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4.14. Lemma. Let U be a c∞ -open subset of a locally convex space, let µn ’ ∞
be a real sequence, and let f : U ’ F be a mapping which is bounded on each
µ-converging sequence in U . Then f is bounded on every bornologically compact
subset (i.e. compact in some EB ) of U .

Proof. By composing with linear functionals we may assume that F = R. Let
K ⊆ EB © U be compact in EB for some bounded absolutely convex set B. Assume
that f (K) is not bounded. So there is a sequence (xn ) in K with |f (xn )| ’ ∞.
Since K is compact in the normed space EB we may assume that (xn ) converges to
x ∈ K. By passing to a subsequence we may even assume that (xn ) is µ-converging.
Contradiction.

4.15. Lemma. Let U be c∞ -open in E — R and K ⊆ R be compact. Then
U0 := {x ∈ E : {x} — K ⊆ U } is c∞ -open in E.

Proof. Let x : R ’ E be a smooth curve in E with x(0) ∈ U0 , i.e. (x(0), t) ∈ U
for all t ∈ K. We have to show that x(s) ∈ U0 for all s near 0. So consider the
smooth map x — R : R — R ’ E — R. By assumption (x — R)’1 (U ) is open in
c∞ (R2 ) = R2 . It contains the compact set {0} — K and hence also a W — K for
some neighborhood W of 0 in R. But this amounts in saying that x(W ) ⊆ U0 .

4.16. The c∞ -topology of a product. Consider the product E — F of two
locally convex vector spaces. Since the projections onto the factors are linear and
continuous, and hence smooth, we always have that the identity mapping c∞ (E —
F ) ’ c∞ (E)—c∞ (F ) is continuous. It is not always a homeomorphism: Just take a
bounded separately continuous bilinear functional, which is not continuous (like the
evaluation map) on a product of spaces where the c∞ -topology is the bornological
topology.
However, if one of the factors is ¬nite dimensional the product is well behaved:

4.16
42 Chapter I. Calculus of smooth mappings 4.19

Corollary. For any locally convex space E the c∞ -topology of E —Rn is the product
topology of the c∞ -topologies of the two factors, so that we have c∞ (E — Rn ) =
c∞ (E) — Rn .

Proof. This follows recursively from the special case E — R, for which we can
proceed as follows. Take a c∞ -open neighborhood U of some point (x, t) ∈ E — R.
Since the inclusion map s ’ (x, s) from R into E — R is continuous and a¬ne, the
inverse image of U in R is an open neighborhood of t. Let™s take a smaller compact
neighborhood K of t. Then by the previous lemma U0 := {y ∈ E : {y} — K ⊆ U }
is a c∞ -open neighborhood of x, and hence U0 — K o is a neighborhood of (x, t) in
c∞ (E) — R, what was to be shown.

4.17. Lemma. Let U be c∞ -open in a locally convex space and x ∈ U . Then the
star stx (U ) := {x + v : x + »v ∈ U for all |»| ¤ 1} with center x in U is again
c∞ -open.

Proof. Let c : R ’ E be a smooth curve with c(0) ∈ stx (U ). The smooth mapping
f : (t, s) ’ (1 ’ s)x + sc(t) maps {0} — {s : |s| ¤ 1} into U . So there exists δ > 0
with f ({(t, s) : |t| < δ, |s| ¤ 1}) ⊆ U . Thus, c(t) ∈ stx (U ) for |t| < δ.

4.18. Lemma. The (absolutely) convex hull of a c∞ -open set is again c∞ -open.

Proof. Let U be c∞ -open in a locally convex vector space E.
For each x ∈ U the set

Ux := {x + t(y ’ x) : t ∈ [0, 1], y ∈ U } = U ∪ (x + t(U ’ x))
0<t¤1


is c∞ -open. The convex hull can be constructed by applying n times the operation
U ’ x∈U Ux and taking the union over all n ∈ N, which respects c∞ -openness.
The absolutely convex hull can be obtained by forming ¬rst {» : |»| = 1}.U =

|»|=1 »U which is c -open, and then forming the convex hull.

4.19. Corollary. Let E be a bornological convenient vector space containing a
nonempty c∞ -open subset which is either locally compact or metrizable in the c∞ -
topology. Then the c∞ -topology on E is locally convex. In the ¬rst case E is ¬nite
dimensional, in the second case E is a Fr´chet space.
e

Proof. Let U ⊆ E be a c∞ -open metrizable subset. We may assume that 0 ∈ U .
Then there exists a countable neighborhood basis of 0 in U consisting of c∞ -open
sets. This is also a neighborhood basis of 0 for the c∞ -topology of E. We take
the absolutely convex hulls of these open sets, which are again c∞ -open by (4.18),
and obtain by (4.5) a countable neighborhood basis for the bornologi¬cation of the
locally convex topology, so the latter is metrizable and Fr´chet, and by (4.11) it
e
equals the c∞ -topology.
If U is locally compact in the c∞ -topology we may ¬nd a c∞ -open neighborhood V
of 0 with compact closure V in the c∞ -topology. By lemma (4.18) the absolutely

4.19
4. The c∞ -topology
4.23 43

convex hull of V is also c∞ -open, and by (4.5) it is also open in the bornologi¬cation
Eborn of E. The set V is then also compact in Eborn , hence precompact. So the
absolutely convex hull of V is also precompact by (52.6). Therefore, the absolutely
convex hull of V is a precompact neighborhood of 0 in Eborn , thus E is ¬nite
dimensional by (52.5). So Eborn = c∞ (E).

Now we describe classes of spaces where c∞ E = E or where c∞ E is not even a
topological vector space. Finally, we give an example where the c∞ -topology is not
completely regular. We begin with the relationship between the c∞ -topology and
the locally convex topology on locally convex vector spaces.

4.20. Proposition. Let E and F be bornological locally convex vector spaces. If
there exists a bilinear smooth mapping m : E — F ’ R that is not continuous with
respect to the locally convex topologies, then c∞ (E — F ) is not a topological vector
space.

We shall show in lemma (5.5) below that multilinear mappings are smooth if and
only if they are bounded.

Proof. Suppose that addition c∞ (E — F ) — c∞ (E — F ) ’ c∞ (E — F ) is continuous
with respect to the product topology. Using the continuous inclusions c∞ E ’
c∞ (E — F ) and c∞ F ’ c∞ (E — F ) we can write m as composite of continuous
+ m
maps as follows: c∞ E — c∞ F ’ c∞ (E — F ) — c∞ (E — F ) ’ c∞ (E — F ) ’ R.
’ ’
Thus, for every µ > 0 there are 0-neighborhoods U and V with respect to the
c∞ -topology such that m(U — V ) ⊆ (’µ, µ). Then also m( U — V ) ⊆ (’µ, µ)
where denotes the absolutely convex hull. By (4.5) one concludes that m is
continuous with respect to the locally convex topology, a contradiction.

4.21. Corollary. Let E be a non-normable bornological locally convex space. Then
c∞ (E — E ) is not a topological vector space.

Proof. By (4.20) it is enough to show that ev : E — E ’ R is not continuous for
the bornological topologies on E and E ; if it were so there was be a neighborhood
U of 0 in E and a neighborhood U of 0 in E such that ev(U — U ) ⊆ [’1, 1]. Since
U is absorbing, U is scalarwise bounded, hence a bounded neighborhood. Thus,
E is normable.

4.22. Remark. In particular, for a Fr´chet Schwartz space E and its dual E we
e
have c∞ (E —E ) = c∞ E —c∞ E , since by (4.11) we have c∞ E = E and c∞ E = E ,
so equality would contradict corollary (4.21).
In order to get a large variety of spaces where the c∞ -topology is not a topological
vector space topology the next three technical lemmas will be useful.

4.23. Lemma. Let E be a locally convex vector space. Suppose a double sequence
bn,k in E exists which satis¬es the following two conditions:
(b™) For every sequence k ’ nk the sequence k ’ bnk ,k has no accumulation
point in c∞ E.
(b”) For all k the sequence n ’ bn,k converges to 0 in c∞ E.

4.23
44 Chapter I. Calculus of smooth mappings 4.25

Suppose furthermore that a double sequence cn,k in E exists that satis¬es the fol-
lowing two conditions:
(c™) For every 0-neighborhood U in c∞ E there exists some k0 such that cn,k ∈ U
for all k ≥ k0 and all n.
(c”) For all k the sequence n ’ cn,k has no accumulation point in c∞ E.
Then c∞ E is not a topological vector space.
Proof. Assume that the addition c∞ E — c∞ E ’ c∞ E is continuous. In this
proof convergence is meant always with respect to c∞ E. We may without loss of
generality assume that cn,k = 0 for all n, k, since by (c”) we may delete all those
cn,k which are equal to 0. Then we consider A := {bn,k + µn,k cn,k : n, k ∈ N} where
the µn,k ∈ {’1, 1} are chosen in such a way that 0 ∈ A.
/
We ¬rst show that A is closed in the sequentially generated topology c∞ E: Let
bni ,ki +µni ,ki cni ,ki ’ x, and assume that (ki ) is unbounded. By passing if necessary
to a subsequence we may even assume that i ’ ki is strictly increasing. Then
cni ,ki ’ 0 by (c™), hence bni ,ki ’ x by the assumption that addition is continuous,
which is a contradiction to (b™). Thus, (ki ) is bounded, and we may assume it to be
constant. Now suppose that (ni ) is unbounded. Then bni ,k ’ 0 by (b”), and hence
µni ,k cni ,k ’ x, and for a subsequence where µ is constant one has cni ,k ’ ±x,
which is a contradiction to (c”). Thus, ni is bounded as well, and we may assume
it to be constant. Hence, x = bn,k + µn,k cn,k ∈ A.
By the assumed continuity of the addition there exists an open and symmetric
0-neighborhood U in c∞ E with U + U ⊆ E \ A. For K su¬ciently large and n
arbitrary one has cn,K ∈ U by (c™). For such a ¬xed K and N su¬ciently large
bN,K ∈ U by (b™). Thus, bN,K + µN,K cN,K ∈ A, which is a contradiction.
/
Let us now show that many spaces have a double sequence cn,k as in the above
lemma.
4.24. Lemma. Let E be an in¬nite dimensional metrizable locally convex space.
Then a double sequence cn,k subject to the conditions (c™) and (c”) of (4.23) exists.
Proof. If E is normable we choose a sequence (cn ) in the unit ball without accu-
1
mulation point and de¬ne cn,k := k cn . If E is not normable we take a countable
increasing family of non-equivalent seminorms pk generating the locally convex
1
topology, and we choose cn,k with pk (cn,k ) = k and pk+1 (cn,k ) > n.
Next we show that many spaces have a double sequence bn,k as in lemma (4.23).
4.25. Lemma. Let E be a non-normable bornological locally convex space hav-
ing a countable basis of its bornology. Then a double sequence bn,k subject to the
conditions (b™) and (b”) of (2.11) exists.
Proof. Let Bn (n ∈ N) be absolutely convex sets forming an increasing basis of
the bornology. Since E is not normable the sets Bn can be chosen such that Bn
1
does not absorb Bn+1 . Now choose bn,k ∈ n Bk+1 with bn,k ∈ Bk .
/
Using these lemmas one obtains the

4.25
4. The c∞ -topology
4.26 45

4.26. Proposition. For the following bornological locally convex spaces the c∞ -
topology is not a vector space topology:
(i) Every bornological locally convex space that contains as c∞ -closed subspaces
an in¬nite dimensional Fr´chet space and a space which is non-normable in
e
the bornological topology and having a countable basis of its bornology.
(ii) Every strict inductive limit of a strictly increasing sequence of in¬nite di-
mensional Fr´chet spaces.
e
(iii) Every product for which at least 2„µ0 many factors are non-zero.
(iv) Every coproduct for which at least 2„µ0 many summands are non-zero.

Proof. (i) follows directly from the last 3 lemmas.
(ii) Let E be the strict inductive limit of the spaces En (n ∈ N). Then E contains
the in¬nite dimensional Fr´chet space E1 as subspace. The subspace generated
e
by points xn ∈ En+1 \ En (n ∈ N) is bornologically isomorphic to R(N) , hence its
bornology has a countable basis. Thus, by (i) we are done.
(iii) Such a product E contains the Fr´chet space RN as complemented subspace.
e
We want to show that R(N) is also a subspace of E. For this we may assume that the
index set J is RN and all factors are equal to R. Now consider the linear subspace
E1 of the product generated by the elements xn ∈ E = RN , where (xn )j := j(n) for
every j ∈ J = RN . The linear map R(N) ’ E1 ⊆ E that maps the n-th unit vector
to xn is injective, since for a given ¬nite linear combination tn xn = 0 the j-th
|tn |. It is a morphism since R(N) carries
coordinate for j(n) := sign(tn ) equals
the ¬nest structure. So it remains to show that it is a bornological embedding.
We have to show that any bounded B ⊆ E1 is contained in a subspace generated
by ¬nitely many xn . Otherwise, there would exist a strictly increasing sequence
(nk ) and bk = n¤nk tk xn ∈ B with tk k = 0. De¬ne an index j recursively by
n n
k ’1 k
j(n) := n|tn | · sign m<n tm j(m) if n = nk and j(n) := 0 if n = nk for all k.
Then the absolute value of the j-th coordinate of bk evaluates as follows:

|(bk )j | = tk j(n) = tk j(n) + tk k j(nk )
n n n
n<nk
n¤nk

tk j(n) + |tk k j(nk )| ≥ |tk k j(nk )| ≥ nk .
= n n n
n<nk


Hence, the j-th coordinates of {bk : k ∈ N} are unbounded with respect to k ∈ N,
thus B is unbounded.
(iv) We can not apply lemma (4.23) since every double sequence has countable
support and hence is contained in the dual R(A) of a Fr´chet Schwartz space RA for
e
some countable subset A ‚ J. It is enough to show (iv) for R(J) where J = N ∪ c0 .
Let A := {jn (en + ej ) : n ∈ N, j ∈ c0 , jn = 0 for all n}, where en and ej denote
the unit vectors in the corresponding summand. The set A is M-closed, since its
intersection with ¬nite subsums is ¬nite. Suppose there exists a symmetric M-open
0-neighborhood U with U + U ⊆ E \ A. Then for each n there exists a jn = 0
with jn en ∈ U . We may assume that n ’ jn converges to 0 and hence de¬nes

4.26
46 Chapter I. Calculus of smooth mappings 4.27

an element j ∈ c0 . Furthermore, there has to be an N ∈ N with jN ej ∈ U , thus
jN (eN + ej ) ∈ (U + U ) © A, in contradiction to U + U ⊆ E \ A.

Remark. A nice and simple example where one either uses (i) or (ii) is RN • R(N) .
The locally convex topology on both factors coincides with their c∞ -topology (the
¬rst being a Fr´chet (Schwartz) space, cf. (i) of (4.11), the second as dual of the
e
¬rst, cf. (ii) of (4.11)); but the c∞ -topology on their product is not even a vector
space topology.

From (ii) it follows also that each space Cc (M, R) of smooth functions with com-
pact support on a non-compact separable ¬nite dimensional manifold M has the
property, that the c∞ -topology is not a vector space topology.

4.27. Although the c∞ -topology on a convenient vector space is always functionally
separated, hence Hausdor¬, it is not always completely regular as the following
example shows.

Example. The c∞ -topology is not completely regular. The c∞ -topology of
RJ is not completely regular if the cardinality of J is at least 2„µ0 .

Proof. It is enough to show this for an index set J of cardinality 2„µ0 , since the
corresponding product is a complemented subspace in every product with larger
index set. We prove the theorem by showing that every function f : RJ ’ R
which is continuous for the c∞ -topology is also continuous with respect to the
locally convex topology. Hence, the completely regular topology associated to the
c∞ -topology is the locally convex topology of E. That these two topologies are
di¬erent was shown in (4.8). We use the following theorem of [Mazur, 1952]: Let
E0 := {x ∈ RJ : supp(x) is countable}, and let f : E0 ’ R be sequentially
continuous. Then there is some countable subset A ‚ J such that f (x) = f (xA ),
where in this proof xA is de¬ned as xA (j) := x(j) for j ∈ A and xA (j) = 0 for
j ∈ A. Every sequence which is converging in the locally convex topology of E0
/
is contained in a metrizable complemented subspace RA for some countable A and
therefore is even M-convergent. Thus, this theorem of Mazur remains true if f is
assumed to be continuous for the M-closure topology. This generalization follows
also from the fact that c∞ E0 = E0 , cf. (4.12). Now let f : RJ ’ R be continuous
for the c∞ -topology. Then f |E0 : E0 ’ R is continuous for the c∞ -topology, and
hence there exists a countable set A0 ‚ J such that f (x) = f (xA0 ) for any x ∈ E0 .
We want to show that the same is true for arbitrary x ∈ RJ . In order to show this
we consider for x ∈ RJ the map •x : 2J ’ R de¬ned by •x (A) := f (xA )’f (xA©A0 )
for any A ⊆ J, i.e. A ∈ 2J . For countable A one has xA ∈ E0 , hence •x (A) = 0.
Furthermore, •x is sequentially continuous where one considers on 2J the product
topology of the discrete factors 2 = {0, 1}. In order to see this consider a converging
sequence of subsets An ’ A, i.e. for every j ∈ J one has for the characteristic
functions χAn (j) = χA (j) for n su¬ciently large. Then {n(xAn ’ xA ) : n ∈ N} is
bounded in RJ since for ¬xed j ∈ J the j-th coordinate equals 0 for n su¬ciently
large. Thus, xAn converges Mackey to xA , and since f is continuous for the c∞ -
topology •x (An ) ’ •x (A). Now we can apply another theorem of [Mazur, 1952]:

4.27
4. The c∞ -topology
4.30 47

Any function f : 2J ’ R that is sequentially continuous and is zero on all countable
subsets of J is identically 0, provided the cardinality of J is smaller than the ¬rst
inaccessible cardinal. Thus, we conclude that 0 = •x (J) = f (x) ’ f (xAn ) for all
x ∈ RJ . Hence, f factors over the metrizable space RA0 and is therefore continuous
for the locally convex topology.

In general, the trace of the c∞ -topology on a linear subspace is not its c∞ -topology.
However, for c∞ -closed subspaces this is true:

4.28. Lemma. Closed embedding lemma. Let E be a linear c∞ -closed sub-
space of F . Then the trace of the c∞ -topology of F on E is the c∞ -topology on
E

Proof. Since the inclusion is continuous and hence bounded it is c∞ -continuous.
Therefore, it is enough to show that it is closed for the c∞ -topologies. So let A ⊆ E
be c∞ E-closed. And let xn ∈ A converge Mackey towards x in F . Then x ∈ E,
since E is assumed to be c∞ -closed, and hence xn converges Mackey to x in E.
Since A is c∞ -closed in E, we have that x ∈ A.

We will give an example in (4.33) below which shows that c∞ -closedness of the
subspace is essential for this result. Another example will be given in (4.36).

4.29. Theorem. The c∞ -completion. For any locally convex space E there
˜
exists a unique (up to a bounded isomorphism) convenient vector space E and a
˜
bounded linear injection i : E ’ E with the following universal property:
Each bounded linear mapping : E ’ F into a convenient vector space F
has a unique bounded extension ˜ : E ’ F such that ˜ —¦ i = .
˜
˜
Furthermore, i(E) is dense for the c∞ -topology in E.

˜
Proof. Let E be the c∞ -closure of E in the locally convex completion Eborn of the
˜
bornologi¬cation Eborn of E. The inclusion i : E ’ E is bounded (not continuous
˜
in general). By (4.28) the c∞ -topology on E is the trace of the c∞ -topology on
˜
Eborn . Hence, i(E) is dense also for the c∞ -topology in E.
Using the universal property of the locally convex completion the mapping has
a unique extension ˆ : Eborn ’ F into the locally convex completion of F , whose
restriction to E has values in F , since F is c∞ -closed in F , so it is the desired ˜.
˜
˜
Uniqueness follows, since i(E) is dense for the c∞ -topology in E.

4.30. Proposition. c∞ -completion via c∞ -dense embeddings. Let E be
c∞ -dense and bornologically embedded into a c∞ -complete locally convex space F .
If E ’ F has the extension property for bounded linear functionals, then F is
bornologically isomorphic to the c∞ -completion of E.

Proof. We have to show that E ’ F has the universal property for extending
bounded linear maps T into c∞ -complete locally convex spaces G. Since we are

4.30
48 Chapter I. Calculus of smooth mappings 4.31

only interested in bounded mappings, we may take the bornologi¬cation of G and
hence may assume that G is bornological. Consider the following diagram

y w F@
E
’@
’u @
’ R@
’ ‘ @@—¦ T
’ »
’δ  pr‘‘@
T

’ 
G

“@
‘A
’ »
u”!
»


wR
G
The arrow δ, given by δ(x)» := »(x), is a bornological embedding, i.e. the image of
a set is bounded if and only if the set is bounded, since B ⊆ G is bounded if and
only if »(B) ⊆ R is bounded for all » ∈ G , i.e. δ(B) ⊆ G R is bounded.
By assumption, the dashed arrow on the right hand side exists, hence by the uni-
˜
versal property of the product the dashed vertical arrow (denoted T ) exists. It
˜
remains to show that it has values in the image of δ. Since T is bounded we have
c∞ c∞
c∞
˜ ˜ ˜
) ⊆ T (E) ⊆ δ(G)
T (F ) = T (E = δ(G),

since G is c∞ -complete and hence also δ(G), which is thus c∞ -closed.
˜

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