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∞ ∞ ∞

xx
C CK (M, R) Cc (R, R)
¢
 
 
x
e€x €
x  

e
 
¨



C (R, R) u {C
∞ ∞
e’1 (K) (R, R)


Then obviously the identity on Cc (M, R) is bounded from the structure (1) to the
structure (2).

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

We claim that γ 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 ∈ Kn such that γ(tn )(xn ) = 0.
/
One may ¬nd a proper smooth curve e : R ’ M with e(n) = xn . Then e— —¦ γ is

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 e— —¦ γ locally factors into some


CI (R, R), but (e— —¦ γ)(tn )(n) = γ(tn )(xn ) = 0, a contradiction. This proves that



6.2
6.4 6. Some spaces of smooth functions 69


the curve γ 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


€
x
Cc (M, R)

xx 99
A
9
x
x
† jk
(M, R) e
y

Cc (J k (M, R))
ee
CK
j
h
h
g
e h
u h
p
k
j
(M, R) R
y
∞ k

RR
C C K (J (M, R))

T
R u
k
j
C(J k (M, R))

Obviously, the identity on Cc (M, R) is bounded from the structure (1) into the
structure (3).

For the converse direction we consider a smooth curve γ : R ’ Cc (M, R) with
structure (3). Then for each k the composition j k —¦ γ is a smooth mapping into
the strict inductive limit Cc (J k (M, R)) = limK CK (J k (M, R)), thus locally factors
’’
k
into some step CK (J (M, R)) where K chosen for k = 0 works for any k. Since we
∞ ∞
have CK (M, R) = ← k CK (J k (M, R)), the curve γ factors locally into CK (M, R)
lim

and is thus smooth for the structure (1). For the uniform boundedness principle
we refer to the ¬rst proof.

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

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

(generalized functions), whereas the continuous dual of (Cc (R, R), (3)) are all dis-
tributions of ¬nite order, i.e., globally ¬nite derivatives of continuous functions.

6.3. De¬nition. A convenient vector space E is called re¬‚exive if the canonical
embedding E ’ E is surjective.
It is then even a bornological isomorphism. Note that re¬‚exivity as de¬ned here is
a bornological concept.
Note that this notion is in general stronger than the usual locally convex notion of
re¬‚exivity, since the continuous functionals on the strong dual are bounded func-
tionals on E but not conversely.

6.4. Result. [Fr¨licher, Kriegl, 1988, 5.4.6]. For a convenient bornological vector
o
space E the following statements are equivalent.
(1) E is re¬‚exive.
(2) E is ·-re¬‚exive, see [Jarchow, 1981, p280].
(3) E is completely re¬‚exive, see [Hogbe-Nlend, 1977, p. 89].

6.4
70 Chapter I. Calculus of smooth mappings 6.7

(4) E is re¬‚exive in the usual locally convex sense, and the strong dual of E is
bornological.
(5) The Schwartzening (or nucleari¬cation) of E is a complete locally convex
space.

6.5. Results. [Fr¨licher, Kriegl, 1988, section 5.4].
o
(1) A Fr´chet space is re¬‚exive if and only if it is re¬‚exive in the locally convex
e
sense.
(2) A convenient vector space with a countable base for its bornology is re¬‚exive
if and only if its bornological topology is re¬‚exive in the locally convex sense.
(3) A bornological re¬‚exive convenient vector space is complete in the locally
convex sense.
(4) A closed (in the locally convex sense) linear subspace of a re¬‚exive conve-
nient vector space is re¬‚exive.
(5) A convenient vector space is re¬‚exive if and only if its bornological topology
is complete and its dual is re¬‚exive.
(6) Products and coproducts of re¬‚exive convenient vector spaces are re¬‚exive
if the index set is of non-measurable cardinality.
(7) If E is a re¬‚exive convenient vector space and M is a ¬nite dimensional
separable smooth manifold then C ∞ (M, E) is re¬‚exive.
(8) Let U be a c∞ -open subset of a dual of a Fr´chet Schwartz space, and let F
e
be a Fr´chet Montel space. Then C ∞ (U, F ) is a Fr´chet Montel space, thus
e e
re¬‚exive.
(9) Let U be a c∞ -open subset of a dual of a nuclear Fr´chet space, and let F
e
be a nuclear Fr´chet space. It has been shown by [Colombeau, Meise, 1981]
e
that C ∞ (U, F ) is not nuclear in general.

6.6. De¬nition. Another important additional property for convenient vector
spaces E is the approximation property, i.e. the denseness of E — 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 — E is dense in L(E, E) with respect to the bornological topology. It is said to
have the c∞ -approximation property if this is true with respect to the c∞ -topology
of L(E, E). Finally the Mackey approximation property is the requirement, that
there is some sequence in E — E Mackey converging towards IdE .
Note that although the ¬rst condition is the weakest one, it is di¬cult to check
directly, since the bornologi¬cation of L(E, E) is hard to describe explicitly.

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

L(C ∞ (R, R), C ∞ (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 “ of non-measurable cardinality the
space E of points in R“ with countable carrier has the bornological approximation
property.

Note. One ¬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. c∞ -,
resp. Mackey) approximation property. Then for every convenient vector space F
we have that E — F is dense in the bornological topology of L(E, F ) (resp. in the
c∞ -topology, resp. every T ∈ L(E, F ) is the limit of a Mackey converging sequence
in E — F ).

Proof. Let T ∈ L(E, F ) and T± ∈ E — E a net converging to IdE in the borno-
logical topology of L(E, F ) (resp. the c∞ -topology, resp. in the sense of Mackey).
Since T— : L(E, E) ’ L(E, F ) is bounded and T —¦ T± ∈ F — F , we get the result
in all three cases.

6.10. Lemma. [Adam, 1995, 2.1.21] Let E be a re¬‚exive convenient vector space.
Then E has the bornological (resp. c∞ -, resp. Mackey) approximation property if
and only if E has it.

Proof. For re¬‚exive convenient vector spaces we have:
L(E , E ) ∼ L2 (E , E; R) ∼ L(E, E ) ∼ L(E, E),
= = =
and E — E corresponds to E — E via this isomorphism. So the result follows.

6.11. Lemma. [Adam, 1995, 2.4.3] Let E be the product k∈N Ek of a sequence
of convenient vector spaces Ek . Then E has the Mackey (resp. c∞ -) approximation
property if and only if all Ek have it.

Proof. (’) follows since one easily checks that these approximation properties are
inherited by direct summands.
(⇐) Let (Tn )n be Mackey convergent to T k in L(Ek , Ek ). Then one easily checks
k

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

result follows for the Mackey approximation property.
To obtain it also for the c∞ -topology, one ¬rst notes that by the argument given
in (6.9) it is enough to approximate the identity. Since the c∞ -closure can be
obtained as iterated Mackey-adherence by (4.32) this follows now by trans¬nite
induction.

6.12. Recall that a set P ⊆ RN of sequences is called a K¨the 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
Λ(P) := {x = (xn )n ∈ RN : (pn xn )n ∈ for all p ∈ P}.
A space Λ(P) is said to be a K¨the sequence space whenever P is a K¨the set.
o o

6.12
72 Chapter I. Calculus of smooth mappings 6.14

Lemma. Let P be a K¨the set for which there exists a sequence µ converging
o
monotonely to +∞ and such that (µn pn )n∈N ∈ P for each p ∈ P. Then the K¨the
o
sequence space Λ(P) has the Mackey approximation property.
n
j=1 ej —ej
Proof. The sequence is Mackey convergent in L(Λ(P), Λ(P)) to
n∈N
idΛ(P) , where ej and ej denote the j-th unit vector in Λ(P) and Λ(P) respectively:
Indeed, a subset B ⊆ Λ(P) is bounded if and only if for each p ∈ P there exists
N (p) ∈ R such that
pk |xk | ¤ N (p)
k∈N

for all x = (xk )k∈N ∈ B. But this implies that
n
µn+1 IdΛ(P) ’ ej — ej : n ∈ N ⊆ L(Λ(P), Λ(P))
j=1


is bounded. In fact
n
for k ¤ n,
0
Id ’ ej — ej (x) =
xk for k > n.
k
j=1


and hence
n
pk µn+1 (Id ’ ej —ej )(x) ¤ pk |µn+1 xk | ¤ pk µk |xk | ¤ N (µ p)
k
j=1
k k>n k



Let ± be an unbounded increasing sequence of positive real numbers and P∞ :=
{(ek±n )n∈N : k ∈ N}. Then the associated K¨the sequence space Λ(P∞ ) is called a
o
power series space of in¬nite type (a Fr´chet space by [Jarchow, 1981, 3.6.2]).
e

6.13. Corollary. Each power series space of in¬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 ∞ (M ← F ) of smooth sections of any smooth ¬nite dimensional
p
vector bundle F ’ M with separable base M , see (6.1) and (30.1).


(2) The space Cc (M ← F ) of smooth sections with compact support any smooth
p
¬nite dimensional vector bundle F ’ M with separable base M , see (6.2)

and (30.4).
(3) The Fr´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 in¬nite type associated to the sequence (log(n))n∈N . So by (6.13), (6.11)
and (6.10) the spaces s, sN and s(N) = (s )N have the Mackey approximation

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

6.14
Historical remarks on the development of smooth calculus 73

C ∞ (M ← F ) ∼ s for compact M and C ∞ (M ← F ) ∼ sN for non-compact M (see
= =
[Valdivia, 1982] or [Adam, 1995, 1.5.16]) and the isomorphism Cc (M ← F ) ∼ s(N)

=
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 in¬nite type associated to the sequence (n)n∈N .



Historical Remarks on Smooth Calculus

Roots in the variational calculus. Soon after the invention of the di¬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 ˜useful and applicable but highly formal calculus; even Gauss warned of its
unre¬‚ected application™ ([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 di¬erential equations. The use of the basic principle
of variational calculus for di¬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 ¬rst more concise notion of the variational deriva-
tive was introduced by [Volterra, 1887], a concept of analysis on in¬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 ¬nite dimensions
was introduced by [Stolz, 1893], [Pierpont, 1905], and ¬nally by [Young, 1910]: A
‚f
function f : Rn ’ R is called di¬erentiable if the partial di¬erentials ‚xi exist and
m
‚f
1 1 n n 1 n
+ µi )hi
f (x + h , . . . , x + h ) ’ f (x , . . . , x ) = (
‚xi
i=1

holds, where µi ’ 0 for ||h|| ’ 0. The idea that the derivative is an approximation
to the function was emphasized frequently by Hadamard. His student [Fr´chet, e
1911] replaced the remainder term by µ. h with µ ’ 0 for h ’ 0. In [Fr´chet, e
1937] he writes:
S.241: “C™est M. Volterra qui a eu le premier l™id´e d™´tendre le champ d™application du
e e
Calcul di¬´rentiel ` l™Analyse fonctionnelle. [ . . . ] Toutefois M. Hadamard a signal´ qu™il
e a e
y aurait grand int´r`t ´ g´n´raliser les d´¬nitions de M. Volterra. [ . . . ] M. Hadamard
ee a e e e
a montr´ le chemin qui devait conduire vers des d´¬nitions satisfaisantes en proposant
e e
d™imposer ` la di¬´rentielle d™une fonctionnelle la condition d™ˆtre lin´aire par rapport ` la
a e e e a
di¬´rentielle de l˜argument.”
e

Fr´chet derivative. In [Fr´chet, 1925a] he de¬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) ’ f (x) ’ A · h
lim = 0.
||h||
||h||’0

At this time it was, however, not so clear what a normed space should be. Fr´chet
e
called his spaces somewhat misleadingly ˜vectoriels abstraits distanci´s™. 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: “Ces espaces [complex vector spaces] constituent le point de d´part de la th´orie
e e
des op´rations lin´aires complexes et d™une classe, encore plus vaste, des op´rations ana-
e e e
lytiques, qui pr´sentent une g´n´ralisation des fonctions analytiques ordinaires (cf. p. ex.
e ee
L. Fantappi´, I. funzionali analitici, Citta di Castello 1930). Nous nous proposons d™en
e
exposer la th´orie dans un autre volume.”
e

Gˆteaux derivative. Another student of Hadamard de¬ned the derivative in
a
[Gˆteaux, 1913] with proofs in [Gˆteaux, 1922] as follows, see also [Gˆteaux, 1922]:
a a a
“Consid´rons U (z + »t1 ) (t1 fonction analogue ` z). Supposons que
e a

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