is continuous at 0. It can be rewritten as t ’ df (c(0) + s(c(t) ’ c(0))).c1 (t) ds,

0

where c1 is the smooth curve given by

c(t)’c(0)

for t = 0

t

t’ .

c (0) for t = 0

3.18

34 Chapter I. Calculus of smooth mappings 4.1

Since h : R2 ’ U — E given by

(t, s) ’ (c(0) + s(c(t) ’ c(0)), c1 (t))

is smooth, the map t ’ s ’ df (c(0) + s(c(t) ’ c(0))).c1 (t) is smooth R ’

1

C ∞ (R, F ), and hence t ’ df (c(0) + s(c(t) ’ c(0))).c1 (t) ds is smooth, and hence

0

continuous.

For general g we have

‚ ‚

d(f —¦ g)(x)(v) = (f —¦ g)(x + tv) = (df )(g(x + 0v))( ‚t (g(x + tv)))

‚t 0 0

= (df )(g(x))(dg(x)(v)).

3.19. Lemma. Two locally convex spaces are locally di¬eomorphic if and only if

they are linearly di¬eomorphic.

Any smooth and 1-homogeneous mapping is linear.

Proof. By the chain rule the derivatives at corresponding points give the linear

di¬eomorphisms.

‚

For a 1-homogeneous mapping f one has df (0)v = f (tv) = f (v), and this is

‚t 0

linear in v.

4. The c∞ -Topology

4.1. De¬nition. A locally convex vector space E is called bornological if and only

if the following equivalent conditions are satis¬ed:

(1) For any locally convex vector space F any bounded linear mapping T : E ’

F is continuous; it is su¬cient to know this for all Banach spaces F .

(2) Every bounded seminorm on E is continuous.

(3) Every absolutely convex bornivorous subset is a 0-neighborhood.

A radial subset U (i.e. [0, 1]U ⊆ U ) of a locally convex space E is called bornivorous

if it absorbs each bounded set, i.e. for every bounded B there exists r > 0 such that

[0, r]U ⊇ B.

Proof. (3 ’ 2), since for a > 0 the inverse images under bounded seminorms of

intervals (’∞, a) are absolutely convex and bornivorous. In fact, let B be bounded

and a > 0. Then by assumption p(B) is bounded, and so there exists a C > 0 with

p(B) ⊆ C · (’∞, a). Hence, B ⊆ C · p’1 (’∞, a).

(2 ’ 1), since p —¦ T is a bounded seminorm, for every continuous seminorm on F .

(2 ’ 3), since the Minkowski-functional p generated by an absolutely convex bor-

nivorous subset is a bounded seminorm.

(1 ’ 2) Since the canonical projection T : E ’ E/ker p is bounded, for any

bounded seminorm p, it is by assumption continuous. Hence, p = p—¦T is continuous,

˜

where p denotes the canonical norm on E/ker p induced from p.

˜

4.1

4. The c∞ -topology

4.3 35

4.2. Lemma. Bornologi¬cation. The bornologi¬cation Eborn of a locally convex

space can be described in the following equivalent ways:

(1) It is the ¬nest locally convex structure having the same bounded sets;

(2) It is the ¬nal locally convex structure with respect to the inclusions EB ’ E,

where B runs through all bounded (closed) absolutely convex subsets.

Moreover, Eborn is bornological. For any locally convex vector space F the contin-

uous linear mappings Eborn ’ F are exactly the bounded linear mappings E ’ F .

The continuous seminorms on Eborn are exactly the bounded seminorms of E. An

absolutely convex set is a 0-neighborhood in Eborn if and only if it is bornivorous,

i.e. absorbs bounded sets.

Proof. Let Eborn be the vector space E supplied with the ¬nest locally convex

structure having the same bounded sets as E.

(‘) Since all bounded absolutely convex sets B in E are bounded in Eborn , the

inclusions EB ’ Eborn are bounded and hence continuous. Thus, the ¬nal structure

on E induced by the inclusions EB ’ E is ¬ner than the structure of Eborn .

(“) Since every bounded subset of E is contained in some absolutely convex bounded

set B ⊆ E it has to be bounded in the ¬nal structure given by all inclusions

EB ’ E. Hence, this ¬nal structure has exactly the same bounded sets as E, and

we have equality between the ¬nal structure and that of Eborn .

A seminorm p on E is bounded, if and only if p(B) is bounded for all bounded B,

and this is exactly the case if p|EB is a bounded (=continuous) seminorm on EB

for all B, or equivalently that p is a continuous seminorm for the ¬nal structure

Eborn on E induced by the inclusions EB ’ E.

As a consequence, all bounded seminorms on Eborn are continuous, and hence Eborn

is bornological.

An absolutely convex subset U is a 0-neighborhood for the ¬nal structure induced

by EB ’ E if and only if U © EB is a 0-neighborhood, or equivalently if U absorbs

B, for all bounded absolutely convex B, i.e. U is bornivorous. All other assertions

follow from (4.1).

4.3. Corollary. Bounded seminorms. For a seminorm p and a sequence µn ’

∞ the following statements are equivalent:

(1) p is bounded;

(2) p is bounded on compact sets;

(3) p is bounded on M -converging sequences;

(4) p is bounded on µ-converging sequences;

p is bounded on images of bounded intervals under Lipk -curves (for ¬xed

(5)

0 ¤ k ¤ ∞).

The corresponding statement for subsets of E is the following. For a radial subset

U ⊆ E (i.e., [0, 1] · U ⊆ U ) the following properties are equivalent:

(1) U is bornivorous.

4.3

36 Chapter I. Calculus of smooth mappings 4.5

(2) For all absolutely convex bounded sets B, the trace U © EB is a 0-neighbor-

hood in EB .

(3) U absorbs all compact subsets in E.

(4) U absorbs all Mackey convergent sequences.

(4™) U absorbs all sequences converging Mackey to 0.

(5) U absorbs all µ-convergent sequences (for a ¬xed µ).

(5™) U absorbs all sequences which are µ-converging to 0.

(6) U absorbs the images of bounded sets under Lipk -curves (for a ¬xed 0 ¤

k ¤ ∞).

Proof. We prove the statement on radial subsets, for seminorms p it then follows

by considering the radial set U := {x ∈ E : p(x) ¤ 1} and using the equality

K · U = {x ∈ E : p(x) ¤ K}.

(1) ” (2) ’ (3) ’ (4) ’ (5) ’ (5™), (4) ’ (4™), (3) ’ (6), (4™) ’ (5™), are trivial.

(6) ’ (5™) Suppose that (xn ) is µ-converging to x but is not absorbed by U . Then

for each m ∈ N there is an nm with xnm ∈ mU and clearly we may suppose that

/

1/µnm is fast falling. The sequence (xnm )m is then fast falling and lies on some

compact part of a smooth curve by the special curve lemma (2.8). The set U

absorbs this by (6), a contradiction.

(5™) ’ (1) Suppose U does not absorb some bounded B. Hence, there are bn ∈ B

b

with bn ∈ µ2 U . However, µn is µ-convergent to 0, so it is contained in KU for

/n n

some K > 0. Equivalently, bn ∈ µn KU ⊆ µ2 U for all µn ≥ K, which gives a

n

contradiction.

4.4. Corollary. Bornologi¬cation as locally convex-i¬cation.

The bornologi¬cation of E is the ¬nest locally convex topology with one (hence all)

of the following properties:

(1) It has the same bounded sets as E.

(2) It has the same Mackey converging sequences as E.

(3) It has the same µ-converging sequences as E (for some ¬xed µ).

It has the same Lipk -curves as E (for some ¬xed 0 ¤ k ¤ ∞).

(4)

(5) It has the same bounded linear mappings from E into arbitrary locally convex

spaces.

(6) It has the same continuous linear mappings from normed spaces into E.

Proof. Since the bornologi¬cation has the same bounded sets as the original topol-

ogy, the other objects are also the same: they depend only on the bornology “ this

would not be true for compact sets. Conversely, we consider a topology „ which has

for one of the above mentioned types the same objects as the original one. Then

„ has by (4.3) the same bornivorous absolutely convex subsets as the original one.

Hence, any 0-neighborhood of „ has to be bornivorous for the original topology,

and hence is a 0-neighborhood of the bornologi¬cation of the original topology.

4.5. Lemma. Let E be a bornological locally convex vector space, U ⊆ E a convex

subset. Then U is open for the locally convex topology of E if and only if U is open

4.5

4. The c∞ -topology

4.8 37

for the c∞ -topology.

Furthermore, an absolutely convex subset U of E is a 0-neighborhood for the locally

convex topology if and only if it is so for the c∞ -topology.

Proof. (’) The c∞ -topology is ¬ner than the locally convex topology, cf. (4.2).

(⇐) Let ¬rst U be an absolutely convex 0-neighborhood for the c∞ -topology. Hence,

U absorbs Mackey-0-sequences. By (4.1.3) we have to show that U is bornivorous,

in order to obtain that U is a 0-neighborhood for the locally convex topology. But

this follows immediately from (4.3).

Let now U be convex and c∞ -open, let x ∈ U be arbitrary. We consider the c∞ -

open absolutely convex set W := (U ’ x) © (x ’ U ) which is a 0-neighborhood of

the locally convex topology by the argument above. Then x ∈ W + x ⊆ U . So U

is open in the locally convex topology.

4.6. Corollary. The bornologi¬cation of a locally convex space E is the ¬nest

locally convex topology coarser than the c∞ -topology on E.

4.7. In (2.12) we de¬ned the c∞ -topology on an arbitrary locally convex space E

as the ¬nal topology with respect to the smooth curves c : R ’ E. Now we will

compare the c∞ -topology with other re¬nements of a given locally convex topology.

We ¬rst specify those re¬nements.

De¬nition. Let E be a locally convex vector space.

(i) We denote by kE the Kelley-¬cation of the locally convex topology of E, i.e.

the vector space E together with the ¬nal topology induced by the inclusions of

the subsets being compact for the locally convex topology.

(ii) We denote by sE the vector space E with the ¬nal topology induced by the

curves being continuous for the locally convex topology, or equivalently the se-

quences N∞ ’ E converging in the locally convex topology. The equivalence holds

since the in¬nite polygon through a converging sequence can be continuously pa-

rameterized by a compact interval.

(iii) We recall that by c∞ E we denote the vector space E with its c∞ -topology, i.e.

the ¬nal topology induced by the smooth curves.

Using that smooth curves are continuous and that converging sequences N∞ ’ E

have compact images, the following identities are continuous: c∞ E ’ sE ’ kE ’

E.

If the locally convex topology of E coincides with the topology of c∞ E, resp. sE,

resp. kE then we call E smoothly generated, resp. sequentially generated, resp.

compactly generated.

4.8. Example. On E = RJ all the re¬nements of the locally convex topology

described in (4.7) above are di¬erent, i.e. c∞ E = sE = kE = E, provided the

cardinality of the index set J is at least that of the continuum.

Proof. It is enough to show this for J equipotent to the continuum, since RJ1 is

a direct summand in RJ2 for J1 ⊆ J2 .

4.8

38 Chapter I. Calculus of smooth mappings 4.9

(c∞ E = sE) We may take as index set J the set c0 of all real sequences converging

to 0. De¬ne a sequence (xn ) in E by (xn )j := jn . Since every j ∈ J is a 0-sequence

we conclude that the xn converge to 0 in the locally convex topology of the product,

hence also in sE. Assume now that the xn converge towards 0 in c∞ E. Then by

(1.8) some subsequence converges Mackey to 0. Thus, there exists an unbounded

sequence of reals »n with {»n xn : n ∈ N} bounded. Let j be a 0-sequence with

{jn »n : n ∈ N} unbounded (e.g. (jn )’2 := 1 + max{|»k | : k ¤ n}). Then the j-th

coordinate jn »n of »n xn is not bounded with respect to n, contradiction.

(sE = kE) Consider in E the subset

A := x ∈ {0, 1}J : xj = 1 for at most countably many j ∈ J .

It is clearly closed with respect to the converging sequences, hence closed in sE.

But it is not closed in kE since it is dense in the compact set {0, 1}J .

(kE = E) Consider in E the subsets

An := x ∈ E : |xj | < n for at most n many j ∈ J .

Each An is closed in E since its complement is the union of the open sets {x ∈ E :

|xj | < n for all j ∈ Jo } where Jo runs through all subsets of J with n + 1 elements.

We show that the union A := n∈N An is closed in kE. So let K be a compact

subset of E; then K ⊆ prj (K), and each prj (K) is compact, hence bounded in

R. Since the family ({j ∈ J : prj (K) ⊆ [’n, n]})n∈N covers J, there has to exist an

N ∈ N and in¬nitely many j ∈ J with prj (K) ⊆ [’N, N ]. Thus K © An = … for all

n > N , and hence, A © K = n∈N An © K is closed. Nevertheless, A is not closed

¯

in E, since 0 is in A but not in A.

4.9. c∞ -convergent sequences. By (2.13) every M -convergent sequence gives

a continuous mapping N∞ ’ c∞ E and hence converges in c∞ E. Conversely, a

sequence converging in c∞ E is not necessarily Mackey convergent, see [Fr¨licher,

o

Kriegl, 1985]. However, one has the following result.

Lemma. A sequence (xn ) is convergent to x in the c∞ -topology if and only if every

subsequence has a subsequence which is Mackey convergent to x.

Proof. (⇐) is true for any topological convergence. In fact if xn would not con-

verge to x, then there would be a neighborhood U of x and a subsequence of xn

which lies outside of U and hence cannot have a subsequence converging to x.

(’) It is enough to show that (xn ) has a subsequence which converges Mackey to x,

since every subsequence of a c∞ -convergent sequence is clearly c∞ -convergent to the

same limit. Without loss of generality we may assume that x ∈ A := {xn : n ∈ N}.

/

Hence, A cannot be c∞ -closed, and thus there is a sequence nk ∈ N such that

(xnk ) converges Mackey to some point x ∈ A. The set {nk : k ∈ N} cannot be

/

bounded, and hence we may assume that the nk are strictly increasing by passing

to a subsequence. But then (xnk ) is a subsequence of (xn ) which converges in c∞ E

to x and Mackey to x hence also in c∞ E. Thus x = x.

4.9

4. The c∞ -topology

4.11 39

Remark. A consequence of this lemma is, that there is no topology having as

convergent sequences exactly the M -convergent ones, since this topology obviously

would have to be coarser than the c∞ -topology.

One can use this lemma also to show that the c∞ -topology on a locally convex

vector space gives a so called arc-generated vector space. See [Fr¨licher, Kriegl,

o

1988, 2.3.9 and 2.3.13] for a discussion of this.

Let us now describe several important situations where at least some of these topolo-

gies coincide. For the proof we will need the following

4.10. Lemma. [Averbukh, Smolyanov, 1968] For any locally convex space E the

following statements are equivalent:

(1) The sequential closure of any subset is formed by all limits of sequences in

the subset.

(2) For any given double sequence (xn,k ) in E with xn,k convergent to some

xk for n ’ ∞ and k ¬xed and xk convergent to some x, there are strictly

increasing sequences i ’ n(i) and i ’ k(i) with xn(i),k(i) ’ x for i ’ ∞.

Proof. (1’2) Take an a0 ∈ E di¬erent from k · (xn+k,k ’ x) and from k · (xk ’ x)

1

for all k and n. De¬ne A := {an,k := xn+k,k ’ k · a0 : n, k ∈ N}. Then x is in the

1 1

sequential closure of A, since xn+k,k ’ k · a0 converges to xk ’ k · a0 as n ’ ∞,

1

and xk ’ k · a0 converges to x ’ 0 = x as k ’ ∞. Hence, by (1) there has to exist

a sequence i ’ (ni , ki ) with ani ,ki convergent to x. By passing to a subsequence

we may suppose that i ’ ki and i ’ ni are increasing. Assume that i ’ ki is

1

bounded, hence ¬nally constant. Then a subsequence xni +ki ,ki ’ ki ·a0 is converging

1 1

to xk ’ k · a0 = x if i ’ ni is unbounded, and to xn+k,k ’ k · a0 = x if i ’ ni

is bounded, which both yield a contradiction. Thus, i ’ ki can be chosen strictly

increasing. But then

1

xni +ki ,ki = ani ,ki + ki a0 ’ x.

(1) ⇐ (2) is obvious.

4.11. Theorem. For any bornological vector space E the following implications

hold:

(1) c∞ E = E provided the closure of subsets in E is formed by all limits of

sequences in the subset; hence in particular if E is metrizable.

(2) c∞ E = E provided E is the strong dual of a Fr´chet Schwartz space;

e

(3) c∞ E = kE provided E is the strict inductive limit of a sequence of Fr´chet

e

spaces.

(4) c∞ E = sE provided E satis¬es the M -convergence condition, i.e. every

sequence converging in the locally convex topology is M-convergent.

(5) sE = E provided E is the strong dual of a Fr´chet Montel space;

e

Proof. (1) Using the lemma (4.10) above one obtains that the closure and the

sequential closure coincide, hence sE = E. It remains to show that sE ’ c∞ E is

(sequentially) continuous. So suppose a sequence converging to x is given, and let

4.11

40 Chapter I. Calculus of smooth mappings 4.13

(xn ) be an arbitrary subsequence. Then xn,k := k(xn ’ x) ’ k · 0 = 0 for n ’ ∞,

and hence by lemma (4.10) there are subsequences ki , ni with ki · (xni ’ x) ’ 0,

i.e. i ’ xni is M-convergent to x. Thus, the original sequence converges in c∞ E

by (4.9).

(3) Let E be the strict inductive limit of the Fr´chet spaces En . By (52.8) every En

e

carries the trace topology of E, hence is closed in E, and every bounded subset of E

is contained in some En . Thus, every compact subset of E is contained as compact

subset in some En . Since En is a Fr´chet space such a subset is even compact in

e

c∞ En and hence compact in c∞ E. Thus, the identity kE ’ c∞ E is continuous.

(4) is valid, since the M-closure topology is the ¬nal one induced by the M-

converging sequences.

(5) Let E be the dual of any Fr´chet Montel space F . By (52.29) E is bornological.

e

First we show that kE = sE. Let K ⊆ E = F be compact for the locally convex

topology. Then K is bounded, hence equicontinuous since F is barrelled by (52.25).

Since F is separable by (52.27) the set K is metrizable in the weak topology σ(E, F )

by (52.21). By (52.20) this weak topology coincides with the topology of uniform

convergence on precompact subsets of F . Since F is a Montel space, this latter

topology is the strong one, and even the bornological one, as remarked at the

beginning. Thus, the (metrizable) topology on K is the initial one induced by the

converging sequences. Hence, the identity kE ’ sE is continuous, and therefore

sE = kE.

It remains to show kE = E. Since F is Montel the locally convex topology of

the strong dual coincides with the topology of uniform convergence on precom-

pact subsets of F . Since F is metrizable this topology coincides with the so-called

equicontinuous weak— -topology, cf. (52.22), which is the ¬nal topology induced by

the inclusions of the equicontinuous subsets. These subsets are by the Alao˜lu- g

Bourbaki theorem (52.20) relatively compact in the topology of uniform conver-

gence on precompact subsets. Thus, the locally convex topology of E is compactly

generated.

Proof of (2) By (5), and since Fr´chet Schwartz spaces are Montel by (52.24), we

e

have sE = E and it remains to show that c∞ E = sE. So let (xn ) be a sequence

converging to 0 in E. Then the set {xn : n ∈ N} is relatively compact, and by

[Fr¨licher, Kriegl, 1988, 4.4.39] it is relatively compact in some Banach space EB .

o

Hence, at least a subsequence has to be convergent in EB . Clearly its Mackey limit

has to be 0. This shows that (xn ) converges to 0 in c∞ E, and hence c∞ E = sE. One

can even show that E satis¬es the Mackey convergence condition, see (52.28).

4.12. Example. We give now a non-metrizable example to which (4.11.1) applies.

Let E denote the subspace of RJ of all sequences with countable support. Then

the closure of subsets of E is given by all limits of sequences in the subset, but

for non-countable J the space E is not metrizable. This was proved in [Balanzat,

1960].

4.13. Remark. The conditions (4.11.1) and (4.11.2) are rather disjoint since every

4.13

4. The c∞ -topology

4.16 41

locally convex space, that has a countable basis of its bornology and for which the

sequential adherence of subsets (the set of all limits of sequences in it) is sequentially

closed, is normable as the following proposition shows:

Proposition. Let E be a non-normable bornological locally convex space that has

a countable basis of its bornology. Then there exists a subset of E whose sequential

adherence is not sequentially closed.

Proof. Let {Bk : k ∈ N0 } be an increasing basis of the von Neumann bornology

with B0 = {0}. Since E is non-normable we may assume that Bk does not absorb

1

Bk+1 for all k. Now choose bn,k ∈ n Bk+1 with bn,k ∈ Bk . We consider the

/

double sequence {bk,0 ’ bn,k : n, k ≥ 1}. For ¬xed k the sequence bn,k converges

by construction (in EBk+1 ) to 0 for n ’ ∞. Thus, bk,0 ’ 0 is the limit of the

sequence bk,0 ’ bn,k for n ’ ∞, and bk,0 converges to 0 for k ’ ∞. Suppose

bk(i),0 ’ bn(i),k(i) converges to 0. So it has to be bounded, thus there must be

an N ∈ N with B1 ’ {bk(i),0 ’ bn(i),k(i) : i ∈ N} ⊆ BN . Hence, bn(i),k(i) =

bk(i),0 ’ (bk(i),0 ’ bn(i),k(i) ) ∈ BN , i.e. k(i) < N . This contradicts (4.10.2).