+ yn 0 n n

1 1 1
¤ +0+ =
2m0 2m0 m0

16.18
16.19 16. Smooth partitions of unity and smooth normality 177

Next choose l0 := l ∈ N such that l0 δ0 ≥ 2 and hence l0 R± x∞ ≥ 2 for all ± ∈ ∆0 .
Then
∆ ∆
h(l0 R± xp ) ’ h(m0 πn00 x∞ )
h(m0 πn00 xp ) h(l0 R± x∞ )
±∈∆0 ±∈∆0
h(l0 R± xp ) ’ h(l0 R± x∞ ) = 1 for ± ∈ ∆0
and

Hence
∆ ∆
h(m0 πn00 xp ) ’ h(m0 πn00 x∞ ) = 0,
and so πn00 xp ¤ 2/m0 < µ for all large p. Thus d(xp , R x∆00 ) ¤ µ, hence {xp : p ∈

n
N} has a ¬nite µ-net, since its projection onto the one dimensional subspace Rx∆00n
is bounded.
Thus {x∞ , xp : p ∈ N} is relatively compact, and hence u restricted to its closure
is a homeomorphism onto the image. So xp ’ x∞ .
Now the result follows from (16.15).

16.19. Corollary. [Deville, Godefroy, Zizler, 1990]. Let c0 (“) ’ E ’ F be a
short exact sequence of Banach spaces and assume F admits C p -partitions of unity.
Then E admits C p -partitions of unity.

Proof. Without loss of generality we may assume that the norm of E restricted
to c0 (“) is the supremum norm. Furthermore there is a linear continuous splitting
T : 1 (“) ’ E by (13.17.3) and a continuous splitting S : F ’ E by (53.22) with
S(0) = 0. We put Tγ := T (eγ ) for all γ ∈ “. For n ∈ N let Fn be a C p -partition
of unity on F with diam(carr(f )) ¤ 1/n for all f ∈ Fn . Let F := n Fn and let
“2 := {∆ ⊆ “ : ∆ is ¬nite}. For any f ∈ F choose xf ∈ S(carr(f )) and for any

∆ ∈ “2 choose a dense sequence {yf,m : m ∈ N} 0 in the linear subspace generated
by {xf + eγ : γ ∈ ∆}. Let ∆ ∈ E be such that ∆ (yf,m ) = ∆ · yf,m = 1.
∆ ∆
f,m f,m f,m
Let πf,m : E ’ E be given by πf,m (x) := x ’ ∆ (x) yf,m . Let h : E ’ R be
∆ ∆ ∆
f,m
C p with h(x) = 0 for x ¤ 1 and h(x) = 1 for x ≥ 2. Let g : R ’ [’1, 1] be
C p with g(t) = 0 for |t| ¤ 1 and injective on {t : |t| > 1}. Now de¬ne a mapping
˜
u : E ’ c0 (“), where
˜
“ := (F — “2 — N2 ) (F — “) (F — N) F N N

by
1 ∆
g(n Tγ (x ’ xf ))
u(x)γ := f (ˆ) h(j πf,m (x))
x
˜
2n+m+j
γ∈∆

for γ = (f, ∆, j, m) ∈ Fn — “2 — N2 , and by
˜
±1
 2n f (ˆ) g(n Tγ (x ’ xf )) for γ = (f, γ) ∈ Fn — “
x ˜

1
 n+j f (ˆ) h(j (x ’ xf )) for γ = (f, j) ∈ Fn — N
x ˜

2

1
= f ∈ Fn ⊆ F
u(x)γ := 2n f (ˆ)
x for γ
˜
˜

1
=n∈N
 n h(n x) for γ
˜
2


1
= n ∈ N.
2n h(x/n) for γ
˜

16.19
178 Chapter III. Partitions of unity 16.19

We ¬rst claim that u is well-de¬ned and continuous. Every coordinate x ’ u(x)γ
is continuous, so it remains to show that for every µ > 0 locally in x the set
of coordinates γ, where |u(x)γ | > µ is ¬nite. We do this for the ¬rst type of
coordinates. For this we may ¬x n, m and j (since the factors are bounded by 1).
Since Fn is a partition of unity, locally f (ˆ) = 0 for only ¬nitely many f ∈ Fn , so we
x
1
may also ¬x f ∈ Fn . For such an f the set ∆0 := {γ : |Tγ (x ’ xf )| ≥ π(x ’ xf ) + n }
is ¬nite by the proof of (13.17.3). Since x ’ xf = π(x ’ xf ) ¤ 1/n be have
ˆ
g(n Tγ (x ’ xf )) = 0 for γ ∈ ∆0 .
/
Thus only for those ∆ contained in the ¬nite set ∆0 , we have that the corresponding
coordinate does not vanish.
Next we show that u is injective. Let x = y ∈ E.
If x = y , then there is some n and a f ∈ Fn such that f (ˆ) = 0 = f (ˆ). Thus this
ˆˆ x y
is detected by the 4th row.
If x = y then S x = S y and since x ’ S x, y ’ S y ∈ c0 (“) there is a γ ∈ “ with
ˆˆ ˆ ˆ ˆ ˆ

Tγ (x ’ S x) = (x ’ S x)γ = (y ’ S y )γ = Tγ (y ’ S y ).
ˆ ˆ ˆ ˆ

We will make use of the following method repeatedly:
For every n there is a fn ∈ Fn with fn (ˆ) = 0 and hence x ’ xfn ¤ 1/n.
x ˆˆ
Since S is continuous we get xfn = S(ˆfn ) ’ S(ˆ) and thus limn Tγ (x ’ xfn ) =
x x
limn Tγ (x ’ S(ˆfn )) = Tγ (x ’ S(ˆ)).
x x
So we get

lim Tγ (x ’ xfn ) = Tγ (x ’ S(ˆ)) = Tγ (y ’ S(ˆ)) = lim Tγ (y ’ xfn ).
x y
n n

If all coordinates for u(x) and u(y) in the second row would be equal, then

g(n Tγ (x ’ xf )) = g(n Tγ (y ’ xf ))

since fγ (ˆ) = 0, and hence Tγ (x ’ xf ) ’ Tγ (y ’ xf ) ¤ 2/n, a contradiction.
x
Now let us show that u is a homeomorphism onto its image. We have to show
xk ’ x provided u(xk ) ’ u(x).
We consider ¬rst the case, where x = S x. As before we choose fn ∈ Fn with
ˆ
fn (ˆ) = 0 and get xfn = S(ˆfn ) ’ S(ˆ) = x. Let µ > 0 and j > 3/µ. Choose an n
x x x
such that xfn ’ x < 1/j. Then h(j (xfn ’ x)) = 0. From the coordinates in the
third and fourth row we conclude

f (ˆk ) h(j (xk ’ xfn )) ’ f (ˆ) h(j (x ’ xfn )) f (ˆk ) ’ f (ˆ) = 0.
x x and x x

Hence
h(j (xk ’ xfn )) ’ h(j (x ’ xfn )) = 0.
Thus xk ’ xfn < 2/j for all large k. But then

3
xk ’ x ¤ xk ’ xfn + xfn ’ x < < µ,
j

16.19
16.19 16. Smooth partitions of unity and smooth normality 179

i.e. xk ’ x.
Now the case, where x = S x. We show ¬rst that {xk : k ∈ N} is bounded. Pick
ˆ
n > x . From the coordinates in the last row we get that limk h(xk /n) = 0, i.e.
xk ¤ 2n for all large k.
We claim that for j ∈ N there is an n ∈ N and an f ∈ Fn with f (ˆ) = 0, a ¬nite
x

set ∆ ⊆ “ with γ∈∆ g(n Tγ (x ’ xf )) = 0 and an m ∈ N with h(j πf,m (x)) = 0.
From 0 = (x ’ S x) ∈ c0 (“) we deduce that there is a ¬nite set ∆ ⊆ “ with
ˆ
Tγ (x ’ S x) = (x ’ S x)γ = 0 for all γ ∈ ∆ and dist(x ’ S x, eγ : γ ∈ ∆ ) < 1/(3j),
ˆ ˆ ˆ
i.e. |(x ’ S x)γ | ¤ 1/(3j) for all γ ∈ ∆. As before we choose fn ∈ Fn with fn (ˆ) = 0
ˆ / x
and get xfn = S(ˆfn ) ’ S(ˆ) and
x x

lim Tγ (x ’ xfn ) = (x ’ S x)γ = 0 for γ ∈ ∆.
ˆ
n

Thus g(n (Tγ (x ’ xfn ))) = 0 for all large n and γ ∈ ∆. Furthermore, dist(x, xfn +

eγ : γ ∈ ∆ ) = dist(x ’ xfn , eγ : γ ∈ ∆ ) < 1/(2j). Since {yfn ,m : m ∈ N} is

dense in xfn + eγ : γ ∈ ∆ there is an m such that x ’ yfn ,m < 1/(2j). Since

πfn ,m ¤ 2 we get

πfn ,m (x) ¤ x ’ yfn ,m + |1 ’ ∆ ,m (x)| yfn ,m
∆ ∆ ∆
fn
1 1 1 1
+ ∆ ,m x ’ yfn ,m yfn ,m ¤
∆ ∆
¤ + =,
fn
2j 2j 2j j

hence h(j πfn ,m (x)) = 0.
We claim that for every µ > 0 there is a ¬nite µ-net of {xk : k ∈ N}. Let µ > 0.
We choose j > 4/µ and we pick n ∈ N, f ∈ Fn , ∆ ⊆ “ ¬nite, and m ∈ N satisfying
the previous claim. From u(xk ) ’ u(x) we deduce from the coordinates in the ¬rst
row, that

g(n Tγ (xk ’ xf )) ’
f (ˆk ) h(j πf,m (xk ))
x
γ∈∆

’ f (ˆ) h(j πf,m (x)) g(n Tγ (x ’ xf )) for k ’ ∞
x
γ∈∆

and since by the coordinates in the fourth row f (ˆk ) ’ f (ˆ) = 0 we obtain from
x x
the coordinates in the second row, that

g(n Tγ (xk ’ xf )) ’ g(n Tγ (x ’ xf )) = 0 for γ ∈ ∆.

Hence
∆ ∆
h(j πf,m (xk )) ’ h(j πf,m (x)) = 0.
Therefore
1 µ
∆ ∆ ∆
xk ’ f,m (xk ) yf,n = πf,m (xk ) < < for all large k.
j 4

Thus there is a ¬nite dimensional subspace in E spanned by yf,n and ¬nitely many
xk , such that all xk have distance ¤ µ/4 from it. Since {xk : k ∈ N} are bounded,

16.19
180 Chapter III. Partitions of unity 16.21

the compactness of the ¬nite dimensional balls implies that {xk : k ∈ N} has an
µ-net, hence {xk : k ∈ N} is relatively compact, and since u is injective we have
limk xk = x.
Now the result follows from (16.15).

Remark. In general, the existence of C ∞ -partitions of unity is not inherited by
the middle term of short exact sequences: Take a short exact sequence of Banach
spaces with Hilbert ends and non-Hilbertizable E in the middle, as in (13.18.6).
If both E and E — admitted C 2 -partitions of unity, then they would admit C 2 -
bump functions, hence E was isomorphic to a Hilbert space by [Meshkov, 1978], a
contradiction.

16.20. Results on C(K). Let K be compact. Then for the Banach space C(K)
we have:
(1) [Deville, Godefroy, Zizler, 1990]. If K (ω) = … then C(K) is C ∞ -paracom-
pact.
(2) [Vanderwer¬, 1992] If K (ω1 ) = … then C(K) is C 1 -paracompact.
(3) [Haydon, 1990] In contrast to (2) there exists a compact space K with
K (ω1 ) = {—}, but such that C(K) has no Gˆteaux-di¬erentiable norm. Nev-
a
ertheless C(K) is C 1 -regular by [Haydon, 1991]. Compare with (13.18.2).
(4) [Namioka, Phelps, 1975]. If there exists an ordinal number ± with K (±) = …
then the Banach space C(K) is Asplund (and conversely), hence it does not
admit a rough norm, by (13.8).
(5) [Ciesielski, Pol, 1984] There exists a compact K with K (3) = …. Conse-
quently, there is a short exact sequence c0 (“1 ) ’ C(K) ’ c0 (“2 ), and the
space C(K) is Lipschitz homeomorphic to some c0 (“). However, there is
no continuous linear injection of C(K) into some c0 (“).

Notes. (1) Applying theorem (16.19) recursively we get the result as in (13.17.5).

16.21. Some radial subsets are di¬eomorphic to the whole space. We are
now going to show that certain subsets of convenient vector spaces are di¬eomorphic
to the whole space. So if these subsets form a base of the c∞ -topology of the
modeling space of a manifold, then we may choose charts de¬ned on the whole
modeling space. The basic idea is to ˜blow up™ subsets U ⊆ E along all rays
starting at a common center. Without loss of generality assume that the center
is 0. In order for this technique to work, we need a positive function ρ : U ’ R,
1
which should give a di¬eomorphism f : U ’ E, de¬ned by f (x) := ρ(x) x. For
this we need that ρ is smooth, and since the restriction of f to U © R+ x ’ R+ x
has to be a di¬eomorphism as well, and since the image set is connected, we need
that the domain U © R+ x is connected as well, i.e., U has to be radial. Let Ux :=
t
{t > 0 : tx ∈ U }, and let fx : Ux ’ R be given by f (tx) = ρ(tx) x =: fx (t)x.
Since up to di¬eomorphisms this is just the restriction of the di¬eomorphism f , we
need that 0 < fx (t) = ‚t ρ(tx) = ρ(tx)’tρ (tx)(x) for all x ∈ U and 0 < t ¤ 1. This
‚ t
ρ(tx)2
means that ρ(y) > ρ (y)(y) for all y ∈ U , which is quite a restrictive condition,

16.21
16.21 16. Smooth partitions of unity and smooth normality 181

and so we want to construct out of an arbitrary smooth function ρ : U ’ R, which
tends to 0 towards the boundary, a new smooth function ρ satisfying the additional
assumption.

Theorem. Let U ⊆ E be c∞ -open with 0 ∈ U and let ρ : U ’ R+ be smooth, such
that for all x ∈ U with tx ∈ U for 0 ¤ t < 1 we have ρ(tx) ’ 0 for t
/ 1. Then
star U := {x ∈ U : tx ∈ U for all t ∈ [0, 1]} is di¬eomorphic to E.

Proof. First remark that star U is c∞ -open. In fact, let c : R ’ E be smooth
with c(0) ∈ star U . Then • : R2 ’ E, de¬ned by •(t, s) := t c(s) is smooth and
maps [0, 1] — {0} into U . Since U is c∞ -open and R2 carries the c∞ -topology there
exists a neighborhood of [0, 1] — {0}, which is mapped into U , and in particular
there exists some µ > 0 such that c(s) ∈ star U for all |s| < µ. Thus c’1 (star U )
is open, i.e., star U is c∞ -open. Note that ρ satis¬es on star U the same boundary
condition as on U . So we may assume without loss of generality that U is radial.
Furthermore, we may assume that ρ = 1 locally around 0 and 0 < ρ ¤ 1 everywhere,
by composing with some function which is constantly 1 locally around [ρ(0), +∞).
Now we are going to replace ρ by a new function ρ, and we consider ¬rst the
˜
case, where E = R. We want that ρ satis¬es ρ (t)t < ρ(t) (which says that the
˜ ˜ ˜
tangent to ρ at t intersects the ρ-axis in the positive part) and that ρ(t) ¤ ρ(t),
˜ ˜ ˜
i.e., log —¦˜ ¤ log —¦ρ, and since we will choose ρ(0) = 1 = ρ(0) it is su¬cient to have
ρ ˜
ρ (t)t
˜ ρ (t)t
ρ˜ ρ
ρ = (log —¦˜) ¤ (log —¦ρ) = ρ or equivalently ρ(t) ¤ ρ(t) for t > 0. In order
ρ
˜ ˜
to obtain this we choose a smooth function h : R ’ R which satis¬es h(t) < 1,
and h(t) ¤ t for all t, and h(t) = t for t near 0, and we take ρ as solution of the
˜
following ordinary di¬erential equation

ρ(t)
˜ ρ (t)t
·h
ρ (t) =
˜ with ρ(0) = 1.
˜
t ρ(t)

Note that for t near 0, we have 1 h ρρ(t) = ρ (t) , and hence locally a unique
(t)t
t ρ(t)
smooth solution ρ exists. In fact, we can solve the equation explicitly, since
˜
s
(log —¦˜) (t) = ρ (t) = 1 · h ρρ(t) , and hence ρ(s) = exp( 0 1 · h( ρρ(t) ) dt), which
˜ (t)t (t)t
ρ ˜
ρ(t)
˜ t t
is smooth on the same interval as ρ is.
Note that if ρ is replaced by ρs : t ’ ρ(ts), then the corresponding solution ρs
satis¬es ρs = ρs . In fact,
˜

(˜s ) (t)
ρ s˜ (st)
ρ 1 st˜ (st)
ρ 1 stρ (st) 1 t(ρs ) (t)
(log —¦˜s ) (t) = =·
ρ = =h =h .
ρs (t)
˜ ρ(st)
˜ t ρ(st)
˜ t ρ(st) t ρs (t)

For arbitrary E and x ∈ E let ρx : Ux ’ R+ be given by ρx (t) := ρ(tx), and let
ρ : U ’ R+ be given by ρ(x) := ρx (1), where ρx is the solution of the di¬erential
˜ ˜
equation above with ρx in place of ρ.
Let us now show that ρ is smooth. Since U is c∞ -open, it is enough to consider
˜
a smooth curve x : R ’ U and show that t ’ ρ(x(t)) = ρ(x(t)) (1) is smooth.
˜ ˜
ρx(t) (s)s ρ (s x(t))(s x(t))
1 1
This is the case, since (t, s) ’ sh = sh is smooth,
ρx(t) (s) ρ(s x(t))


16.21
182 Chapter III. Partitions of unity 16.21

ρ (s x(t))(s x(t)) •(t,s)
satis¬es •(t, 0) = 0, and hence 1 h(•(t, s)) =
since •(t, s) := =
ρ(s x(t)) s s
ρ (s x(t))(x(t))
locally.
ρ(s x(t))
From ρsx (t) = ρ(tsx) = ρx (ts) we conclude that ρsx (t) = ρx (ts), and hence ρ(sx) =
˜
‚ ‚
ρx (s). Thus, ρ (x)(x) = ‚t |t=1 ρ(tx) = ‚t |t=1 ρx (t) = ρx (1) < ρx (1) = ρ(x). This
˜ ˜ ˜ ˜ ˜ ˜
shows that we may assume without loss of generality that ρ : U ’ (0, 1] satis¬es
the additional assumption ρ (x)(x) < ρ(x).
t
Note that fx : t ’ ρ(tx) is bijective from Ux := {t > 0 : tx ∈ U } to R+ , since 0
t
is mapped to 0, the derivative is positive, and ρ(tx) ’ ∞ if either ρ(tx) ’ 0 or
t ’ ∞ since ρ(tx) ¤ 1.
1
It remains to show that the bijection x ’ ρ(x) x is a di¬eomorphism. Obviously,
its inverse is of the form y ’ σ(y)y for some σ : E ’ R+ . They are inverse
1
to each other so ρ(σ(y)y) σ(y)y = y, i.e., σ(y) = ρ(σ(y)y) for y = 0. This is
an implicit equation for σ. Note that σ(y) = 1 for y near 0, since ρ has this
property. In order to show smoothness, let t ’ y(t) be a smooth curve in E.
Then it su¬ces to show that the implicit equation (σ —¦ y)(t) = ρ((σ —¦ y)(t) · y(t))
satis¬es the assumptions of the 2-dimensional implicit function theorem, i.e., 0 =

‚σ (σ ’ ρ(σ · y(t))) = 1 ’ ρ (σ · y(t))(y(t)), which is true, since multiplied with σ > 0
it equals σ ’ ρ (σ · y(t))(σ · y(t)) < σ ’ ρ(σ · y(t)) = 0.




16.21
183




Chapter IV
Smoothly Realcompact Spaces


17. Basic Concepts and Topological Realcompactness . . . . . . . . . . 184
18. Evaluation Properties of Homomorphisms . . . . . . . . . . . . . 188
19. Stability of Smoothly Realcompact Spaces . . . . . . . . . . . . . 203
20. Sets on which all Functions are Bounded . . . . . . . . . . . . . . 217

As motivation for the developments in this chapter let us tell a mathematical short
story which was posed as an exercise in [Milnor, Stashe¬, 1974, p.11]. For a ¬nite
dimensional Hausdor¬ second countable manifold M , one can prove that the space
of algebra homomorphisms Hom(C ∞ (M, R), R) equals M as follows. The kernel of
a homomorphism • : C ∞ (M, R) ’ R is an ideal of codimension 1 in C ∞ (M, R).
The zero sets Zf := f ’1 (0) for f ∈ ker • form a ¬lter of closed sets, since Zf © Zg =
Zf 2 +g2 , which contains a compact set Zf for a function f which is proper (i.e.,
compact sets have compact inverse images). Thus f ∈ker • Zf is not empty, it
contains at least one point x0 ∈ M . But then for any f ∈ C ∞ (M, R) the function
f ’ •(f )1 belongs to the kernel of •, so vanishes on x0 and we have f (x0 ) = •(f ).
This question has many rather complicated (partial) answers in any in¬nite dimen-
sional setting which are given in this chapter. One is able to prove that the answer
is positive surprisingly often, but the proofs are involved and tied intimately to the
class of spaces under consideration. The existing counter-examples are based on
rather trivial reasons. We start with setting up notation and listing some interesting
algebras of functions on certain spaces.
First we recall the topological theory of realcompact spaces from the literature and
discuss the connections to the concept of smooth realcompactness. For an algebra
homomorphism • : A ’ R on some algebra of functions on a space X we investigate
when •(f ) = f (x) for some x ∈ X for one function f , later for countably many, and
¬nally for all f ∈ A. We study stability of smooth realcompactness under pullback
along injective mappings, and also under (left) exact sequences. Finally we discuss
the relation between smooth realcompactness and bounding sets, i.e. sets on which
every function of the algebra is bounded. In this chapter, the ordering principle for
sections and results is based on the amount of evaluating properties obtained and
we do not aim for linearly ordered proofs. So we will often use results presented
later in the text. We believe that this is here a more transparent presentation than
the usual one. Most of the material in this chapter can also be found in the theses™
[Bistr¨m, 1993] and [Adam, 1993].
o
184 Chapter IV. Smoothly realcompact spaces 17.1

17. Basic Concepts and Topological Realcompactness

17.1. The setting. In [Hewitt, 1948, p.85] those completely regular topological
spaces were considered under the name Q-spaces, for which each real valued alge-
bra homomorphism on the algebra of all continuous functions is the evaluation at
some point of the space. Later on these spaces where called realcompact spaces.
Accordingly, we call a ˜space™ smoothly realcompact if this is true for ˜the™ algebra
of smooth functions. There are other algebras for which this question is interest-
ing, like polynomials, real analytic functions, C k -functions. So we will treat the
question in the following setting. Let
X be a set;
A ⊆ RX a point-separating subalgebra with unit; If X is a topological space we
also require that A ⊆ C(X, R); If X = E is a locally convex vector space we
also assume that A is invariant under all translations and contains the dual
E — of all continuous linear functionals;
XA the set X equipped with the initial topology with respect to A;
• : A ’ R an algebra homomorphism preserving the unit;
Zf := {x ∈ X : f (x) = •(f )} for f ∈ A;
Hom A be the set of all real valued algebra homomorphisms A ’ R preserving
the unit.
Moreover,
• is called F-evaluating for some subset F ⊆ A if there exists an x ∈ X with
•(f ) = f (x) for all f ∈ F; equivalently f ∈F Zf = …;
• is called m-evaluating for a cardinal number m if • is F-evaluating for all
F ⊆ A with cardinality of F at most m; This is most important for m = 1
and for m = ω, the ¬rst in¬nite cardinal number;
• is said to be ¯
1-evaluating if •(f ) ∈ f (X) for all f ∈ A.
• is said to be evaluating if • is A-evaluating, i.e., • = evx for some x ∈ X;
Homω A is the set of all ω-evaluating homomorphisms in Hom A;
A is called m-evaluating if • is m-evaluating for each algebra homomorphism
• ∈ Hom A;
A is called evaluating if • is evaluating for algebra homomorphism • ∈ Hom A;
X is called A-realcompact if A is evaluating; i.e., each algebra homomorphism
• ∈ Hom A is the evaluation at some point in X.
The algebra A is called
inversion closed if 1/f ∈ A for all f ∈ A with f (x) > 0 for every x ∈ X;
equivalently, if 1/f ∈ A for all f ∈ A with f nowhere 0 (use f 2 > 0).
bounded inversion closed if 1/f ∈ A for f ∈ A with f (x) > µ for some µ > 0 and
all x ∈ X;
C (∞) -algebra if h —¦ f ∈ A for all f ∈ A and h ∈ C ∞ (R, R);
C ∞ -algebra if h —¦ (f1 , . . . , fn ) ∈ A for all fj ∈ A and h ∈ C ∞ (Rn , R);
Clfs -algebra if it is a C ∞ -algebra which is closed under locally ¬nite sums, with


respect to a speci¬ed topology on X. This holds if A is local, i.e., it contains

17.1
17.1 17. Basic concepts and topological realcompactness 185

any function f such that for each x ∈ X there is some fx ∈ A with f = fx
near x.
Clfcs -algebra if it is a C ∞ -algebra which is closed under locally ¬nite countable


sums.
Interesting algebras are the following, where in this chapter in the notation we shall
generally omit the range space R.

y wC


‘‘
Cb


‘‘

y wC y wC © Ce
se
∞ ∞ ∞

g
e   
¢
Clfcs
ee
j
h
c

h ee
ee g
 
h  
¨
h
p e
s
Py w P 99
y wC y wC © C&
4&
G

x C∞
ω ω

&&
f

99
c


( x xx
A
9 &†
y wC
ω ω
Cconv

C(X) = C(X, R), the algebra of continuous functions on a topological space X.
It has all the properties from above.
Cb (X) = Cb (X, R), the algebra of bounded continuous functions on a topological
space X. It is only bounded inversion closed and a C ∞ -algebra, in general.
C ∞ (X) = C ∞ (X, R), the algebra of smooth functions on a Fr¨licher space X,
o
see (23.1), or on a smooth manifold X, see section (27). It has all properties
from above, where we may use the c∞ -topology.
C ∞ (E) © C(E), the algebra of smooth and continuous functions on a locally
convex space E. It has all properties from above, where we use the locally
convex topology on E.
∞ ∞
Cc (E) = Cc (E, R), the algebra of smooth functions, all of whose derivatives
are continuous on a locally convex space E. It has all properties from above,
again for the locally convex topology on E.
C ω (X) = C ω (X, R), the algebra of real analytic functions on a real analytic
manifold X. It is only inversion closed.
C ω (E) © C(E), the algebra of real analytic and continuous functions on a locally
convex space E. It is only inversion closed.
ω ω
Cc (E) = Cc (E, R), the algebra of real analytic functions, all of whose derivatives
are continuous on a locally convex space E. It is only inversion closed.
ω ω
Cconv (E) = Cconv (E, R), the algebra of globally convergent power series on a
locally convex space E.
Pf (E) = Polyf (E, R), the algebra of ¬nite type polynomials on a locally convex
space E, i.e. the algebra E Alg generated by E . This is the free commu-
tative algebra generated by the vector space E , see (18.12). It has none of
the properties from above.

17.1
186 Chapter IV. Smoothly realcompact spaces 17.3

P (E) = Poly(E, R), the algebra of polynomials on a locally convex space E, see
(5.15), (5.17), i.e. the homogeneous parts are given by bounded symmetric
multilinear mappings. No property from above holds.
∞ ∞ ∞
Clfcs (E) = Clfcs (E, R), the Clfcs -algebra (see below) generated by E , and hence
also called (E )∞ . Only the Clfs -property does not hold.

lfcs

17.2. Results. For completely regular topological spaces X and A = C(X) the
following holds:
(1) Due to [Hewitt, 1948, p.85 + p.60] & [Shirota, 1952, p.24], see also [Engel-
king, 1989, 3.12.22.g & 3.11.3]. The space X is called realcompact if all
algebra homomorphisms in Hom C(X) are evaluations at points of X, equiv-
alently, if X is a closed subspace of a product of R™s.
(2) Due to [Hewitt, 1948, p.61] & [Katˇtov, 1951, p.82], see also [Engelking,
e
1989, 3.11.4 & 3.11.5]. Hence every closed subspace of a product of realcom-
pact spaces is realcompact.
(3) Due to [Hewitt, 1948, p.85], see also [Engelking, 1989, 3.11.12]. Each Lin-
del¨f space is realcompact.
o
(4) Due to [Katˇtov, 1951, p.82], see also [Engelking, 1989, 5.5.10]. Paracom-
e
pact spaces are realcompact if and only if all closed discrete subspaces are
realcompact.
(5) Due to [Hewitt, 1950, p.170, p.175] & [Mackey, 1944], see also [Engelk-
ing, 1989, 3.11.D.a]. Discrete spaces are realcompact if and only if their
cardinality is non-measurable.
(6) Hence Banach spaces are realcompact if and only if their density (i.e., the
cardinality of a maximal discrete or of a minimal dense subset) or their
cardinality is non-measurable.
(7) [Shirota, 1952], see also [Engelking, 1989, 5.5.10 & 8.5.13.h]. A space of
non-measurable cardinality is realcompact if and only if it admits a complete
uniformity.
(8) Due to [Dieudonn´, 1939,] see also [Engelking, 1989, 8.5.13.a]. A space
e
admits a complete uniformity, i.e. is Dieudonn´ complete, if and only if it
e
is a closed subspace of a product of metrizable spaces

Realcompact spaces where introduced by [Hewitt, 1948, p.85] under the name Q-
compact spaces. The equivalence in (1) is due to [Shirota, 1952, p.24]. The results
(1) and (2) are proved in [Engelking, 1989] for a di¬erent notion of realcompactness,
which was shown to be equivalent to the original one by [Katˇtov, 1951], see also
e
[Engelking, 1989, 3.12.22.g].

17.3. Lemma. [Kriegl, Michor, Schachermayer, 1989, 2.2, 2.3]. Let A be ¯
1-eval-
uating. Then we have a topological embedding

δ : XA ’ prf —¦δ := f,
R,
A

with dense image in the closed subset Hom A ⊆ R. Hence X is A-realcompact
A
if and only if δ has closed image.

17.3
17.6 17. Basic concepts and topological realcompactness 187

Proof. The topology of XA is by de¬nition initial with respect to all f = prf —¦δ,
hence δ is an embedding. Obviously Hom A ⊆ A R is closed. Let • : A ’ R be an
algebra-homomorphism. For f ∈ A consider Zf . If A is 1-evaluating then by (18.8)
for any ¬nite subset F ⊆ A there exists an xF ∈ f ∈F Zf . Thus δ(xF )f = •(f )
for all f ∈ F. If A is only ¯1-evaluating, then we get as in the proof of (18.3) for
every µ > 0 a point xF ∈ X such that |f (xF ) ’ •(f )| < µ for all f ∈ F. Thus δ(xF )
lies in the corresponding neighborhood of (•(f ))f . Thus δ(X) is dense in Hom A.
Now X is A-realcompact if and only if δ has Hom A as image, and hence if and
only if the image of δ is closed.

17.4. Theorem. [Kriegl, Michor, Schachermayer, 1989, 2.4] & [Adam, Bistr¨m, o
Kriegl, 1995, 3.1]. The topology of pointwise convergence on Homω A is realcom-
pact. If XA is not realcompact then there exists an ω-evaluating homomorphism •
which is not evaluating.

Proof. We ¬rst show the weaker statement, that: If XA is not realcompact then
there exists a non-evaluating •, i.e., X is not A-realcompact.
Assume that X is A-realcompact, then A is 1-evaluating and hence by lemma
(17.3) δ : XA ’ A R is a closed embedding. Thus by (17.2.1) the space XA is
realcompact.
Now we give a proof of the stronger statement that Homω (A, R) is realcompact:
Assume that all sets of homomorphisms are endowed with the pointwise topology.
Let M ⊆ 2A be the family of all countable subsets of A containing the unit.
For M ∈ M, consider the topological space Homω M , where M denotes the
subalgebra generated by M . Obviously the family (δf )f ∈M , where δf (•) = •(f ), is
a countable subset of C(Homω M ) that separates the points in Homω M . Hence
Homω M = Hom C(Homω M ) by (18.25), since C(Homω M ) is ω-evaluating by
(18.11), i.e. Homω M is realcompact. Now Homω A is an inverse limit of the spaces
Homω M for M ∈ M. Since Homω M is Hausdor¬, we obtain that Homω A as
a closed subset of a product of realcompact spaces is realcompact by (17.2.2).
Since X is not realcompact in the topology XA , which is that induced from the em-
bedding into Homω A, we have that X = Homω A and the statement is proved.

17.5. Counter-example. [Kriegl, Michor, 1993, 3.6.2]. The locally convex space

R“
count of all points in the product with countable carrier is not C -realcompact, if
“ is uncountable and none-measurable.

Proof. By [Engelking, 1989, 3.10.17 & 3.11.2] the space R“ count is not realcompact,
in fact every c∞ -continuous function on it extends to a continuous function on R“ ,
see the proof of (4.27). Since the projections are smooth, XC ∞ is the product
topology. So the result follows from (17.4).

17.6. Theorem. [Kriegl, Michor, Schachermayer, 1989, 3.2] & [Garrido, G´mez,
o
Jaramillo, 1994, 1.8]. Let X be a realcompact and completely regular topological
space, let A be uniformly dense in C(X) and ¯
1-evaluating.

17.6
188 Chapter IV. Smoothly realcompact spaces 18.1

Then X is A-realcompact. Moreover, if X is A-paracompact then A is uniformly
dense in C(X).

In [Kriegl, Michor, Schachermayer, 1989] it is shown that Clfcs -algebra A is uni-
formly dense in C(X) if and only if A © Cb (X) is uniformly dense in Cb (X). One
may ¬nd also other equivalent conditions there.

Proof. Since A ⊆ C(X) we have that the identity XA ’ X is continuous, and
hence A ⊆ C(XA ) ⊆ C(X). For each of these point-separating algebras we consider
the natural inclusion δ of X into the product of factors R over the algebra, given
by prf —¦δ = f . It is a uniform embedding for the uniformity induced on X by this
algebra and the complete product uniformity on R with basis formed by the sets
Uf,µ := {(u, v) : | prf (u) ’ prf (v)| < µ} with µ > 0.
The condition that A ⊆ C is dense implies that the uniformities generated by
C(X), by C(XA ) and by A coincide and hence we will consider X as a uniform
space endowed with this uniform structure in the sequel. In fact for an arbitrarily
given continuous map f and µ > 0 choose a g ∈ A such that |g ’ f | < µ. Then

{(x, y) : |f (x) ’ f (y)| < µ} ⊆ {(x, y) : |g(x) ’ g(y)| < 3µ}
⊆ {(x, y) : |f (x) ’ f (y)| < 5µ}.

Since XA is realcompact, δC (XA ) = Hom(C(XA )) and hence is closed and so the
uniform structure on X is complete. And similarly also if X is realcompact. Thus,
the image δA (X) is a complete uniform subspace of A R and so it is closed with
respect to the product topology, i.e. X is A-realcompact by (17.3).

17.7. In the case of a locally convex vector space the last result (17.6) can be
slightly generalized to:

Result. [Bistr¨m, Lindstr¨m, 1993b, Thm.6]. For E a realcompact locally convex
o o
vector space, let E ⊆ A ⊆ C(E) be a ω-evaluating C (∞) -algebra which is invari-
ant under translations and homotheties. Moreover, we assume that there exists
a 0-neighborhood U in E such that for each f ∈ C(E) there exists g ∈ A with
supx∈U |f (x) ’ g(x)| < ∞.
Then E is A-realcompact.



18. Evaluation Properties of Homomorphisms

In this section we consider ¬rst properties near the evaluation property at single
functions, then evaluation properties for homomorphisms on countable many func-
tions, and ¬nally direct situations where all homomorphisms are point evaluations.

18.1. Remark. If • in Hom A is 1-evaluating (i.e., •(f ) ∈ f (X) for all f in A),
then • is ¯
1-evaluating.


18.1
18.3 18. Evaluation properties of homomorphisms 189

18.2. Lemma. [Bistr¨m, Bjon, Lindstr¨m, 1991, p.181]. For a topological space
o o
X the following assertions are equivalent:
(1) • is ¯
1-evaluating;
˜x
ˇ
(2) There exists x in the Stone-Cech compacti¬cation βX with •(f ) = f (˜) for
˜
all f ∈ A.
˜ ˇ
Here f denotes the extension of f : X ’ R ’ R∞ to the Stone-Cech-compacti¬-
cation βX with values in the 1-point compacti¬cation R∞ of R.
In [Garrido, G´mez, Jaramillo, 1994, 1.3] it is shown for a subalgebra of Cb (R) that
o
x need not be unique.
˜

Proof. For f ∈ A and µ > 0 let U (f, µ) := {x ∈ X : |•(f ) ’ f (x)| < µ}. Then U :=
{U (f, µ) : f ∈ A, µ > 0} is a ¬lter basis on X. Consider X as embedded into βX and
˜
take an ultra¬lter U on βX that is ¬ner than U. For f := (f1 ’•(f1 ))2 +(f2 ’•(f2 ))2
we have in fact
U (f1 , µ1 ) © U (f2 , µ2 ) ⊇ U (f, min{µ1 , µ2 }2 ).
˜
Let x ∈ βX be the point to which U converges. For an arbitrary function f in A
˜
˜˜ ˜
the ¬lter f (U) converges to •(f ) by construction. But f (U) ≥ f (U) = f (U), so
˜x ˜x ˜
•(f ) = f (˜). The converse is obvious since •(f ) = f (˜) ∈ f (βX) ⊆ f (X) ⊆ R∞ ,
and •(f ) ∈ R.

18.3. Lemma. [Adam, Bistr¨m, Kriegl, 1995, 4.1]. An algebra homomorphism •
o
is ¯
1-evaluating if and only if • extends (uniquely) to an algebra homomorphism on
A∞ , the C ∞ -algebra generated by A.

Proof. For C ∞ -algebras A, we have that

•(h —¦ (f1 , . . . , fn )) = h(•(f1 ), . . . , •(fn ))

for all h ∈ C ∞ (Rn , R) and f1 , . . . , fn in A.
In fact set a := (•(f1 ), . . . , •(fn )) ∈ Rn . Then
1
ha (x) · (xj ’ aj ),
h(x) ’ h(a) = ‚j h(a + t(x ’ a)) dt · (xj ’ aj ) = j
0 j¤n j¤n

1
where ha (x) := 0 ‚j h(a + t(x ’ a))dt. Applying • to this equation composed with
j
the fi one obtains

•(h —¦ (f1 , . . . , fn )) ’ h(•(f1 ), . . . , •(fn )) =
•(ha —¦ (f1 , . . . , fn )) · (•(fj ) ’ •(fj )) = 0.
= j
j¤n


(’) We de¬ne •(h —¦ (f1 , . . . , fn )) := h(•(f1 ), . . . , •(fn )). By what we have shown
˜
above (1-preserving) algebra homomorphisms are C ∞ -algebra homomorphisms and
hence this is the only candidate for an extension. This map is well de¬ned. Indeed,

18.3
190 Chapter IV. Smoothly realcompact spaces 18.5

let h —¦ (f1 , . . . , fn ) = k —¦ (g1 , . . . , gm ). For each µ > 0 there is a point x ∈ E such
that |•(fi ) ’ fi (x)| < µ for i = 1, ..., n, and |•(gj ) ’ gj (x)| < µ for j = 1, ..., m. In
˜x
fact by (18.2) there is a point x ∈ βX with •(f ) = f (˜) for
˜

n m
(fi ’ •(fi ))2 + (gj ’ •(gj ))2 ,
f :=
i=1 j=1


˜x
and hence •(fi ) = fi (˜) and •(gj ) = gj (˜). Now approximate x by x ∈ X.
˜x ˜
By continuity of h and k we obtain that

h(•(f1 ), . . . , •(fn )) = k(•(f1 ), . . . , •(fm )),

and we therefore have a well de¬ned extension of •. This extension is a homo-
morphism, since for every polynomial θ on Rm (or even for θ ∈ C ∞ (Rm )) and
gi := hi —¦ (f1 , . . . , fni ) ∈ A∞ we have
i i


1 m
•(θ —¦ (g1 , . . . , gm )) = •(θ —¦ (h1 — . . . — hm ) —¦ (f1 , . . . , fnm ))
˜ ˜
1 m
= (θ —¦ (h1 — . . . — hm ))(•(f1 ), . . . , •(fnm ))
1 1 m m
= θ(h1 (•(f1 ), . . . , •(fn1 )), . . . , hm (•(f1 ), . . . , •(fnm ))
= θ(•(g1 ), . . . , •(gm )).
˜ ˜


(⇐) Suppose there is some f ∈ A with •(f ) ∈ f (X). Then we may ¬nd an
/
h ∈ C (R) with h(•(f )) = 1 and carr h © f (X) = …. Since A∞ is a C ∞ -algebra,


we conclude from what we said above that •(h —¦ f ) = h(•(f )) = 1. But since
˜
h —¦ f = 0 we arrive at a contradiction.

18.4. Proposition. [Garrido, G´mez, Jaramillo, 1994, 1.2]. If A is bounded
o
inversion closed and • ∈ Hom A then • is ¯
1-evaluating.

Proof. We assume indirectly that there is a function f ∈ A with •(f ) ∈ f (X).
Let µ := inf x∈X |•(f ) ’ f (x)| and g(x) := 1 (•(f ) ’ f (x)). Then g ∈ A, •(g) = 0
µ
1
and |g(x)| = µ |•(f ) ’ f (x)| ≥ 1 for each x ∈ X. Thus 1/g ∈ A. But then
1 = •(g · 1/g) = •(g)•(1/g) = 0 gives a contradiction.

18.5. Lemma. Any C (∞) -algebra is bounded inversion closed.
Moreover, it is stable under composition with smooth locally de¬ned functions, which
contain the closure of the image in its domain of de¬nition.

Proof. Let A be a C ∞ -algebra (resp. C (∞) -algebra), n a natural number (resp.
n = 1), U ⊆ Rn open, h ∈ C ∞ (U, R), f := (f1 , . . . , fn ), with fi ∈ A such that
f (X) ⊆ U , then h —¦ f ∈ A . Indeed, choose ρ ∈ C ∞ (R) with ρ|f (X) = 1 and
supp ρ ⊆ U . Then k := ρ · h is a globally smooth function and h —¦ f = k —¦ f ∈ A.


18.5
18.9 18. Evaluation properties of homomorphisms 191

18.6. Lemma. Any inverse closed algebra A is 1-evaluating.

By (18.10) the converse is wrong.

Proof. Let f ∈ A and assume indirectly that Zf = …. Let g := f ’ •(f ). Then
g ∈ A and g(x) = 0 for all x ∈ X, by which 1/g ∈ A since A is inverse-closed. But
then 1 = •(g · 1/g) = •(g)•(1/g) = 0, which is a contradiction.

18.7. Proposition. [Bistr¨m, Jaramillo, Lindstr¨m, 1995, Lem.14] & [Adam, Bi-
o o
str¨m, Kriegl, 1995, 4.2]. For • in Hom A the following statements are equivalent:
o
(1) • is 1-evaluating.
(2) • extends to a unique (1-evaluating) homomorphism on the algebra RA :=
{f /g : f, g ∈ A, 0 ∈ g(X)}.
/
(3) • extends to a unique (1-evaluating) homomorphism on the following C ∞ -
algebra A ∞ constructed from A:

A := {h —¦ (f1 , . . . , fn ) :fi ∈ A, (f1 , . . . , fn )(X) ⊆ U,
U open in some Rn , h ∈ C ∞ (U )}.

Proof. (1) ’ (3) We de¬ne •(h —¦ (f1 , . . . , fn )) := h(•(f1 ), . . . , •(fn )). Since there
exists by (18.8) an x with •(fi ) = fi (x), we have (•(f1 ), . . . , •(fn )) ∈ U , hence the
right side makes sense. The rest follows in the same way as in the proof of (18.3).

(3) ’ (2) Existence is obvious, since RA ⊆ A , and uniqueness follows from the
de¬nition of RA.
(2) ’ (1) Since RA is inverse-closed, the extension of • to this algebra is 1-
evaluating by (18.6), hence the same is true for • on A.

18.8. Lemma. Every 1-evaluating homomorphism is ¬nitely evaluating.

Proof. Let F be a ¬nite subset of A. De¬ne a function f : X ’ R by

(g ’ •(g))2 .
f :=
g∈F

Then f ∈ A and •(f ) = 0. By assumption there is a point x ∈ X with •(f ) = f (x).
Hence g(x) = •(g) for all g ∈ F.

18.9. Theorem. Automatic boundedness. [Kriegl, Michor, 1993] & [Arias-
de-Reyna, 1988] Every 1-evaluating homomorphism • ∈ Hom A is positive, i.e.,
0 ¤ •(f ) for all 0 ¤ f ∈ A. Moreover we even have •(f ) > 0 for f ∈ A with
f (x) > 0 for all x ∈ X.
Every positive homomorphism • ∈ Hom A is bounded for any convenient algebra
structure on A.

A convenient algebra structure on A is a locally convex topology, which turns A
into a convenient vector space and such that the multiplication A — A ’ A is
bounded, compare (5.21).

18.9
192 Chapter IV. Smoothly realcompact spaces 18.11

Proof. Positivity: Let f1 ¤ f2 . By (17) and (18.8) there exists an x ∈ X such
that •(fi ) = fi (x) for i = 1, 2. Thus •(f1 ) = f1 (x) ¤ f2 (x) = •(f2 ). Note that if
f (x) > 0 for all x, then •(f ) > 0.
Boundedness: Suppose fn is a bounded sequence, but |•(fn )| is unbounded. By
2
replacing fn by fn we may assume that fn ≥ 0 and hence also •(fn ) ≥ 0. Choosing
a subsequence we may even assume that •(fn ) ≥ 2n . Now consider n 21 fn . This
n

series converges Mackey, and since the bornology on A is by assumption complete
the limit is an element f ∈ A. Applying • yields

N N
1 1 1 1

•(f ) = • fn + fn = •(fn ) + • fn
2n 2n 2n 2n
n=0 n=0
n>N n>N
N N
1 1
≥ •(fn ) + 0 = •(fn ),
2n 2n
n=0 n=0


where we used the monotonicity of • applied to n>N 21 fn ≥ 0. Thus the series
n
N
N ’ n=0 21 •(fn ) is bounded and increasing, hence converges, but its summands
n

are bounded by 1 from below. This is a contradiction.

18.10. Lemma. For a locally convex vector space E the algebra Pf (E) is 1-
evaluating.

More on the algebra Pf (E) can be found in (18.27), (18.28), and (18.12).

Proof. Every ¬nite type polynomial p is a polynomial in a ¬nite number of linearly
independent functionals 1 , . . . , n in E . So there is for each i = 1, . . . , n some point
ai ∈ E such that i (ai ) = •( i ) and j (ai ) = 0 for all j = i. Let a = a1 +· · ·+an ∈ E.
Then i (a) = i (ai ) = •( i ) for i = 1, . . . , n hence •(p) = p(a).


Countably Evaluating Homomorphisms

18.11. Theorem. Idea of [Arias-de-Reyna, 1988, proof of thm.8], [Adam, Bis-

tr¨m, Kriegl, 1995, 2.5]. For a topological space X any Clfcs -algebra A ⊆ C(X) is
o
closed under composition with local smooth functions and is ω-evaluating.

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