ńņš. 11 |

xx

C CK (M, R) Cc (R, R)

Ā¢

Ā

Ā

x

eĀx Ā

x Ā

ā—

e

Ā

ĀØ

ā—

C (R, R) u {C

ā ā

eā’1 (K) (R, R)

ā

Then obviously the identity on Cc (M, R) is bounded from the structure (1) to the

structure (2).

ā

For the converse we consider a smooth curve Ī³ : R ā’ Cc (M, R) in the structure (2).

ā

We claim that Ī³ locally factors into some CKn (M, R) where (Kn ) is an exhaustion

of M by compact subsets such that Kn is contained in the interior of Kn+1 . If not

there exist a bounded sequence (tn ) in R and xn ā Kn such that Ī³(tn )(xn ) = 0.

/

One may ļ¬nd a proper smooth curve e : R ā’ M with e(n) = xn . Then eā— ā—¦ Ī³ is

ā

a smooth curve into Cc (R, R). Since the latter space is a strict inductive limit of

spaces CI (R, R) for compact intervals I, the curve eā— ā—¦ Ī³ locally factors into some

ā

CI (R, R), but (eā— ā—¦ Ī³)(tn )(n) = Ī³(tn )(xn ) = 0, a contradiction. This proves that

ā

6.2

6.4 6. Some spaces of smooth functions 69

ā

the curve Ī³ is also smooth into the structure (1), and so the identity on Cc (M, R)

is bounded from the structure (2) to the structure (1).

For the comparison of the structures (3) and (1) we consider the diagram:

99

ā

Ā

x

Cc (M, R)

xx 99

A

9

x

x

Ā jk

(M, R) e

y

ā

Cc (J k (M, R))

ee

CK

j

h

h

g

e h

u h

p

k

j

(M, R) R

y

ā k

RR

C C K (J (M, R))

T

R u

k

j

C(J k (M, R))

ā

Obviously, the identity on Cc (M, R) is bounded from the structure (1) into the

structure (3).

ā

For the converse direction we consider a smooth curve Ī³ : R ā’ Cc (M, R) with

structure (3). Then for each k the composition j k ā—¦ Ī³ is a smooth mapping into

the strict inductive limit Cc (J k (M, R)) = limK CK (J k (M, R)), thus locally factors

ā’ā’

k

into some step CK (J (M, R)) where K chosen for k = 0 works for any k. Since we

ā ā

have CK (M, R) = ā k CK (J k (M, R)), the curve Ī³ factors locally into CK (M, R)

lim

ā’

and is thus smooth for the structure (1). For the uniform boundedness principle

we refer to the ļ¬rst proof.

Remark. Note that the locally convex topologies described in (1) and (3) are

ā

distinct: The continuous dual of (Cc (R, R), (1)) is the space of all distributions

ā

(generalized functions), whereas the continuous dual of (Cc (R, R), (3)) are all dis-

tributions of ļ¬nite order, i.e., globally ļ¬nite derivatives of continuous functions.

6.3. Deļ¬nition. A convenient vector space E is called reļ¬‚exive if the canonical

embedding E ā’ E is surjective.

It is then even a bornological isomorphism. Note that reļ¬‚exivity as deļ¬ned here is

a bornological concept.

Note that this notion is in general stronger than the usual locally convex notion of

reļ¬‚exivity, since the continuous functionals on the strong dual are bounded func-

tionals on E but not conversely.

6.4. Result. [FrĀØlicher, Kriegl, 1988, 5.4.6]. For a convenient bornological vector

o

space E the following statements are equivalent.

(1) E is reļ¬‚exive.

(2) E is Ī·-reļ¬‚exive, see [Jarchow, 1981, p280].

(3) E is completely reļ¬‚exive, see [Hogbe-Nlend, 1977, p. 89].

6.4

70 Chapter I. Calculus of smooth mappings 6.7

(4) E is reļ¬‚exive in the usual locally convex sense, and the strong dual of E is

bornological.

(5) The Schwartzening (or nucleariļ¬cation) of E is a complete locally convex

space.

6.5. Results. [FrĀØlicher, Kriegl, 1988, section 5.4].

o

(1) A FrĀ“chet space is reļ¬‚exive if and only if it is reļ¬‚exive in the locally convex

e

sense.

(2) A convenient vector space with a countable base for its bornology is reļ¬‚exive

if and only if its bornological topology is reļ¬‚exive in the locally convex sense.

(3) A bornological reļ¬‚exive convenient vector space is complete in the locally

convex sense.

(4) A closed (in the locally convex sense) linear subspace of a reļ¬‚exive conve-

nient vector space is reļ¬‚exive.

(5) A convenient vector space is reļ¬‚exive if and only if its bornological topology

is complete and its dual is reļ¬‚exive.

(6) Products and coproducts of reļ¬‚exive convenient vector spaces are reļ¬‚exive

if the index set is of non-measurable cardinality.

(7) If E is a reļ¬‚exive convenient vector space and M is a ļ¬nite dimensional

separable smooth manifold then C ā (M, E) is reļ¬‚exive.

(8) Let U be a cā -open subset of a dual of a FrĀ“chet Schwartz space, and let F

e

be a FrĀ“chet Montel space. Then C ā (U, F ) is a FrĀ“chet Montel space, thus

e e

reļ¬‚exive.

(9) Let U be a cā -open subset of a dual of a nuclear FrĀ“chet space, and let F

e

be a nuclear FrĀ“chet space. It has been shown by [Colombeau, Meise, 1981]

e

that C ā (U, F ) is not nuclear in general.

6.6. Deļ¬nition. Another important additional property for convenient vector

spaces E is the approximation property, i.e. the denseness of E ā— E in L(E, E).

There are at least 3 successively stronger requirements, which have been studied in

[Adam, 1995]:

A convenient vector space E is said to have the bornological approximation property

if E ā— E is dense in L(E, E) with respect to the bornological topology. It is said to

have the cā -approximation property if this is true with respect to the cā -topology

of L(E, E). Finally the Mackey approximation property is the requirement, that

there is some sequence in E ā— E Mackey converging towards IdE .

Note that although the ļ¬rst condition is the weakest one, it is diļ¬cult to check

directly, since the bornologiļ¬cation of L(E, E) is hard to describe explicitly.

6.7. Result. [Adam, 1995, 2.2.9] The natural topology on

L(C ā (R, R), C ā (R, R))

of uniform convergence on bounded sets is not bornological.

6.7

6.12 6. Some spaces of smooth functions 71

6.8. Result. [Adam, 1995, 2.5.5] For any set Ī“ of non-measurable cardinality the

space E of points in RĪ“ with countable carrier has the bornological approximation

property.

Note. One ļ¬rst shows that for this space E the topology of uniform convergence

on bounded sets is bornological, and the classical approximation property holds for

this topology by [Jarchow, 1981, 21.2.2], since E is nuclear.

6.9. Lemma. Let E be a convenient vector space with the bornological (resp. cā -,

resp. Mackey) approximation property. Then for every convenient vector space F

we have that E ā— F is dense in the bornological topology of L(E, F ) (resp. in the

cā -topology, resp. every T ā L(E, F ) is the limit of a Mackey converging sequence

in E ā— F ).

Proof. Let T ā L(E, F ) and TĪ± ā E ā— E a net converging to IdE in the borno-

logical topology of L(E, F ) (resp. the cā -topology, resp. in the sense of Mackey).

Since Tā— : L(E, E) ā’ L(E, F ) is bounded and T ā—¦ TĪ± ā F ā— F , we get the result

in all three cases.

6.10. Lemma. [Adam, 1995, 2.1.21] Let E be a reļ¬‚exive convenient vector space.

Then E has the bornological (resp. cā -, resp. Mackey) approximation property if

and only if E has it.

Proof. For reļ¬‚exive convenient vector spaces we have:

L(E , E ) ā¼ L2 (E , E; R) ā¼ L(E, E ) ā¼ L(E, E),

= = =

and E ā— E corresponds to E ā— E via this isomorphism. So the result follows.

6.11. Lemma. [Adam, 1995, 2.4.3] Let E be the product kāN Ek of a sequence

of convenient vector spaces Ek . Then E has the Mackey (resp. cā -) approximation

property if and only if all Ek have it.

Proof. (ā’) follows since one easily checks that these approximation properties are

inherited by direct summands.

(ā) Let (Tn )n be Mackey convergent to T k in L(Ek , Ek ). Then one easily checks

k

the Mackey convergence of (Tn )k ā’ (T k )n in k L(Ek , Ek ) ā L(E, E). So the

k

result follows for the Mackey approximation property.

To obtain it also for the cā -topology, one ļ¬rst notes that by the argument given

in (6.9) it is enough to approximate the identity. Since the cā -closure can be

obtained as iterated Mackey-adherence by (4.32) this follows now by transļ¬nite

induction.

6.12. Recall that a set P ā RN of sequences is called a KĀØthe set if it is directed

o

+

upwards with respect to the componentwise partial ordering, see (52.35). To P we

may associate the set

1

Ī(P) := {x = (xn )n ā RN : (pn xn )n ā for all p ā P}.

A space Ī(P) is said to be a KĀØthe sequence space whenever P is a KĀØthe set.

o o

6.12

72 Chapter I. Calculus of smooth mappings 6.14

Lemma. Let P be a KĀØthe set for which there exists a sequence Āµ converging

o

monotonely to +ā and such that (Āµn pn )nāN ā P for each p ā P. Then the KĀØthe

o

sequence space Ī(P) has the Mackey approximation property.

n

j=1 ej ā—ej

Proof. The sequence is Mackey convergent in L(Ī(P), Ī(P)) to

nāN

idĪ(P) , where ej and ej denote the j-th unit vector in Ī(P) and Ī(P) respectively:

Indeed, a subset B ā Ī(P) is bounded if and only if for each p ā P there exists

N (p) ā R such that

pk |xk | ā¤ N (p)

kāN

for all x = (xk )kāN ā B. But this implies that

n

Āµn+1 IdĪ(P) ā’ ej ā— ej : n ā N ā L(Ī(P), Ī(P))

j=1

is bounded. In fact

n

for k ā¤ n,

0

Id ā’ ej ā— ej (x) =

xk for k > n.

k

j=1

and hence

n

pk Āµn+1 (Id ā’ ej ā—ej )(x) ā¤ pk |Āµn+1 xk | ā¤ pk Āµk |xk | ā¤ N (Āµ p)

k

j=1

k k>n k

Let Ī± be an unbounded increasing sequence of positive real numbers and Pā :=

{(ekĪ±n )nāN : k ā N}. Then the associated KĀØthe sequence space Ī(Pā ) is called a

o

power series space of inļ¬nite type (a FrĀ“chet space by [Jarchow, 1981, 3.6.2]).

e

6.13. Corollary. Each power series space of inļ¬nite type has the Mackey approx-

imation property.

6.14. Theorem. The following convenient vector spaces have the Mackey approx-

imation property:

(1) The space C ā (M ā F ) of smooth sections of any smooth ļ¬nite dimensional

p

vector bundle F ā’ M with separable base M , see (6.1) and (30.1).

ā’

ā

(2) The space Cc (M ā F ) of smooth sections with compact support any smooth

p

ļ¬nite dimensional vector bundle F ā’ M with separable base M , see (6.2)

ā’

and (30.4).

(3) The FrĀ“chet space of holomorphic functions H(C, C), see (8.2).

e

Proof. The space s of rapidly decreasing sequences coincides with the power series

space of inļ¬nite type associated to the sequence (log(n))nāN . So by (6.13), (6.11)

and (6.10) the spaces s, sN and s(N) = (s )N have the Mackey approximation

ā

property. Now assertions (1) and (2) follow from the isomorphisms Cc (M ā F ) =

6.14

Historical remarks on the development of smooth calculus 73

C ā (M ā F ) ā¼ s for compact M and C ā (M ā F ) ā¼ sN for non-compact M (see

= =

[Valdivia, 1982] or [Adam, 1995, 1.5.16]) and the isomorphism Cc (M ā F ) ā¼ s(N)

ā

=

for non-compact M (see [Valdivia, 1982] or [Adam, 1995, 1.5.16]).

(3) follows since by [Jarchow, 1981, 2.10.11] the space H(C, C) is isomorphic to the

(complex) power series space of inļ¬nite type associated to the sequence (n)nāN .

Historical Remarks on Smooth Calculus

Roots in the variational calculus. Soon after the invention of the diļ¬eren-

tial calculus ideas were developed which would later lead to variational calculus.

Bernoulli used them to determine the shape of a rope under gravity. It evolved

into a ā˜useful and applicable but highly formal calculus; even Gauss warned of its

unreļ¬‚ected applicationā™ ([Bemelmans, Hildebrand, von Wahl, 1990, p. 151]). In his

Lecture courses Weierstrass gave more reliable foundations to the theory, which

was made public by [Kneser, 1900], see also [Bolza, 1909] and [Hadamard, 1910].

Further development concerned mainly the relation between the calculus of varia-

tions and the theory of partial diļ¬erential equations. The use of the basic principle

of variational calculus for diļ¬erential calculus itself appeared only in the search for

the exponential law, i.e. a cartesian closed setting for calculus, see below.

The notion of derivative. The ļ¬rst more concise notion of the variational deriva-

tive was introduced by [Volterra, 1887], a concept of analysis on inļ¬nite dimensional

spaces; and this happened even before the modern concept of the total derivative

of a function of several variables was born: only partial derivatives were used at

that time. The derivative of a function in several variables in ļ¬nite dimensions

was introduced by [Stolz, 1893], [Pierpont, 1905], and ļ¬nally by [Young, 1910]: A

ā‚f

function f : Rn ā’ R is called diļ¬erentiable if the partial diļ¬erentials ā‚xi exist and

m

ā‚f

1 1 n n 1 n

+ Īµi )hi

f (x + h , . . . , x + h ) ā’ f (x , . . . , x ) = (

ā‚xi

i=1

holds, where Īµi ā’ 0 for ||h|| ā’ 0. The idea that the derivative is an approximation

to the function was emphasized frequently by Hadamard. His student [FrĀ“chet, e

1911] replaced the remainder term by Īµ. h with Īµ ā’ 0 for h ā’ 0. In [FrĀ“chet, e

1937] he writes:

S.241: āCā™est M. Volterra qui a eu le premier lā™idĀ“e dā™Ā“tendre le champ dā™application du

e e

Calcul diļ¬Ā“rentiel ` lā™Analyse fonctionnelle. [ . . . ] Toutefois M. Hadamard a signalĀ“ quā™il

e a e

y aurait grand intĀ“r`t Ā“ gĀ“nĀ“raliser les dĀ“ļ¬nitions de M. Volterra. [ . . . ] M. Hadamard

ee a e e e

a montrĀ“ le chemin qui devait conduire vers des dĀ“ļ¬nitions satisfaisantes en proposant

e e

dā™imposer ` la diļ¬Ā“rentielle dā™une fonctionnelle la condition dā™Ėtre linĀ“aire par rapport ` la

a e e e a

diļ¬Ā“rentielle de lā˜argument.ā

e

FrĀ“chet derivative. In [FrĀ“chet, 1925a] he deļ¬ned the derivative of a mapping

e e

f between normed spaces as follows: There exists a continuous linear operator A

74 Chapter I. Calculus of smooth mappings

such that

f (x + h) ā’ f (x) ā’ A Ā· h

lim = 0.

||h||

||h||ā’0

At this time it was, however, not so clear what a normed space should be. FrĀ“chet

e

called his spaces somewhat misleadingly ā˜vectoriels abstraits distanciĀ“sā™. Banach

e

spaces were introduced by Stefan Banach in his Dissertation in 1920, with a view

also to a non-linear theory, as he wrote in [Banach, 1932]:

S.231: āCes espaces [complex vector spaces] constituent le point de dĀ“part de la thĀ“orie

e e

des opĀ“rations linĀ“aires complexes et dā™une classe, encore plus vaste, des opĀ“rations ana-

e e e

lytiques, qui prĀ“sentent une gĀ“nĀ“ralisation des fonctions analytiques ordinaires (cf. p. ex.

e ee

L. FantappiĀ“, I. funzionali analitici, Citta di Castello 1930). Nous nous proposons dā™en

e

exposer la thĀ“orie dans un autre volume.ā

e

GĖteaux derivative. Another student of Hadamard deļ¬ned the derivative in

a

[GĖteaux, 1913] with proofs in [GĖteaux, 1922] as follows, see also [GĖteaux, 1922]:

a a a

āConsidĀ“rons U (z + Ī»t1 ) (t1 fonction analogue ` z). Supposons que

e a

ńņš. 11 |