This is bounded, and so for f1 = f0 + f2 the expression |δeq (f1 —¦ c)(xn ; 2n vn )| is

k+2

also bounded, with respect to n. However, on the other hand we get

δeq (f1 —¦ c)(xn ; 2n vn ) = δeq (f1 —¦ cn )(0; 2n vn )

k+2 k+2

δ k+2 (f1 —¦ en )(xn ; vn )

1

= 2n(k+2) eq

k+2

f1 ((i ’ 1)vn + xn , ivn )

(k+2)! 1

= 2n(k+2) i’j

(k+2)

vn

i=0 0¤j¤k+2

j=i

12.6

12.8 12. Di¬erentiability of ¬nite order 123

k+2

g((i ’ 1)vn + xn )ivn

(k+2)! 1

= 2n(k+2) i’j

(k+2)

vn

i=0 0¤j¤k+2

j=i

k+1

g(lvn + xn )

(k+2)! 1

= 2n(k+2) l’j

(k+1)

vn

l=0 0¤j¤k+1

j=l

δ k+1 g(xn ; vn ),

k+2

= 2n(k+2) eq

which in absolute value is larger than (k + 2)n by (1), a contradiction.

12.7. Lemma. Let E be a normed space and F be a convenient vector space, U

open in E. Then, a mapping f : U ’ F is Lip0 if and only if f is locally Lipschitz,

i.e., f (x)’f (y) is locally bounded.

x’y

Proof. (’) If f is Lip0 but not locally Lipschitz near z ∈ U , there are ∈ F

and points xn = yn in U with xn ’ z ¤ 1/2n and yn ’ z ¤ 1/2n , such that

(f (yn ) ’ f (xn )) ≥ n. yn ’ xn . Now we apply the general curve lemma (12.2)

with sn := yn ’ xn and cn (t) := xn ’ z + t(yn ’ xn ) to get a smooth curve c with

c(t + tn ) = cn (t) for 0 ¤ t ¤ sn . Then s1 (( —¦ f —¦ c)(tn + sn ) ’ ( —¦ f —¦ c)(tn )) =

n

1

yn ’xn (f (yn ) ’ f (xn )) ≥ n.

(⇐) This is obvious, since the composition of locally Lipschitzian mappings is again

locally Lipschitzian.

12.8. Theorem. Let f : E ⊇ U ’ F be a mapping, where E and F are convenient

vector spaces, and U ⊆ E is c∞ -open. Then the following assertions are equivalent

for each k ≥ 0:

(1) f is Lipk+1 .

(2) The directional derivative

‚

‚t |t=0 (f (x

(dv f )(x) := + tv))

exists for x ∈ U and v ∈ E and de¬nes a Lipk -mapping U — E ’ F .

Note that this result gives a di¬erent (more algebraic) proof of Boman™s theorem

(3.4) and (3.14).

Proof. (1) ’ (2) Clearly, t ’ f (x+tv) is Lipk+1 , and so the directional derivative

exists and is the Mackey-limit of the di¬erence quotients, by lemma (1.7). In order

to show that df : (x, v) ’ dv f (x) is Lipk we take a smooth curve (x, v) : R ’

U — E and ∈ F , and we consider g(t, s) := x(t) + s.v(t), g : R2 ’ E. Then

—¦ f —¦ g : R2 ’ R is Lipk+1 , so by lemma (12.6) the curve

‚ ‚

t ’ (df (x(t), v(t))) = ‚s |0 f (g(t, s)) ‚s |0

= (f (g(t, s)))

is of class Lipk .

12.8

124 Chapter III. Partitions of unity 12.10

(2) ’ (1) If c ∈ C ∞ (R, U ) then

f (c(t)) ’ f (c(0))

’ df (c(0), c (0)) =

t

1

df (c(0) + s(c(t) ’ c(0)), c(t)’c(0) ) ’ df (c(0), c (0)) ds

= t

0

converges to 0 for t ’ 0 since g : (t, s) ’ df (c(0) + s(c(t) ’ c(0)), c(t)’c(0) ) ’

t

k

df (c(0), c (0)) is Lip , thus by lemma (12.7) g is locally Lipschitz, so the set of all

1

g(t1 ,s)’g(t2 ,s)

is locally bounded, and ¬nally t ’ 0 g(t, s)ds is locally Lipschitz.

t1 ’t2

Thus, f —¦ c is di¬erentiable with derivative (f —¦ c) (0) = df (c(0), c (0)).

Since df is Lipk and (c, c ) is smooth we get that (f —¦ c) is Lipk , hence f —¦ c is

Lipk+1 .

12.9. Corollary. Chain rule. The composition of Lipk -mappings is again Lipk ,

and the usual formula for the derivative of the composite holds.

Proof. We have to compose f —¦ g with a smooth curve c, but then g —¦ c is a Lipk -

curve, thus it is su¬cient to show that the composition of a Lipk curve c : R ’ U ⊆

E with a Lipk -mapping f : U ’ F is again Lipk , and that (f —¦c) (t) = df (c(t), c (t)).

This follows by induction on k for k ≥ 1 in the same way as we proved theorem

(12.8.2) ’ (12.8.1), using theorem (12.8) itself.

12.10. De¬nition and Proposition. Let F be a convenient vector space. The

space Lipk (R, F ) of all Lipk -curves in F is again a convenient vector space with

the following equivalent structures:

(1) The initial structure with respect to the k + 2 linear mappings (for 0 ¤ j ¤

k + 1) c ’ δ j c from Lipk (R, F ) into the space of all F -valued maps in j + 1

pairwise di¬erent real variables (t0 , . . . , tj ) which are bounded on bounded

subsets, with the c∞ -complete locally convex topology of uniform convergence

on bounded subsets. In fact, the mappings δ 0 and δ k+1 are su¬cient.

(2) The initial structure with respect to the k + 2 linear mappings (for 0 ¤ j ¤

k +1) c ’ δeq c from Lipk (R, F ) into the space of all maps from R—(R\{0})

j

into F which are bounded on bounded subsets, with the c∞ -complete locally

convex topology of uniform convergence on bounded subsets. In fact, the

0 k+1

mappings δeq and δeq are su¬cient.

(3) The initial structure with respect to the derivatives of order j ¤ k considered

as linear mappings into the space of Lip0 -curves, with the locally convex

topology of uniform convergence of the curve on bounded subsets of R and

of the di¬erence quotient on bounded subsets of {(t, s) ∈ R2 : t = s}.

The convenient vector space Lipk (R, F ) satis¬es the uniform boundedness principle

with respect to the point evaluations.

Proof. All three structures describe closed embeddings into ¬nite products of

spaces, which in (1) and (2) are obviously c∞ -complete. For (3) this follows, since

by (1) the structure on Lip0 (R, E) is convenient.

12.10

12.13 12. Di¬erentiability of ¬nite order 125

All structures satisfy the uniform boundedness principle for the point evaluations

by (5.25), and since spaces of all bounded mappings on some (bounded) set satisfy

this principle. This can be seen by composing with — for all ∈ E , since Banach

spaces do this by (5.24).

By applying this uniform boundedness principle one sees that all these structures

are indeed equivalent.

12.11. De¬nition and Proposition. Let E and F be convenient vector spaces

and U ⊆ E be c∞ -open. The space Lipk (U, F ) of all Lipk -mappings from U to F

is again a convenient vector space with the following equivalent structures:

(1) The initial structure with respect to the linear mappings c— : Lipk (U, F ) ’

Lipk (R, F ) for all c ∈ C ∞ (R, F ).

(2) The initial structure with respect to the linear mappings c— : Lipk (U, F ) ’

Lipk (R, F ) for all c ∈ Lipk (R, F ).

This space satis¬es the uniform boundedness principle with respect to the evaluations

evx : Lipk (U, F ) ’ F for all x ∈ U .

Proof. The structure (1) is convenient since by (12.1) it is a closed subspace of the

product space which is convenient by (12.10). The structure in (2) is convenient

since it is closed by (12.9). The uniform boundedness principle for the point evalu-

ations now follows from (5.25) and (12.10), and this in turn gives us the equivalence

of the two structures.

12.12. Remark. We want to call the attention of the reader to the fact that there

is no general exponential law for Lipk -mappings. In fact, if f ∈ Lipk (R, Lipk (R, F ))

then ( ‚t )p ( ‚s )q f § (t, s) exists if max(p, q) ¤ k. This describes a smaller space than

‚ ‚

Lipk (R2 , F ), which is not invariantly describable.

However, some partial results still hold, namely for convenient vector spaces E, F ,

and G, and for c∞ -open sets U ⊆ E, V ⊆ F we have

Lipk (U, L(F, G)) ∼ L(F, Lipk (U, G)),

=

Lipk (U, Lipl (V, G)) ∼ Lipl (V, Lipk (U, G)),

=

see [Fr¨licher, Kriegl, 1988, 4.4.5, 4.5.1, 4.5.2]. For a mapping f : U — F ’ G which

o

is linear in F we have: f ∈ Lipk (U — F, G) if and only if f ∨ ∈ Lipk (U, L(E, F )),

see [Fr¨licher, Kriegl, 1988, 4.3.5]. The last property fails if we weaken Lipschitz to

o

continuous, see the following example.

1

12.13. Smolyanov™s Example. Let f : 2 ’ R be de¬ned by f := k≥1 k2 fk ,

where fk (x) := •(k(kxk ’ 1)) · j<k •(jxj ) and • : R ’ [0, 1] is smooth with

•(0) = 1 and •(t) = 0 for |t| ≥ 1 . We shall show that

4

2

’ R is Fr´chet di¬erentiable.

(1) f : e

2

’ ( 2 ) is not continuous.

(2) f :

2

— 2 ’ R is continuous.

(3) f :

12.13

126 Chapter III. Partitions of unity 12.13

1

2 2

Proof. Let A := {x ∈ : |kxk | ¤ for all k}. This is a closed subset of .

4

2

(1) Remark that for x ∈ at most one fk (x) can be unequal to 0. In fact fk (x) = 0

1 1 3

implies that |kxk ’ 1| ¤ 4k ¤ 4 , and hence kxk ≥ 4 and thus fj (x) = 0 for j > k.

For x ∈ A there exists a k > 0 with |kxk | > 1 and the set of points satisfying this

/ 4

condition is open. It follows that •(kxk ) = 0 and hence f = j<k j1 fj is smooth

2

on this open set.

On the other hand let x ∈ A. Then |kxk ’ 1| ≥ 4 > 1 and hence •(k(kxk ’ 1)) = 0

3

4

2

for all k and thus f (x) = 0. Let v ∈ be such that f (x + v) = 0. Then there exists

a unique k such that fk (x + v) = 0 and therefore |j(xj + vj )| < 1 for j < k and

4

1 1 1

|k(xk +vk )’1| < 4k ¤ 4 . Since |kxk | ¤ 4 we conclude |kvk | ≥ 1’|k(xk +vk )’1|’

|kxk | ≥ 1 ’ 1 ’ 1 = 2 . Hence |f (x + v)| = k2 |fk (x + v)| ¤ k2 ¤ (2|vk |)2 ¤ 4 v 2 .

1 1 1

4 4

Thus f (x+v)’0’0 ¤ 4 v ’ 0 for v ’ 0, i.e. f is Fr´chet di¬erentiable at x

e

v

with derivative 0.

d1

(2) If fact take a ∈ R with • (a) = 0. Then f (t ek )(ek ) = dt k2 fk (t ek ) =

d1 1a

2

dt k2 •(k t ’ k) = • (k (k t ’ 1)) = • (a) if t = tk := k k + 1 , which goes to

0 for k ’ ∞. However f (0)(ek ) = 0 since 0 ∈ A.

(3) We have to show that f (xn )(v n ) ’ f (x)(v) for (xn , v n ) ’ (x, v). For x ∈ A

/

this is obviously satis¬ed, since then there exists a k with |kxk | > 1 and hence

4

1

f = j¤k j 2 fj locally around x.

If x ∈ A then f (x) = 0 and thus it remains to consider the case, where xn ∈ A. /

Let µ > 0 be given. Locally around xn at most one summand fk does not vanish:

If xn ∈ A then there is some k with |kxk | > 1/4 which we may choose minimal.

/

Thus |jxj | ¤ 1/4 for all j < k, so |j(jxj ’ 1)| ≥ 3j/4 and hence fj = 0 locally since

the ¬rst factor vanishes. For j > k we get fj = 0 locally since the second factor

vanishes. Thus we can evaluate the derivative:

1 •∞ 2n

|f (xn )(v n )| = 2 fk (xn )(v n ) ¤ n

k |vk | + j|vj | .

2

k k

j<k

Since v ∈ 2 we ¬nd a K1 such that ( j≥K1 |vj |2 )1/2 ¤ 2 •µ ∞ . Thus we conclude

from v n ’v 2 ’ 0 that |vj | ¤ • µ ∞ for j ≥ K1 and large n. For the ¬nitely many

n

small n we can increase K1 such that for these n and j ≥ K1 also |vj | ¤ • µ ∞ .

n

Furthermore there is a constant K2 ≥ 1 such that v n ∞ ¤ v n 2 ¤ K2 for all n.

Now choose N ≥ K1 so large that N 2 ≥ 1 • ∞ K2 K1 . Obviously n<N n2 fn is 1

2

µ

smooth. So it remains to consider those n for which the non-vanishing term has

index k ≥ N . For those terms we have

1 1

|f (xn )(v n )| = 2 fk (xn )(v n ) ¤ • ∞ |vk | + 2

n n

j|vj |

k k

j<k

1 1

n n n

¤ |vk | • j|vj | + j|vj | •

∞+ • ∞2 ∞

k2

k

K1 ¤j<k

j<K1

2

K1 n 1

¤µ+ • ∞ 2 v j µ ¤ µ + µ + µ = 3µ

∞+

k2

N

K1 ¤j<k

This shows the continuity.

12.13

127

13. Di¬erentiability of Seminorms

A desired separation property is that the smooth functions generate the topology.

Since a locally convex topology is generated by the continuous seminorms it is

natural to look for smooth seminorms. Note that every seminorm p : E ’ R on a

vector space E factors over Ep := E/ ker p and gives a norm on this space. Hence, it

˜˜ ˜

can be extended to a norm p : Ep ’ R on the completion Ep of the space Ep which

is normed by this factorization. If E is a locally convex space and p is continuous,

then the canonical quotient mapping E ’ Ep is continuous. Thus, smoothness of

p o¬ 0 implies smoothness of p on its carrier, and so the case where E is a Banach

˜

space is of central importance.

Obviously, every seminorm is a convex function, and hence we can generalize our

treatment slightly by considering convex functions instead. The question of their

di¬erentiability properties is exactly the topic of this section.

Note that since the smooth functions depend only on the bornology and not on

the locally convex topology the same is true for the initial topology induced by all

smooth functions. Hence, it is appropriate to make the following

Convention. In this chapter the locally convex topology on all convenient vector

spaces is assumed to be the bornological one.

13.1. Remark. It can be easily seen that for a function f : E ’ R on a vector

space E the following statements are equivalent (see for example [Fr¨licher, Kriegl,

o

1988, p. 199]):

n n

(1) The function f is convex, i.e. f ( i=1 »i xi ) ¤ i=1 »i f (xi ) for »i ≥ 0

n

with i=1 »i = 1;

(2) The set Uf := {(x, t) ∈ E — R : f (x) ¤ t} is convex;

(3) The set Af := {(x, t) ∈ E — R : f (x) < t} is convex.

Moreover, for any translation invariant topology on E (and hence in particular for

the locally convex topology or the c∞ -topology on a convenient vector space) and

any convex function f : E ’ R the following statements are equivalent:

(1) The function f is continuous;

(2) The set Af is open in E — R;

(3) The set f<t := {x ∈ E : f (x) < t} is open in E for all t ∈ R.

13.2. Result. Convex Lipschitz functions. Let f : E ’ R be a convex func-

tion on a convenient vector space E. Then the following statements are equivalent:

(1) It is locally Lipschitzian;

(2) It is continuous for the locally convex topology;

continuous for the c∞ -topology;

(3) It is

(4) It is bounded on Mackey converging sequences;

If f is a seminorm, then these further are equivalent to

(5) It is bounded on bounded sets.

13.2

128 Chapter III. Partitions of unity 13.3

If E is normed this further is equivalent to

(6) It is locally bounded.

The proof is due to [Aronszajn, 1976] for Banach spaces and [Fr¨licher, Kriegl,

o

1988, p. 200], for convenient vector spaces.

13.3. Basic de¬nitions. Let f : E ⊇ U ’ F be a mapping de¬ned on a c∞ -open

subset of a convenient vector space E with values in another one F . Let x ∈ U

and v ∈ E. Then the (one sided) directional derivative of f at x in direction v is

de¬ned as

f (x + t v) ’ f (x)

f (x)(v) = dv f (x) := lim .

t

t0

Obviously, if f (x)(v) exists, then so does f (x)(s v) for s > 0 and equals s f (x)(v).

Even if f (x)(v) exists for all v ∈ E the mapping v ’ f (x)(v) may not be linear

in general, and if it is linear it will not be bounded in general. Hence, f is called

Gˆteaux-di¬erentiable at x, if the directional derivatives f (x)(v) exist for all v ∈ E

a

and v ’ f (x)(v) is a bounded linear mapping from E ’ F .

Even for Gˆteaux-di¬erentiable mappings the di¬erence quotient f (x+t v)’f (x) need

a t

not converge uniformly for v in bounded sets (or even in compact sets). Hence, one

de¬nes f to be Fr´chet-di¬erentiable at x if f is Gˆteaux-di¬erentiable at x and

e a

f (x+t v)’f (x)

’ f (x)(v) ’ 0 uniformly for v in any bounded set. For a Banach

t

space E this is equivalent to the existence of a bounded linear mapping denoted

f (x) : E ’ F such that

f (x + v) ’ f (x) ’ f (x)(v)

lim = 0.

v

v’0

If f : E ⊇ U ’ F is Gˆteaux-di¬erentiable and the derivative f : E ⊇ U ’

a

L(E, F ) is continuous, then f is Fr´chet-di¬erentiable, and we will call such a

e

function C 1 . In fact, the fundamental theorem applied to t ’ f (x + t v) gives us