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.

˜