We may assume that the bounded sets of X are formed by those subsets B, for

which f (B) is bounded for all f ∈ ∞ (X). Obviously, any bounded set has this

property, and the space ∞ (X) is not changed by adding these sets. Furthermore,

the restriction map ιB : ∞ (X) ’ ∞ (B) is also bounded for such a B, since us-

ing the closed graph theorem (52.10) we only have to show that evb —¦ιB = ι{b} is

bounded for every b ∈ B, which is obviously the case.

By proposition (4.30) it is enough to show the universal property for bounded

linear functionals. In analogy to Banach-theory, we only have to show that the

dual Cc (X) is just

1

(X) := {g : X ’ R : supp g is bounded and g is absolutely summable}.

4.35

4. The c∞ -topology

4.36 51

In fact, any such g acts even as bounded linear functional on ∞ (X, R) by g, f :=

∞

x g(x) f (x), since a subset is bounded in (X) if and only if it is uniformly

bounded on all bounded sets B ⊆ X. Conversely, let : Cc (X) ’ R be bounded

and linear and de¬ne g : X ’ R, by g(x) := (ex ), where ex denotes the function

given by ex (y) := 1 for x = y and 0 otherwise. Obviously (f ) = g, f for all

f ∈ Cc (X). Suppose indirectly supp g = {x : (ex ) = 0} is not bounded. Then

there exists a sequence xn ∈ supp g and a function f ∈ ∞ (X) such that |f (xn )| ≥ n.

In particular, the only bounded subsets of {xn : n ∈ N} are the ¬nite ones. Hence

n

{ |g(xn )| exn } is bounded in Cc (X), but the image under is not. Furthermore, g

has to be absolutely summable since the set of ¬nite subsums of x sign g(x) ex is

bounded in Cc (X) and its image under are the subsums of x |g(x)|.

4.36. Corollary. Counter-examples on c∞ -topology. The following state-

ments are false:

(1) The c∞ -closure of a subset (or of a linear subspace) is given by the Mackey

adherence, i.e. the set formed by all limits of sequences in this subset which

are Mackey convergent in the total space.

(2) A subset U of E that contains a point x and has the property, that every

sequence which M -converges to x belongs to it ¬nally, is a c∞ -neighborhood

of x.

(3) A c∞ -dense subspace of a c∞ -complete space has this space as c∞ -comple-

tion.

(4) If a subspace E is c∞ -dense in the total space, then it is also c∞ -dense in

each linear subspace lying in between.

(5) The c∞ -topology of a linear subspace is the trace of the c∞ -topology of the

whole space.

(6) Every bounded linear functional on a linear subspace can be extended to such

a functional on the whole space.

(7) A linear subspace of a bornological locally convex space is bornological.

(8) The c∞ -completion preserves embeddings.

Proof. (1) For this we give an example, where the Mackey adherence of Cc (X) is

not all of c0 (X).

Let X = N — N, and take as bounded sets all sets of the form Bµ := {(n, k) : n ¤

µ(k)}, where µ runs through all functions N ’ N. Let f : X ’ R be de¬ned by

1

f (n, k) := k . Obviously, f ∈ c0 (X), since for given j ∈ N and function µ the set of

points (n, k) ∈ Bµ for which f (n, k) = k ≥ 1 is the ¬nite set {(n, k) : k ¤ j, n ¤

1

j

µ(k)}.

Assume there is a sequence fn ∈ Cc (X) Mackey convergent to f . By passing to a

subsequence we may assume that n2 (f ’ fn ) is bounded. Now choose µ(k) to be

larger than all of the ¬nitely many n, with fk (n, k) = 0. If k 2 (f ’ fk ) is bounded

on Bµ , then in particular {k 2 (f ’ fk )(µ(k), k) : k ∈ N} has to be bounded, but

1

k 2 (f ’ fk )(µ(k), k) = k 2 k ’ 0 = k.

(2) Let A be a set for which (1) fails, and choose x in the c∞ -closure of A but not

in the M -adherence of A. Then U := E \ A satis¬es the assumptions of (2). In

4.36

52 Chapter I. Calculus of smooth mappings 5.1

fact, let xn be a sequence which converges Mackey to x, and assume that it is not

¬nally in U . So we may assume without loss of generality that xn ∈ U for all n, but

/

then A xn ’ x would imply that x is in the Mackey adherence of A. However,

U cannot be a c∞ -neighborhood of x. In fact, such a neighborhood must meet A

since x is assumed to be in the c∞ -closure of A.

(3) Let F be a locally convex vector space whose Mackey adherence in its c∞ -com-

pletion E is not all of E, e.g. Cc (X) ⊆ c0 (X) as in the previous counter-example.

Choose a y ∈ E that is not contained in the Mackey adherence of F , and let F1

be the subspace of E generated by F ∪ {y}. We claim that F1 ⊆ E cannot be the

c∞ -completion although F1 is obviously c∞ -dense in the convenient vector space

E. In order to see this we consider the linear map : F1 ’ R characterized by

(F ) = 0 and (y) = 1. Clearly is well de¬ned.

: F1 ’ R is bornological: For any bounded B ⊆ F1 there exists an N such that

B ⊆ F + [’N, N ]y. Otherwise, bn = xn + tn y ∈ B would exist with tn ’ ∞ and

xn ∈ F . This would imply that bn = tn ( xn + y), and thus ’ xn would converge

n n

t t

Mackey to y; a contradiction.

Now assume that a bornological extension ¯ to E exists. Then F ⊆ ker( ¯) and

ker( ¯) is c∞ -closed, which is a contradiction to the c∞ -denseness of F in E. So

F1 ⊆ E does not have the universal property of a c∞ -completion.

This shows also that (6) fails.

(4) Furthermore, it follows that F is c∞ F1 -closed in F1 , although F and hence F1

are c∞ -dense in E.

(5) The trace of the c∞ -topology of E to F1 cannot be the c∞ -topology of F1 , since

for the ¬rst one F is obviously dense.

(7) Obviously, the trace topology of the bornological topology on E cannot be

bornological on F1 , since otherwise the bounded linear functionals on F1 would be

continuous and hence extendable to E.

(8) Furthermore, the extension of the inclusion ι : F • R ∼ F1 ’ E to the com-

=

pletion is given by (x, t) ∈ E • R ∼ F • R = F1 ’ x + ty ∈ E and has as kernel

=˜ ˜

the linear subspace generated by (y, ’1). Hence, the extension of an embedding to

the c∞ -completions need not be an embedding anymore, in particular the inclusion

functor does not preserve injectivity of morphisms.

5. Uniform Boundedness Principles and Multilinearity

5.1. The category of locally convex spaces and smooth mappings. The

category of all smooth mappings between bornological vector spaces is a subcate-

gory of the category of all smooth mappings between locally convex spaces which

is equivalent to it, since a locally convex space and its bornologi¬cation (4.4) have

the same bounded sets and smoothness depends only on the bornology by (1.8).

So it is also cartesian closed, but the topology on C ∞ (E, F ) from (3.11) has to be

5.1

5.3 5. Uniform boundedness principles and multilinearity 53

bornologized. For an example showing the necessity see [Kriegl, 1983, p. 297] or

[Fr¨licher, Kriegl, 1988, 5.4.19]: The topology on C ∞ (R, R(N) ) is not bornological.

o

We will in general, however, work in the category of locally convex spaces and

smooth mappings, so function spaces carry the topology of (3.11).

The category of bounded (equivalently continuous) linear mappings between bor-

nological vector spaces is in the same way equivalent to the category of all bounded

linear mappings between all locally convex spaces, since a linear mapping is smooth

if and only if it is bounded, by (2.11). It is closed under formation of colimits and

under quotients (this is an easy consequence of (4.1.1)). The Mackey-Ulam theo-

rem [Jarchow, 1981, 13.5.4] tells us that a product of non trivial bornological vector

spaces is bornological if and only if the index set does not admit a Ulam measure,

i.e. a non trivial {0, 1}-valued measure on the whole power set. A cardinal admit-

ting a Ulam measure has to be strongly inaccessible, so we can restrict set theory

to exclude measurable cardinals.

Let L(E1 , . . . , En ; F ) denote the space of all bounded n-linear mappings from E1 —

. . . — En ’ F with the topology of uniform convergence on bounded sets in E1 —

. . . — En .

5.2. Proposition. Exponential law for L. There are natural bornological

isomorphisms

L(E1 , . . . , En+k ; F ) ∼ L(E1 , . . . , En ; L(En+1 , . . . , En+k ; F )).

=

Proof. We proof this for bilinear maps, the general case is completely analogous.

We already know that bilinearity translates into linearity into the space of linear

functions. Remains to prove boundedness. So let B ⊆ L(E1 , E2 ; F ) be given. Then

B is bounded if and only if B(B1 — B2 ) ⊆ F is bounded for all bounded Bi ⊆ Ei .

This however is equivalent to B ∨ (B1 ) is contained and bounded in L(E2 , F ) for all

bounded B ⊆ E1 , i.e. B ∨ is contained and bounded in L(E1 , L(E2 , F )).

Recall that we have already put a structure on L(E, F ) in (3.17), namely the initial

one with respect to the inclusion in C ∞ (E, F ). Let us now show that bornologically

these de¬nitions agree:

5.3. Lemma. Structure on L. A subset is bounded in L(E, F ) ⊆ C ∞ (E, F )

if and only if it is uniformly bounded on bounded subsets of E, i.e. L(E, F ) ’

C ∞ (E, F ) is initial.

Proof. Let B ⊆ L(E, F ) be bounded in C ∞ (E, F ), and assume that it is not

uniformly bounded on some bounded set B ⊆ E. So there are fn ∈ B, bn ∈ B, and

∈ F — with | (fn (bn ))| ≥ nn . Then the sequence n1’n bn converges fast to 0, and

hence lies on some compact part of a smooth curve c by the special curve lemma

(2.8). So B cannot be bounded, since otherwise C ∞ ( , c) = — —¦ c— : C ∞ (E, F ) ’

C ∞ (R, R) ’ ∞ (R, R) would have bounded image, i.e. { —¦ fn —¦ c : n ∈ N} would

be uniformly bounded on any compact interval.

5.3

54 Chapter I. Calculus of smooth mappings 5.6

Conversely, let B ⊆ L(E, F ) be uniformly bounded on bounded sets and hence

in particular on compact parts of smooth curves. We have to show that dn —¦ c— :

L(E, F ) ’ C ∞ (R, F ) ’ ∞ (R, F ) has bounded image. But for linear smooth maps

we have by the chain rule (3.18), recursively applied, that dn (f —¦ c)(t) = f (c(n) (t)),

and since c(n) is still a smooth curve we are done.

Let us now generalize this result to multilinear mappings. For this we ¬rst charac-

terize bounded multilinear mappings in the following two ways:

5.4. Lemma. A multilinear mapping is bounded if and only if it is bounded on

each sequence which converges Mackey to 0.

Proof. Suppose that f : E1 — . . . — Ek ’ F is not bounded on some bounded set

B ⊆ E1 — . . . — Ek . By composing with a linear functional we may assume that

F = R. So there are bn ∈ B with »k+1 := |f (bn )| ’ ∞. Then |f ( »1 bn )| = »n ’ ∞,

n n

1

but ( »n bn ) is Mackey convergent to 0.

5.5. Lemma. Bounded multilinear mappings are smooth. Let f : E1 —

. . . — En ’ F be a multilinear mapping. Then f is bounded if and only if it is

smooth. For the derivative we have the product rule:

n

df (x1 , . . . , xn )(v1 , . . . , vn ) = f (x1 , . . . , xi’1 , vi , xi+1 , . . . , xn ).

i=1

In particular, we get for f : E ⊇ U ’ R, g : E ⊇ U ’ F and x ∈ U , v ∈ E the

Leibniz formula

(f · g) (x)(v) = f (x)(v) · g(x) + f (x) · g (x)(v).

Proof. We use induction on n. The case n = 1 is corollary (2.11). The induction

goes as follows:

f is bounded

f (B1 — . . . — Bn ) = f ∨ (B1 — . . . — Bn’1 )(Bn ) is bounded for all bounded

⇐’

sets Bi in Ei ;

f ∨ (B1 — . . . — Bn’1 ) ⊆ L(En , F ) ⊆ C ∞ (En , F ) is bounded, by (5.3);

⇐’

f ∨ : E1 — . . . — En’1 ’ C ∞ (En , F ) is bounded;

⇐’

f ∨ : E1 — . . . — En’1 ’ C ∞ (En , F ) is smooth by the inductive assumption;

⇐’

f ∨ : E1 — . . . — En ’ F is smooth by cartesian closedness (3.13).

⇐’

The particular case follows by application to the scalar multiplication R — F ’ F .

Now let us show that also the structures coincide:

5.6. Proposition. Structure on space of multilinear maps. The injection

of L(E1 , . . . , En ; F ) ’ C ∞ (E1 — . . . — En , F ) is a bornological embedding.

Proof. We can show this by induction. In fact, let B ⊆ L(E1 , . . . , En ; F ). Then

B is bounded

5.6

5.7 5. Uniform boundedness principles and multilinearity 55

⇐’ B(B1 — . . . — Bn ) = B ∨ (B1 — . . . — Bn’1 )(Bn ) is bounded for all bounded

Bi in Ei ;

⇐’ B ∨ (B1 — . . . — Bn’1 ) ⊆ L(En , F ) ⊆ C ∞ (En , F ) is bounded, by (5.3);

⇐’ B ∨ ⊆ C ∞ (E1 — . . . — En’1 , C ∞ (En , F )) is bounded by the inductive as-

sumption;

⇐’ B ⊆ C ∞ (E1 — . . . — En , F ) is bounded by cartesian closedness (3.13).

5.7. Bornological tensor product. It is natural to consider the universal prob-

lem of linearizing bounded bilinear mappings. The solution is given by the bornolo-

gical tensor product E —β F , i.e. the algebraic tensor product with the ¬nest locally

convex topology such that E — F ’ E — F is bounded. A 0-neighborhood basis of

this topology is given by those absolutely convex sets, which absorb B1 — B2 for all

bounded B1 ⊆ E1 and B2 ⊆ E2 . Note that this topology is bornological since it is

the ¬nest locally convex topology with given bounded linear mappings on it.

Theorem. The bornological tensor product is the left adjoint functor to the Hom-

functor L(E, ) on the category of bounded linear mappings between locally convex

spaces, and one has the following bornological isomorphisms:

L(E —β F, G) ∼ L(E, F ; G) ∼ L(E, L(F, G))

= =

E —β R ∼ E

=

E —β F ∼ F —β E

=

(E —β F ) —β G ∼ E —β (F —β G)

=

Furthermore, the bornological tensor product preserves colimits. It neither preserves

embeddings nor countable products.

Proof. We show ¬rst that this topology has the universal property for bounded

bilinear mappings f : E1 — E2 ’ F . Let U be an absolutely convex zero neighbor-

hood in F , and let B1 , B2 be bounded sets. Then f (B1 — B2 ) is bounded, hence

˜ ˜

it is absorbed by U . Then f ’1 (U ) absorbs —(B1 — B2 ), where f : E1 — E2 ’ F is

˜

the canonically associated linear mapping. So f ’1 (U ) is in the zero neighborhood

˜

basis of E1 —β E2 described above. Therefore, f is continuous.

A similar argument for sets of mappings shows that the ¬rst isomorphism L(E —β

F, G) ∼ L(E, F ; G) is bornological.

=

The topology on E1 —β E2 is ¬ner than the projective tensor product topology, and

so it is Hausdor¬. The rest of the positive results is clear.

The counter-example for embeddings given for the projective tensor product works

also, since all spaces involved are Banach.

Since the bornological tensor-product preserves coproducts it cannot preserve prod-

ucts. In fact (R —β R(N) )N ∼ (R(N) )N whereas RN —β R(N) ∼ (RN —β R)(N) ∼

= = =

(RN )(N) .

5.7

56 Chapter I. Calculus of smooth mappings 5.9

5.8. Proposition. Projective versus bornological tensor product. If every

bounded bilinear mapping on E — F is continuous then E —π F = E —β F . In

particular, we have E —π F = E —β F for any two metrizable spaces, and for a

normable space F we have Eborn —π F = E —β F .

Proof. Recall that E —π F carries the ¬nest locally convex topology such that

— : E — F ’ E — F is continuous, whereas E —β F carries the ¬nest locally

convex topology such that — : E — F ’ E — F is bounded. So we have that

— : E — F ’ E —β F is bounded and hence by assumption continuous, and thus

the topology of E —π F is ¬ner than that of E —β F . Since the converse is true in

general, we have equality.

In (52.23) it is shown that in metrizable locally convex spaces the convergent se-

quences coincide with the Mackey-convergent ones. Now let T : E — F ’ G be

bounded and bilinear. We have to show that T is continuous. So let (xn , yn ) be

a convergent sequence in E — F . Without loss of generality we may assume that

its limit is (0, 0). So there are µn ’ ∞ such that {µn (xn , yn ) : n ∈ N} is bounded

and hence also T {µn (xn , yn ) : n ∈ N} = µ2 T (xn , yn ) : n ∈ N , i.e. T (xn , yn )

n

converges even Mackey to 0.

If F is normable and T : Eborn — F ’ G is bounded bilinear then T ∨ : Eborn ’

L(F, G) is bounded, and since Eborn is bornological it is even continuous. Clearly,

for normed spaces F the evaluation map ev : L(F, G) — F ’ G is continuous, and

hence T = ev —¦(T ∨ — F ) : Eborn — F ’ G is continuous. Thus, Eborn —π F =

E —β F .

Note that the bornological tensor product is invariant under bornologi¬cation, i.e.

Eborn —β Fborn ∼ E —β F . So it is no loss of generality to assume that both spaces

=

are bornological. Keep however in mind that the corresponding identity for the

projective tensor product does not hold. Another possibility to obtain the identity

E —π F = E —β F is to assume that E and F are bornological and every separately

continuous bilinear mapping on E —F is continuous. In fact, every bounded bilinear

mapping is obviously separately bounded, and since E and F are assumed to be

bornological, it has to be separately continuous. We want to ¬nd another class

beside the Fr´chet spaces (see (52.9)) which satis¬es these assumptions.

e

5.9. Corollary. The following mappings are bounded multilinear.

(1) lim : Nat(F, G) ’ L(lim F, lim G), where F and G are two functors on

the same index category, and where Nat(F, G) denotes the space of all

natural transformations with the structure induced by the embedding into

i L(F(i), G(i)).

(2) colim : Nat(F, G) ’ L(colim F, colim G).

(3)

L : L(E1 , F1 ) — . . . —L(En , Fn ) — L(F, E) ’

’ L(L(F1 , . . . , Fn ; F ), L(E1 , . . . , En ; E))

(T1 , . . . , Tn , T ) ’ (S ’ T —¦ S —¦ (T1 — . . . — Tn ));

5.9

5.9 5. Uniform boundedness principles and multilinearity 57

n

: L(E1 , F1 ) — . . . — L(En , Fn ) ’ L(E1 —β · · · —β En , F1 —β · · · —β Fn ).

(4) β

n n n n

: L(E, F ) ’ L(

(5) E, F ), where E is the linear subspace of all

n

alternating tensors in β E. It is the universal solution of

n

∼ Ln (E; F ),

L E, F = alt

where Ln (E; F ) is the space of all bounded n-linear alternating mappings

alt

E — ... — E ’ F.

n n n n

: L(E, F ) ’ L(

(6) E, F ), where E is the linear subspace of all

n

symmetric tensors in β E. It is the universal solution of

n

∼ Ln (E; F ),

L E, F = sym

where Ln (E; F ) is the space of all bounded n-linear symmetric mappings

sym

E — ... — E ’ F.

n

∞

β : L(E, F ) ’ L( n=0 —β E is the tensor

(7) β E, β F ), where β E :=

algebra of E. Note that is has the universal property of prolonging bounded

linear mappings with values in locally convex spaces, which are algebras with

bounded operations, to continuous algebra homomorphisms:

L(E, F ) ∼ Alg( E, F ).

=

β

∞ n

: L(E, F ) ’ L( E, F ), where E :=

(8) E is the exterior

n=0

algebra. It has the universal property of prolonging bounded linear mappings

to continuous algebra homomorphisms into graded-commutative algebras,

i.e. algebras in the sense above, which are as vector spaces a coproduct

n∈N En and the multiplication maps Ek — El ’ Ek+l and for x ∈ Ek and

y ∈ El one has x · y = (’1)kl y · x.

∞ n

: L(E, F ) ’ L( E, F ) , where E := n=0

(9) E is the symmetric

algebra. It has the universal property of prolonging bounded linear mappings

to continuous algebra homomorphisms into commutative algebras.

Recall that the symmetric product is given as the image of the symmetrizer sym :

E —β · · · —β E ’ E —β · · · —β E given by

1

x1 — · · · — xn ’ xσ(1) — · · · — xσ(n) .

n!

σ∈Sn

Similarly the wedge product is given as the image of the alternator

alt : E —β · · · —β E ’ E —β · · · —β E

1

given by x1 — · · · — xn ’ sign(σ) xσ(1) — · · · — xσ(n) .

n!

σ∈Sn

5.9

58 Chapter I. Calculus of smooth mappings 5.11

Symmetrizer and alternator are bounded projections, so both subspaces are com-

plemented in the tensor product.

Proof. All results follow easily by ¬‚ipping coordinates until only a composition of

products of evaluation maps remains.

That the spaces in (5), and similar in (6), are universal solutions can be seen from

the following diagram:

w E — · · · — E ew

k

— alt

E — ... — E E

˜ ee

β β

f eef˜|

uh

e e

f k

E

F

5.10. Lemma. Let E be a convenient vector space. Then E ’ Pf (E) :=

E alg ⊆ C ∞ (E, R) is the free commutative algebra over the vector space E , i.e. to

every linear mapping f : E ’ A in a commmutative algebra, there exists a unique

˜

algebra homomorphism f : Pf (E) ’ A.

Elements of the space Pf (E) are called polynomials of ¬nite type on E.

Proof. The solution of this universal problem is given by the symmetric alge-

∞ k

bra E := E described in (5.9.9). In particular we have an algebra

k=0

homomorphism ˜ : E ’ Pf (E), which is onto, since by de¬nition Pf (E) is gen-

ι

N

k=1 ±k ∈

erated by E . It remains to show that it is injective. So let E,

k N

i.e. ±k ∈ E , with ˜( k=1 ±k ) = 0. Thus all derivatives ι(±k ) at 0 of this

ι

mapping in Pf (E) ⊆ P (E) ⊆ C ∞ (E, R) vanish. So it remains to show that

k

E ’ L(E, . . . , E; R) is injective, since then by the polarization identity also

k

E ’ Pf (E) ⊆ C ∞ (E, R) is injective. Let ± ∈ E — F be zero as element on

L(E, F ; R). We have ¬nitely many xk ∈ E and y k ∈ F with ± = k xk — y k and

we may assume that the {xk } are linearly independent. So we may choose vectors

xj ∈ E with xk (xj ) = δj . Then 0 = ±(xj , y) = k xk (xj ) · y k (y) = y j (y), so y j = 0

k

for all j and hence ± = 0.

Now for the mapping E1 — · · · — En ’ L(E1 , . . . , En ; R). We proceed by induction.

Let ± = k ±k — xk , where ±k ∈ E1 — En’1 and xk ∈ En . We may assume that

(xk )k is linearly independent. So, as before, we choose xj ∈ En with xk (xj ) = δj k

and get 0 = ±(y 1 , . . . , y n’1 , xj ) = ±j (y 1 , . . . , y n’1 ), hence ±j = 0 for all j and so

± = 0.

5.11. Corollary. Symmetry of higher derivatives. Let f : E ⊇ U ’ F

be smooth. The n-th derivative f (n) (x) = dn f (x), considered as an element of

Ln (E; F ), is symmetric, so has values in the space Lsym (E, . . . , E; F ) ∼ L( E; F )

k

=

5.11

5.13 5. Uniform boundedness principles and multilinearity 59

Proof. Recall that we can form iterated derivatives as follows:

f :E⊇U ’F

df : E ⊇ U ’ L(E, F )

d(df ) : E ⊇ U ’ L(E, L(E, F )) ∼ L(E, E; F )

=

.

.

.

d(. . . (d(df )) . . . ) : E ⊇ U ’ L(E, . . . , L(E, F ) . . . ) ∼ L(E, . . . , E; F )

=

Thus, the iterated derivative dn f (x)(v1 , . . . , vn ) is given by

˜

‚ ‚

‚t1 |t1 =0 · · · ‚tn |tn =0 f (x + t1 v1 + · · · + tn vn ) = ‚1 . . . ‚n f (0, . . . , 0),

˜

where f (t1 , . . . , tn ) := f (x + t1 v1 + · · · + tn vn ). The result now follows from the

¬nite dimensional property.

5.12. Theorem. Taylor formula. Let f : U ’ F be smooth, where U is c∞ -

open in E. Then for each segment [x, x + y] = {x + ty : 0 ¤ t ¤ 1} ⊆ U we

have

n 1

(1 ’ t)n n+1

1k k

f (x + ty)y n+1 dt,

f (x + y) = d f (x)y + d

k! n!

0

k=0

where y k = (y, . . . , y) ∈ E k .

Proof. This is an assertion on the smooth curve t ’ f (x + ty). Using functionals

we can reduce it to the scalar valued case, or we proceed directly by induction on

n: The ¬rst step is (6) in (2.6), and the induction step is partial integration of the

remainder integral.

5.13. Corollary. The following subspaces are direct summands:

L(E1 , . . . , En ; F ) ⊆ C ∞ (E1 — . . . — En , F ),

Ln (E; F ) ⊆ Ln (E; F ) := L(E, . . . , E; F ),

sym

Ln (E; F ) ⊆ Ln (E; F ),

alt

Ln (E; F ) ’ C ∞ (E, F ).

sym

Note that direct summand is meant in the bornological category, i.e. the embedding

admits a left-inverse in the category of bounded linear mappings, or, equivalently,

with respect to the bornological topology it is a topological direct summand.

Proof. The projection for L(E, F ) ⊆ C ∞ (E, F ) is f ’ df (0). The statement on

Ln follows by induction using cartesian closedness and (5.2). The projections for

the next two subspaces are the symmetrizer and alternator, respectively.

The last embedding is given by — , which is bounded and linear C ∞ (E — . . . —

E, F ) ’ C ∞ (E, F ). Here ∆ : E ’ E — . . . — E denotes the diagonal mapping

5.13

60 Chapter I. Calculus of smooth mappings 5.15

x ’ (x, . . . , x). A bounded linear left inverse C ∞ (E, F ) ’ Lk (E; F ) is given by

sym

1k

f ’ k! d f (0). See the following diagram:

y w L(E, . . .y , E; F )

Lk (E; F )

sym

u u

u

∞ ∞

C (E — . . . — E, F )

C (E, F )

∆—

5.14. Remark. We are now going to discuss polynomials between locally convex

spaces. Recall that for ¬nite dimensional spaces E = Rn a polynomial in a locally

convex vector space F is just a ¬nite sum

ak xk ,

k∈Nn

n

where ak ∈ F and xk := i=1 xki . Thus, it is just an element in the algebra gener-

i

ated by the coordinate projections pri tensorized with F . Since every (continuous)

linear functional on E = Rn is a ¬nite linear combination of coordinate projections,

this algebra is also the algebra generated by E — . For a general locally convex space

E we de¬ne the algebra of ¬nite type polynomials to be the one generated by E — .

However, there is also another way to de¬ne polynomials, namely as those smooth

functions for which some derivative is equal to 0. Take for example the square

2

: E ’ R on some in¬nite dimensional Hilbert space E. Its

of the norm

derivative is given by x ’ (v ’ 2 x, v ), and hence is linear. The second derivative

is x ’ ((v, w) ’ 2 v, w ) and hence constant. Thus, the third derivative vanishes.

This function is not a ¬nite type polynomial. Otherwise, it would be continuous

for the weak topology σ(E, E — ). Hence, the unit ball would be a 0-neighborhood

for the weak topology, which is not true, since it is compact for it.

Note that for (xk ) ∈ 2 the series k x2 converges pointwise and even uniformly

k

on compact sets. In fact, every compact set is contained in the absolutely convex

hull of a 0-sequence xn . In particular µk := sup{|xn | : n ∈ N} ’ 0 for k ’ ∞

k

nj

(otherwise, we can ¬nd an µ > 0 and kj ’ ∞ and nj ∈ N with xnj 2 ≥ |xkj | ≥ µ.

Since xn ∈ 2 ⊆ c0 , we conclude that nj ’ ∞, which yields a contradiction to

xn 2 ’ 0.) Thus

K ⊆ xn : n ∈ N ⊆ µn en absolutely convex ,

absolutely convex

and hence k≥n |xk | ¤ max{µk : k ≥ n} for all x ∈ K.

The series does not converge uniformly on bounded sets. To see this choose x = ek .

5.15. De¬nition. A smooth mapping f : E ’ F is called a polynomial if some

derivative dn f vanishes on E. The largest p such that dp f = 0 is called the degree

of the polynomial. The mapping f is called a monomial of degree p if it is of the

˜ ˜

form f (x) = f (x, . . . , x) for some f ∈ Lp (E; F ).

sym

5.15

5.18 5. Uniform boundedness principles and multilinearity 61

5.16. Lemma. Polynomials versus monomials.

(1) The smooth p-homogeneous maps are exactly the monomials of degree p.

(2) The symmetric multilinear mapping representing a monomial is unique.

(3) A smooth mapping is a polynomial of degree ¤ p if and only if its restriction

to each one dimensional subspace is a polynomial of degree ¤ p.

(4) The polynomials are exactly the ¬nite sums of monomials.

Proof. (1) Every monomial of degree p is clearly smooth and p-homogeneous. If

f is smooth and p-homogeneous, then

(dp f )(0)(x, . . . , x) = ( ‚t )p f (tx) = ( ‚t )p tp f (x) = p!f (x).

‚ ‚

t=0 t=0

(2) The symmetric multilinear mapping g ∈ Lp (E; F ) representing f is uniquely

sym

p

determined, since we have (d f )(0)(x1 , . . . , xp ) = p!g(x1 , . . . , xp ).

(3) & (4) Let the restriction of f to each one dimensional subspace be a polynomial

k

p

of degree ¤ p, i.e., we have (f (tx)) = k=0 t ( ‚t )k t=0 (f (tx)) for x ∈ E and ∈

‚

k!

p 1

F . So f (x) = k=0 k! dk f (0.x)(x, . . . , x) and hence is a ¬nite sum of monomials.

For the derivatives of a monomial q of degree k we have q (j) (tx)(v1 , . . . , vj ) =

k(k ’ 1) . . . (k ’ j + 1)tk’j q (x, . . . , x, v1 , . . . , vj ). Hence, any such ¬nite sum is a

˜

polynomial in the sense of (5.15).

Finally, any such polynomial has a polynomial as trace on each one dimensional

subspace.

5.17. Lemma. Spaces of polynomials. The space Polyp (E, F ) of polynomi-

k

als of degree ¤ p is isomorphic to k¤p L( E; F ) and is a direct summand in

C ∞ (E, F ) with a complement given by the smooth functions which are p-¬‚at at 0.

k

Proof. We have already shown that L( E; F ) embeds into C ∞ (E, F ) as a di-

rect summand, where a retraction is given by the derivative of order k at 0. Fur-

thermore, we have shown that the polynomials of degree ¤ p are exactly the di-

k

rect sums of homogeneous terms in L( E; F ). A retraction to the inclusion

k

E; F ) ’ C ∞ (E, F ) is hence given by k¤p k! dk |0 .

1

k¤p L(

Remark. The corresponding statement is false for in¬nitely ¬‚at functions. I.e.

the sequence E ’ C ∞ (R, R) ’ RN does not split, where E denotes the space of

smooth functions which are in¬nitely ¬‚at at 0. Otherwise, RN would be a subspace

of C ∞ ([0, 1], R) (compose the section with the restriction map from C ∞ (R, R) ’

C ∞ ([0, 1], R)) and hence would have a continuous norm. This is however easily

seen to be not the case.

5.18. Theorem. Uniform boundedness principle. If all Ei are convenient

vector spaces, and if F is a locally convex space, then the bornology on the space

L(E1 , . . . , En ; F ) consists of all pointwise bounded sets.

So a mapping into L(E1 , . . . , En ; F ) is smooth if and only if all composites with

evaluations at points in E1 — . . . — En are smooth.

5.18

62 Chapter I. Calculus of smooth mappings 5.20

Proof. Let us ¬rst consider the case n = 1. So let B ⊆ L(E, F ) be a pointwise

bounded subset. By lemma (5.3) we have to show that it is uniformly bounded on

each bounded subset B of E. We may assume that B is closed absolutely convex,

and thus EB is a Banach space, since E is convenient. By the classical uniform

boundedness principle, see (52.25), the set B|EB is bounded in L(EB , F ), and thus

B is bounded on B.

The smoothness detection principle: Clearly it su¬ces to recognize smooth curves.

If c : R ’ L(E, F ) is such that evx —¦c : R ’ F is smooth for all x ∈ E, then

j

c

clearly R ’ L(E, F ) ’

’ ’ E F is smooth. We will show that (j —¦ c) has values

in L(E, F ) ⊆ E F . Clearly, (j —¦ c) (s) is linear E ’ F . The family of mappings

c(s+t)’c(s)

: E ’ F is pointwise bounded for s ¬xed and t in a compact interval,

t

so by the ¬rst part it is uniformly bounded on bounded subsets of E. It converges

pointwise to (j —¦ c) (s), so this is also a bounded linear mapping E ’ F . By the

¬rst part j : L(E, F ) ’ E F is a bornological embedding, so c is di¬erentiable

into L(E, F ). Smoothness follows now by induction on the order of the derivative.

The multilinear case follows from the exponential law (5.2) by induction on n.

5.19. Theorem. Multilinear mappings on convenient vector spaces. A

multilinear mapping from convenient vector spaces to a locally convex space is boun-

ded if and only if it is separately bounded.

Proof. Let f : E1 — . . . — En ’ F be n-linear and separately bounded, i.e. xi ’

f (x1 , . . . , xn ) is bounded for each i and all ¬xed xj for j = i. Then f ∨ : E1 — . . . —

En’1 ’ L(En , F ) is (n ’ 1)-linear. By (5.18) the bornology on L(En , F ) consists

of the pointwise bounded sets, so f ∨ is separately bounded. By induction on n

it is bounded. The bornology on L(En , F ) consists also of the subsets which are

uniformly bounded on bounded sets by lemma (5.3), so f is bounded.

We will now derive an in¬nite dimensional version of (3.4), which gives us minimal

requirements for a mapping to be smooth.

5.20. Theorem. Let E be a convenient vector space. An arbitrary mapping f :

E ⊇ U ’ F is smooth if and only if all unidirectional iterated derivatives dp f (x) =

v

‚p p

( ‚t ) |0 f (x + tv) exist, x ’ dv f (x) is bounded on sequences which are Mackey

converging in U , and v ’ dp f (x) is bounded on fast falling sequences.

v

Proof. A smooth mapping obviously satis¬es this requirement. Conversely, from

(3.4) we see that f is smooth restricted to each ¬nite dimensional subspace, and

the iterated directional derivatives dv1 . . . dvn f (x) exist and are bounded multilinear

mappings in v1 , . . . , vn by (5.4), since they are universal linear combinations of the

unidirectional iterated derivatives dp f (x), compare with the proof of (3.4). So

v

n n

d f : U ’ L (E; F ) is bounded on Mackey converging sequences with respect to

the pointwise bornology on Ln (E; F ). By the uniform boundedness principle (5.18)

together with lemma (4.14) the mapping dn f : U — E n ’ F is bounded on sets

which are contained in a product of a bornologically compact set in U - i.e. a set

in U which is contained and compact in some EB - and a bounded set in E n .

5.20

5.21 5. Uniform boundedness principles and multilinearity 63

f (c(t))’f (c(0))

Now let c : R ’ U be a smooth curve. We have to show that converges

t

to f (c(0))(c (0)). It su¬ces to check that

f (c(t)) ’ f (c(0))

1

’ f (c(0))(c (0))

t t

is locally bounded with respect to t. Integrating along the segment from c(0) to

c(t) we see that this expression equals

1

c(t) ’ c(0)

1

f c(0) + s(c(t) ’ c(0)) ’ f (c(0))(c (0)) ds =

t t

0

c(t)’c(0)

1

’ c (0)

t

f c(0) + s(c(t) ’ c(0))

= ds

t

0

1 1

c(t) ’ c(0)

c(0) + rs(c(t) ’ c(0))

+ f s , c (0) dr ds.

t

0 0

The ¬rst integral is bounded since df : U — E ’ F is bounded on the product of

the bornologically compact set {c(0) + s(c(t) ’ c(0)) : 0 ¤ s ¤ 1, t near 0} in U and

the bounded set { 1 c(t)’c(0) ’ c (0) : t near 0} in E (use (1.6)).

t t

The second integral is bounded since d2 f : U — E 2 ’ F is bounded on the product

of the bornologically compact set {c(0) + rs(c(t) ’ c(0)) : 0 ¤ r, s ¤ 1, t near 0} in

U and the bounded set { s c(t)’c(0) , c (0) : 0 ¤ s ¤ 1, t near 0} in E 2 .

t

Thus f —¦ c is di¬erentiable in F with derivative df —¦ (c, c ). Now df : U — E ’ F

satis¬es again the assumptions of the theorem, so we may iterate.

5.21. The following result shows that bounded multilinear mappings are the right

ones for uses like homological algebra, where multilinear algebra is essential and

where one wants a kind of ˜continuity™. With continuity itself it does not work.

The same results hold for convenient algebras and modules, one just may take

c∞ -completions of the tensor products.

So by a bounded algebra A we mean a (real or complex) algebra which is also

a locally convex vector space, such that the multiplication is a bounded bilinear

mapping. Likewise, we consider bounded modules over bounded algebras, where the

action is bounded bilinear.

Lemma. [Cap et. al., 1993]. Let A be a bounded algebra, M a bounded right A-

module and N a bounded left A-module.

(1) There are a locally convex vector space M —A N and a bounded bilinear map

b : M — N ’ M —A N , (m, n) ’ m —A n such that b(ma, n) = b(m, an) for

all a ∈ A, m ∈ M and n ∈ N which has the following universal property: If

E is a locally convex vector space and f : M — N ’ E is a bounded bilinear

map such that f (ma, n) = f (m, an) then there is a unique bounded linear

˜ ˜

map f : M —A N ’ E with f —¦ b = f . The space of all such f is denoted

by LA (M, N ; E), a closed linear subspace of L(M, N ; E).

5.21

64 Chapter I. Calculus of smooth mappings 5.21

(2) We have a bornological isomorphism

LA (M, N ; E) ∼ L(M —A N, E).

=

(3) Let B be another bounded algebra such that N is a bounded right B-module

and such that the actions of A and B on N commute. Then M —A N is in

a canonical way a bounded right B-module.

(4) If in addition P is a bounded left B-module then there is a natural bibounded

isomorphism M —A (N —B P ) ∼ (M —A N ) —B P .

=

Proof. We construct M —A N as follows: Let M —β N be the algebraic tensor

product of M and N equipped with the (bornological) topology mentioned in (5.7)

and let V be the locally convex closure of the subspace generated by all elements of

the form ma — n ’ m — an, and de¬ne M —A N to be M —A N := (M —β N )/V . As

M —β N has the universal property that bounded bilinear maps from M — N into

arbitrary locally convex spaces induce bounded and hence continuous linear maps

on M — N , (1) is clear.

(2) By (1) the bounded linear map b— : L(M —A N, E) ’ LA (M, N ; E) is a bijection.

Thus, it su¬ces to show that its inverse is bounded, too. From (5.7) we get a

bounded linear map • : L(M, N ; E) ’ L(M —β N, E) which is inverse to the

map induced by the canonical bilinear map. Now let Lann(V ) (M —β N, E) be the

closed linear subspace of L(M —β N, E) consisting of all maps which annihilate V .

Restricting • to LA (M, N ; E) we get a bounded linear map • : LA (M, N ; E) ’

Lann(V ) (M —β N, E).

Let ψ : M —β N ’ M —A N be the the canonical projection. Then ψ induces a

ˆ ˆ

well de¬ned linear map ψ : Lann(V ) (M —β N, E) ’ L(M —A N, E), and ψ —¦ • is

ˆ

inverse to b— . So it su¬ces to show that ψ is bounded.

This is the case if and only if the associated map Lann(V ) (M —β N, E)—(M —A N ) ’

E is bounded. This in turn is equivalent to boundedness of the associated map

M —A N ’ L(Lann(V ) (M —β N, E), E) which sends x to the evaluation at x and is

clearly bounded.

(3) Let ρ : B op ’ L(N, N ) be the right action of B on N and let ¦ : LA (M, N ; M —A

N ) ∼ L(M —A N, M —A N ) be the isomorphism constructed in (2). We de¬ne the

=

right module structure on M —A N as:

ρ Id — b

B op ’ L(N, N ) ’ ’ ’ L(M — N, M — N ) ’—

’ ’’ ’

¦

’ LA (M, N ; M —A N ) ’ L(M —A N, M —A N ).

’ ’

This map is obviously bounded and easily seen to be an algebra homomorphism.

(4) Straightforward computations show that both spaces have the following uni-

versal property: For a locally convex vector space E and a trilinear map f : M —

N — P ’ E which satis¬es f (ma, n, p) = f (m, an, p) and f (m, nb, p) = f (m, n, bp)

there is a unique linear map prolonging f .

5.21

5.23 5. Uniform boundedness principles and multilinearity 65

5.22. Lemma. Uniform S-boundedness principle. Let E be a locally convex

space, and let S be a point separating set of bounded linear mappings with common

domain E. Then the following conditions are equivalent.

(1) If F is a Banach space (or even a c∞ -complete locally convex space) and

f : F ’ E is a linear mapping with » —¦ f bounded for all » ∈ S, then f is

bounded.

(2) If B ⊆ E is absolutely convex such that »(B) is bounded for all » ∈ S and

the normed space EB generated by B is complete, then B is bounded in E.

(3) Let (bn ) be an unbounded sequence in E with »(bn ) bounded for all » ∈ S,

then there is some (tn ) ∈ 1 such that tn bn does not converge in E for

the initial locally convex topology induced by S.

De¬nition. We say that E satis¬es the uniform S-boundedness principle if these

equivalent conditions are satis¬ed.

Proof. (1) ’ (3) : Suppose that (3) is not satis¬ed. So let (bn ) be an unbounded

sequence in E such that »(bn ) is bounded for all » ∈ S, and such that for all

(tn ) ∈ 1 the series tn bn converges in E for the initial locally convex topology

induced by S. We de¬ne a linear mapping f : 1 ’ E by f ((tn )n ) = tn bn , i.e.

f (en ) = bn . It is easily checked that » —¦ f is bounded, hence by (1) the image of

the closed unit ball, which contains all bn , is bounded. Contradiction.

(3) ’ (2): Let B ⊆ E be absolutely convex such that »(B) is bounded for all

» ∈ S and that the normed space EB generated by B is complete. Suppose that B

is unbounded. Then B contains an unbounded sequence (bn ), so by (3) there is some

(tn ) ∈ 1 such that tn bn does not converge in E for the weak topology induced

m m

by S. But tn bn is a Cauchy sequence in EB , since k=n tn bn ∈ ( k=n |tn |) · B,

and thus converges even bornologically, a contradiction.

(2) ’ (1): Let F be convenient, and let f : F ’ E be linear such that » —¦ f is

bounded for all » ∈ S. It su¬ces to show that f (B), the image of an absolutely

convex bounded set B in F with FB complete, is bounded. By assumption, »(f (B))

is bounded for all » ∈ S, the normed space Ef (B) is a quotient of the Banach space

FB , hence complete. By (2) the set f (B) is bounded.

5.23. Lemma. A convenient vector space E satis¬es the uniform S-boundedness

principle for each point separating set S of bounded linear mappings on E if and

only if there exists no strictly weaker ultrabornological topology than the bornological

topology of E.

Proof. (’) Let „ be an ultrabornological topology on E which is strictly weaker

than the natural bornological topology. Since every ultrabornological space is an

inductive limit of Banach spaces, cf. (52.31), there exists a Banach space F and

a continuous linear mapping f : F ’ (E, „ ) which is not continuous into E. Let

S = {Id : E ’ (E, „ )}. Now f does not satisfy (5.22.1).

(⇐) If S is a point separating set of bounded linear mappings, the ultrabornological

topology given by the inductive limit of the spaces EB with B satisfying (5.22.2)

equals the natural bornological topology of E. Hence, (5.22.2) is satis¬ed.

5.23

66 Chapter I. Calculus of smooth mappings 6.1

5.24. Theorem. Webbed spaces have the uniform boundedness prop-

erty. A locally convex space which is webbed satis¬es the uniform S-boundedness

principle for any point separating set S of bounded linear functionals.

Proof. Since the bornologi¬cation of a webbed space is webbed, cf. (52.14), we

may assume that E is bornological, and hence that every bounded linear functional

is continuous, see (4.1). Now the closed graph principle (52.10) applies to any

mapping satisfying the assumptions of (5.22.1).

5.25. Lemma. Stability of the uniform boundedness principle. Let F be

a set of bounded linear mappings f : E ’ Ef between locally convex spaces, let Sf

be a point separating set of bounded linear mappings on Ef for every f ∈ F, and

let S := f ∈F f — (Sf ) = {g —¦ f : f ∈ F, g ∈ Sf }. If F generates the bornology and

Ef satis¬es the uniform Sf -boundedness principle for all f ∈ F, then E satis¬es

the uniform S-boundedness principle.

Proof. We check the condition (1) of (5.22). So assume h : F ’ E is a linear

mapping for which g —¦ f —¦ h is bounded for all f ∈ F and g ∈ Sf . Then f —¦ h

is bounded by the uniform Sf - boundedness principle for Ef . Consequently, h is

bounded since F generates the bornology of E.

5.26. Theorem. Smooth uniform boundedness principle. Let E and F be

convenient vector spaces, and let U be c∞ -open in E. Then C ∞ (U, F ) satis¬es the

uniform S-boundedness principle where S := {evx : x ∈ U }.

Proof. For E = F = R this follows from (5.24), since C ∞ (U, R) is a Fr´chet space.

e

The general case then follows from (5.25).

6. Some Spaces of Smooth Functions

6.1. Proposition. Let M be a smooth ¬nite dimensional paracompact manifold.

Then the space C ∞ (M, R) of all smooth functions on M is a convenient vector space

in any of the following (bornologically) isomorphic descriptions, and it satis¬es the

uniform boundedness principle for the point evaluations.

(1) The initial structure with respect to the cone

c—

C ∞ (M, R) ’ C ∞ (R, R)

’

for all c ∈ C ∞ (R, M ).

(2) The initial structure with respect to the cone

(u’1 )—

C (M, R) ’ ’ ’ C ∞ (Rn , R),

∞ ±

’’

where (U± , u± ) is a smooth atlas with u± (U± ) = Rn .

6.1

6.2 6. Some spaces of smooth functions 67

(3) The initial structure with respect to the cone

jk

∞

C (M, R) ’ C(J k (M, R))

’

for all k ∈ N, where J k (M, R) is the bundle of k-jets of smooth functions on

M , where j k is the jet prolongation, and where all the spaces of continuous

sections are equipped with the compact open topology.

It is easy to see that the cones in (2) and (3) induce even the same locally con-

vex topology which is sometimes called the compact C ∞ topology, if C ∞ (Rn , R)

is equipped with its usual Fr´chet topology. From (2) we see also that with the

e

bornological topology C ∞ (M, R) is nuclear by (52.35), and is a Fr´chet space if and

e

only if M is separable.

Proof. For all three descriptions the initial locally convex topology is convenient,

since the spaces are closed linear subspaces in the relevant products of the right

hand sides. Thus, the uniform boundedness principle for the point evaluations holds

for all structures since it holds for all right hand sides (for C(J k (M, R)) we may

reduce to a connected component of M , and we then have a Fr´chet space). So the

e

identity is bibounded between all structures.

6.2. Spaces of smooth functions with compact supports. For a smooth

∞

separable ¬nite dimensional Hausdor¬ manifold M we denote by Cc (M, R) the

vector space of all smooth functions with compact supports in M .

∞

Lemma. The following convenient structures on the space Cc (M, R) are all iso-

morphic:

∞

(1) Let CK (M, R) be the space of all smooth functions on M with supports

contained in the ¬xed compact subset K ⊆ M , a closed linear subspace of

C ∞ (M, R). Let us consider the ¬nal convenient vector space structure on

∞

the space Cc (M, R) induced by the cone

∞ ∞

CK (M, R) ’ Cc (M, R)

where K runs through a basis for the compact subsets of M . Then the

∞

space Cc (M, R) is even the strict inductive limit of a sequence of spaces

∞

CK (M, R).

∞

(2) We equip Cc (M, R) with the initial structure with respect to the cone:

e—

∞

’∞

Cc (M, R) ’ Cc (R, R),

∞

where e ∈ Cprop (R, M ) runs through all proper smooth mappings R ’ M ,

∞

and where Cc (R, R) carries the usual inductive limit topology on the space

∞

of test functions, with steps CI (R, R) for compact intervals I.

(3) The initial structure with respect to the cone

jk

∞

Cc (J k (M, R))

Cc (M, R) ’’

6.2

68 Chapter I. Calculus of smooth mappings 6.2

for all k ∈ N, where J k (M, R) is the bundle of k-jets of smooth functions

on M , where j k is the jet prolongation, and where the spaces of continuous

sections with compact support are equipped with the inductive limit topology

with steps CK (J k (M, R)).

∞

The space Cc (M, R) satis¬es the uniform boundedness principle for the point eval-

uations.

∞

First Proof. We note ¬rst that in all descriptions the space Cc (M, R) is conve-

nient and satis¬es the uniform boundedness principle for point evaluations:

∞ ∞

In (1) we have Cc (M, R) = Cc (Mi , R) where Mi are the connected components

of M , which are separable, so the inductive limit is a strict inductive limit of a

∞

sequence of Fr´chet spaces, hence each Cc (Mi , R) is convenient and webbed by

e

(52.13) and (52.12), hence satis¬es the uniform boundedness principle by (5.24).

∞

So Cc (M, R) is convenient and satis¬es also the uniform boundedness principle for

the point evaluations, by [Fr¨licher, Kriegl, 1988, 3.4.4].

o

In (2) and (3) the space is a closed subspace of the product of the right hand side

spaces, which are convenient and satisfy the uniform boundedness principle, shown

as for (1).

Hence, the identity is bibounded for all structures.

∞

Second Proof. In all three structures the space Cc (M, R) is the direct sum of

∞

the spaces Cc (M± , R) for all connected components M± of M . So we may assume

that M is connected and thus separable.

We consider the diagram

RR

∞

Cc (M, R)

‘“

‘ RT

R

‘

‘

™ e—

u{

(M, R)x xx

∞ ∞ ∞

xx

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

¢

x

ex

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