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