∞

(2) follows directly from (19.15) applied to the algebra A = Clfs .

∞

(3) By (19.13.2) the space E has ω-small Clfs -zerosets. By (19.14.1) we have as-

∞

sumption (iv) in (19.13), and then by (19.13.4) we have E = Homω (Clfs (E)). Thus

E belongs to RZ.

(4) Since every space E in c0 -ext is obtained by applying ¬nitely many constructions

as in (2) and a last one as in (3) we get it for E.

19.25. Remark. [Adam, Bistr¨m, Kriegl, 1995]. The class RZ is ˜quite big™. By

o

(19.24.4) we have that c0 -ext is a subclass of RZ. Also the following spaces are in

RZ:

The space C(K) where K is the one-point compacti¬cation of the topological

(ω)

disjoint union of a sequence of compact spaces Kn with Kn = …. In fact we

have a continuous injection given by the countable product of the restriction maps

C(K) ’ C(Kn ). Hence the result follows from (19.24.4) using also the remark in

(19.16) for the C(Kn ), followed by (19.20) for the product and by (19.21) for C(K).

Remark that in such a situation we might have K (ω) = {∞} = ….

The space D[0, 1] of all functions f : [0, 1] ’ R which are right continuous and have

left limits and endowed with the sup norm is in RZ. Indeed it contains C[0, 1] as a

subspace and D[0, 1]/C[0, 1] ∼ c0 [0, 1] according to [Corson, 1961]. By (18.27) we

=

have that C[0, 1] is weakly Lindel¨f and Pf is ω-isolating, since {evt : t ∈ Q © [0, 1]}

o

are point-separating. Now we use (19.24.3).

Open Problem. Is ∞ (“) in RZ for |“| non-measurable, i.e. is Clfs (

∞ ∞

(“)) ω-

isolating on ∞ (“) and is Homω Clfs ( ∞ (“)) = ∞ (“)?

∞

If this is true, then every complete locally convex space E of non-measurable cardi-

nality would be in RZ, since every Banach space E is a closed subspace of ∞ (“),

where “ is the closed unit-ball of E .

19.25

217

20. Sets on which all Functions are Bounded

In this last section the relationship of evaluation properties and bounding sets, i.e.

sets on which every function of the algebra is bounded, are studied.

20.1. Proposition. [Kriegl, Nel, 1990, 2.2]. Let A be a convenient algebra, and

B ⊆ X be A-bounding. Then pB : f ’ sup{|f (x)| : x ∈ B} is a bounded seminorm

on A.

A subset B ⊆ X is called A-bounding if f (B) ⊆ R is bounded for all f ∈ A .

Proof. Since B is bounding, we have that pB (f ) < ∞. Now assume there is

some bounded set F ⊆ A, for √ which pB (F) is not bounded. Then we may choose

fn ∈ F, such that pB (fn ) ≥ n2n . Note that {f 2 : f ∈ F} is bounded, since

multiplication is assumed to be bounded. Furthermore pB (f 2 ) = sup{|f (x)|2 :

x ∈ B} = sup{|f (x)| : x ∈ B}2 = pB (f )2 , since t ’ t2 is a monotone bijection

∞

R+ ’ R+ , hence pB (fn ) ≥ n2n . Now consider the series n=0 21 fn . This series is

2 2

n

Mackey-Cauchy, since (2’n )n ∈ 1 and {fn : n ∈ N} is bounded. Since A is assumed

2

to be convenient, we have that this series is Mackey convergent. Let f ∈ A be its

limit. Since all summands are non-negative we have

∞

12 12 1

fn ≥ pB ( n fn ) = n pB (fn )2 ≥ n,

pB (f ) = pB

2n 2 2

n=0

for all n ∈ N, a contradiction.

20.2. Proposition. [Kriegl, Nel, 1990, 2.3] for A-paracompact, [Bistr¨m, Bjon,

o

Lindstr¨m, 1993, Prop.2]. If X is A-realcompact then every A-bounding subset of

o

X is relatively compact in XA .

Proof. Consider the diagram

y w Hom(A) y w

∼

=

XA R

A

and let B ⊆ X be A-bounding. Then its image in A R is relatively compact by

Tychono¬™s theorem. Since Hom(A) ⊆ A R is closed, we have that B is relatively

compact in XA .

20.3. Proposition. [Bistr¨m, Jaramillo, Lindstr¨m, 1995, Prop.7]. Every func-

o o

∞

tion f = n=0 pn ∈ Cconv ( ∞ ) converges uniformly on the bounded sets in c0 . In

ω

particular, each bounded set in c0 is Cconv -bounding in l∞ .

ω

∞

Proof. Take f = n=0 pn ∈ Cconv ( ∞ ). According to (7.14), the function f may

ω

˜

be extended to a holomorphic function f ∈ H( ∞ — C) on the complexi¬cation.

[Josefson, 1978] showed that each holomorphic function on ∞ — C is bounded on

˜

every bounded set in c0 —C. Hence, the restriction f |c0 —C is a holomorphic function

on c0 — C which is bounded on bounded subsets. By (7.15) its Taylor series at zero

∞

n=0 pn converges uniformly on each bounded subset of c0 — C. The statement

then follows by restricting to the bounded subsets of the real space c0 .

20.3

218 Chapter IV. Smoothly realcompact spaces 20.6

20.4. Result. [Bistr¨m, Jaramillo, Lindstr¨m, 1995, Corr.8]. Every weakly com-

o o

pact set in c0 , in particular the set {en : n ∈ N} ∪ {0} with en the unit vectors, is

RCconv -bounding in l∞ .

ω

20.5. Result. [Bistr¨m, Jaramillo, Lindstr¨m, 1995, Thm.5]. Let A be a functo-

o o

rial algebra on the category of continuous linear maps between Banach spaces with

RP ⊆ A. Then, for every Banach space E, the A-bounding sets are relatively

compact in E if there is a function in A( ∞ ) that is unbounded on the set of unit

vectors in ∞ .

20.6. Result.

(1) [Bistr¨m, Jaramillo, 1994, Thm.2] & [Bistr¨m, 1993, p.73, Thm.5.23]. In

o o

∞

all Banach spaces the Clfcs -bounding sets are relatively compact.

∞

(2) [Bistr¨m, Jaramillo, 1994, p.5] & [Bistr¨m, 1993, p.74,Cor.5.24]. Any Clfcs -

o o

bounding set in a locally convex space E is precompact and therefore rela-

tively compact if E, in addition, is quasi-complete.

(3) [Bistr¨m, Jaramillo, 1994, Cor.4] & [Bistr¨m, 1993, p.74, 5.25]. Let E be

o o

a quasi-complete locally convex space. Then E and EClfcs have the same

∞

compact sets. Furthermore xn ’ x in E if and only if f (xn ) ’ f (x) for all

∞

f ∈ Clfcs (E).

20.6

219

Chapter V

Extensions and Liftings of Mappings

21. Extension and Lifting Properties . . . . . . . . . . . . . . . . . 220

22. Whitney™s Extension Theorem Revisited . . . . . . . . . . . . . . 226

23. Fr¨licher Spaces and Free Convenient Vector Spaces

o . . . . . . . . . 238

24. Smooth Mappings on Non-Open Domains . . . . . . . . . . . . . 247

25. Real Analytic Mappings on Non-Open Domains . . . . . . . . . . 254

26. Holomorphic Mappings on Non-Open Domains . . . . . . . . . . . 261

In this chapter we will consider various extension and lifting problems. In the

¬rst section we state the problems and give several counter-examples: We consider

the subspace F of all functions which vanish of in¬nite order at 0 in the nuclear

Fr´chet space E := C ∞ (R, R), and we construct a smooth function on F that

e

has no smooth extension to E, and a smooth curve R ’ F that has not even

locally a smooth lifting along E ’ F . These results are based on E. Borel™s

theorem which tells us that RN is isomorphic to the quotient E/F and the fact

that this quotient map E ’ RN has no continuous right inverse. Also the result

(16.8) of [Seeley, 1964] is used which says that, in contrast to F , the subspace

{f ∈ C ∞ (R, R) : f (t) = 0 for t ¤ 0} of E is complemented.

In section (22) we characterize in terms of a simple boundedness condition on the

di¬erence quotients those functions f : A ’ R on an arbitrary subset A ⊆ R which

˜

admit a smooth extension f : R ’ R as well as those which admit an m-times

˜

di¬erentiable extension f having locally Lipschitzian derivatives. This results are

due to [Fr¨licher, Kriegl, 1993] and are much stronger than Whitney™s extension

o

theorem, which holds for closed subsets only and needs the whole jet and conditions

on it. There is, however, up to now no analog in higher dimensions, since di¬erence

quotients are de¬ned only on lattices.

Section (23) gives an introduction to smooth spaces in the sense of Fr¨licher. These

o

are sets together with curves and functions which compose into C ∞ (R, R) and

determine each other by this. They are very useful for chasing smoothness of

mappings which sometimes leave the realm of manifolds.

In section (23) it is shown that there exist free convenient vector spaces over

Fr¨licher spaces, this means that to every such space X one can associate a con-

o

venient vector space »X together with a smooth map ιX : X ’ »X such that

for any convenient vector space E the map (ιX )— : L(»X, E) ’ C ∞ (X, E) is a

bornological isomorphism. The space »X can be obtained as the c∞ -closure of the

linear subspace spanned by the image of the canonical map X ’ C ∞ (X, R) . In

220 Chapter V. Extensions and liftings of mappings 21.1

the case where X is a ¬nite dimensional smooth manifold we prove that the linear

subspace generated by { —¦ evx : x ∈ X, ∈ E } is c∞ -dense in C ∞ (X, E) . From

this we conclude that the free convenient vector space over a manifold X is the

space of distributions with compact support on X.

In the last 3 sections we discuss germs of smooth, holomorphic, and real analytic

functions on convex sets with non-empty interior, following [Kriegl, 1997]. Let us

recall the ¬nite dimensional situation for smooth maps, so let ¬rst E = F = R

and X be a non-trivial closed interval. Then a map f : X ’ R is usually called

smooth, if it is in¬nite often di¬erentiable on the interior of X and the one-sided

derivatives of all orders exist. The later condition is equivalent to the condition,

that all derivatives extend continuously from the interior of X to X. Furthermore,

by Whitney™s extension theorem these maps turn out to be the restrictions to X of

smooth functions on (some open neighborhood of X in) R. In case where X ⊆ R

is more general, these conditions fall apart. Now what happens if one changes

to X ⊆ Rn . For closed convex sets with non-empty interior the corresponding

conditions to the one dimensional situation still agree. In case of holomorphic and

real analytic maps the germ on such a subset is already de¬ned by the values on

the subset. Hence, we are actually speaking about germs in this situation. In

in¬nite dimensions we will consider maps on just those convex subsets. So we do

not claim greatest achievable generality, but rather restrict to a situation which is

quite manageable. We will show that even in in¬nite dimensions the conditions

above often coincide, and that real analytic and holomorphic maps on such sets

are often germs of that class. Furthermore, we have exponential laws for all three

classes, more precisely, the maps on a product correspond uniquely to maps from

the ¬rst factor into the corresponding function space on the second.

21. Extension and Lifting Properties

21.1. Remark. The extension property. The general extension problem is to

˜

¬nd an arrow f making a diagram of the following form commutative:

w

RR i

ii

X Y

R i f˜

Tk

Ri

f

Z

We will consider problems of this type for smooth, for real-analytic and for holo-

morphic mappings between appropriate spaces, e.g., Fr¨licher spaces as treated in

o

section (23).

Let us ¬rst sketch a step by step approach to the general problem for the smooth

mappings at hand.

˜

If for a given mapping i : X ’ Y an extension f : Y ’ Z exists for all f ∈

C ∞ (X, Z), then this says that the restriction operator i— : C ∞ (Y, Z) ’ C ∞ (X, Z)

is surjective.

21.1

21.2 21. Extension and lifting properties 221

Note that a mapping i : X ’ Y has the extension property for all f : X ’ Z

with values in an arbitrary space Z if and only if i is a section, i.e. there exists a

˜

mapping IdX : Y ’ X with IdX —¦ i = IdX . (Then f := f —¦ IdX is the extension of

a general mapping f ).

A particularly interesting case is Z = R. A mapping i : X ’ Y with the extension

property for all f : X ’ R is said to have the scalar valued extension property.

Such a mapping is necessarily initial: In fact let g : Z ’ X be a mapping with

˜

i —¦ g : X ’ Y being smooth. Then f —¦ g = f —¦ i —¦ g is smooth for all f ∈ C ∞ (X, R)

and hence g is smooth, since the functions f ∈ C ∞ (X, R) generate the smooth

structure on the Fr¨licher space X.

o

More generally, we consider the same question for any convenient vector space

Z = E. Let us call this the vector valued extension property. Assume that we have

already shown the scalar valued extension property for i : X ’ Y , and thus we have

an operator S : C ∞ (X, R) ’ C ∞ (Y, R) between convenient vector spaces, which is

a right inverse to i— : C ∞ (Y, R) ’ C ∞ (X, R). It is reasonable to hope that S will be

linear (which can be easily checked). So the next thing would be to check, whether

it is bounded. By the uniform boundedness theorem it is enough to show that

˜

evy —¦S : C ∞ (X, R) ’ C ∞ (Y, R) ’ Y given by f ’ f (y) is smooth, and usually

this is again easily checked. By dualization we get a bounded linear operator S — :

C ∞ (Y, R) ’ C ∞ (X, R) which is a left inverse to i—— : C ∞ (X, R) ’ C ∞ (Y, R) .

Now in order to solve the vector valued extension problem we use the free convenient

vector space »X over a smooth space X given in (23.6). Thus any f ∈ C ∞ (X, E)

˜ ˜

corresponds to a bounded linear f : »X ’ E. It is enough to extend f to a bounded

˜

linear operator »Y ’ E given by f —¦ S — . So we only need that S — |»Y has values

in »X, or equivalently, that S — —¦ δY : Y ’ C ∞ (Y, R) ’ C ∞ (X, R) , given by

˜

y ’ (f ’ f (y)), has values in »X. In the important cases (e.g. ¬nite dimensional

manifolds X), where »X = C ∞ (X, R) , this is automatically satis¬ed. Otherwise it

is by the uniform boundedness principle enough to ¬nd for given y ∈ Y a bounding

sequence (xk )k in X (i.e. every f ∈ C ∞ (X, R) is bounded on {xk : k ∈ N}) and

˜

an absolutely summable sequence (ak )k ∈ 1 such that f (y) = k ak f (xk ) for all

f ∈ C ∞ (X, R). Again we can hope that this can be achieved in many cases.

21.2. Proposition. Let i : X ’ Y be a smooth mapping, which satis¬es the vector

valued extension property. Then there exists a bounded linear extension operator

C ∞ (X, E) ’ C ∞ (Y, E).

Proof. Since i is smooth, the mapping i— : C ∞ (Y, E) ’ C ∞ (X, E) is a bounded

linear operator between convenient vector spaces. Its kernel is ker(i— ) = {f ∈

C ∞ (Y, E) : f —¦ i = 0}. And we have to show that the sequence

w ker(i ) y wC wC w0

i—

— ∞ ∞

0 (Y, E) (X, E)

˜

splits via a bounded linear operator σ : C ∞ (X, E) f ’ f ∈ C ∞ (Y, E), i.e. a

bounded linear extension operator.

By the exponential law (3.13) a mapping σ ∈ L(C ∞ (X, E), C ∞ (Y, E)) would cor-

respond to σ ∈ C ∞ (Y, L(C ∞ (X, E), E)) and σ —¦ i— = Id translates to σ —¦ i = Id =

˜ ˜

21.2

222 Chapter V. Extensions and liftings of mappings 21.4

δ : X ’ L(C ∞ (X, E), E), given by x ’ (f ’ f (x)), i.e. σ must be a solution of

˜

the following vector valued extension problem:

wY

RR i

ii

X

R T

R i

δ

kσ

i˜

L(C ∞ (X, E), E)

By the vector valued extension property such a σ exists.

˜

21.3. The lifting property. Dual to the extension problem, we have the lifting

˜

problem, i.e. we want to ¬nd an arrow f making a diagram of the following form

uU

commutative:

R p

R j

i

X Y

RR ii

f

i

˜

f

Z

Note that in this situation it is too restrictive to search for a bounded linear or even

just a smooth lifting operator T : C ∞ (Z, X) ’ C ∞ (Z, Y ). If such an operator

exists for some Z = …, then p : Y ’ X has a smooth right inverse namely the

dashed arrow in the following diagram:

T w Xu

¡¡¡

RRR

Id

X

¡¡¡ R

¢ RRR

¡ p

Yu

evz evz

const—

ee

C ∞ (Z, Y )

j

h

h eg

e

hT

u h

p—

wC

Id

∞ ∞

C (Z, X) (Z, X)

Again the ¬rst important case is, when Z = R. If X and Y are even convenient

vector spaces, then we know that the image of a convergent sequence tn ’ t under

a smooth curve c : R ’ Y is Mackey convergent. And since one can ¬nd by

the general curve lemma a smooth curve passing through su¬ciently fast falling

subsequences of a Mackey convergent sequence, the ¬rst step could be to check

whether such sequences can be lifted. If bounded sets (or at least sequences) can

be lifted, then the same is true for Mackey convergent sequences. However, this is

not always true as we will show in (21.9).

21.4. Remarks. The scalar valued extension property for bounded linear map-

pings on a c∞ -dense linear subspace is true if and only if the embedding represents

21.4

21.5 21. Extension and lifting properties 223

the c∞ -completion by (4.30). In this case it even has the vector valued extension

property by (4.29).

That in general bounded linear functionals on a (dense or c∞ -closed subspace) may

not be extended to bounded (equivalently, smooth) linear functionals on the whole

space was shown in (4.36.6).

The scalar valued extension problem is true for the c∞ -closed subspace of an un-

countable product formed by all points with countable support, see (4.27) (and

(4.12)). As a consequence this subspace is not smoothly real compact, see (17.5).

Let E be not smoothly regular and U be a corresponding 0-neighborhood. Then

the closed subset X := {0}∪(E \U ) ⊆ Y := E does not have the extension property

for the smooth mapping f = χ{0} : X ’ R.

Let E be not smoothly normal and A0 , A1 be the corresponding closed subsets.

Then the closed subset X := A1 ∪ A2 ⊆ Y := E does not have the extension

property for the smooth mapping f = χA1 : X ’ R.

If q : E ’ F is a quotient map of convenient vector spaces one might expect that

for every smooth curve c : R ’ F there exists (at least locally) a smooth lifting,

i.e. a smooth curve c : R ’ E with q —¦ c = c. And if ι : F ’ E is an embedding of

a convenient subspace one might expect that for every smooth function f : F ’ R

there exists a smooth extension to E. In this section we give examples showing

that both properties fail. As convenient vector spaces we choose spaces of smooth

real functions and their duals. We start with some lemmas.

21.5. Lemma. Let E := C ∞ (R, R), let q : E ’ RN be the in¬nite jet mapping at

ι

0, given by q(f ) := (f (n) (0))n∈N , and let F ’ E be the kernel of q, consisting of

’

all smooth functions which are ¬‚at of in¬nite order at 0.

Then the following sequence is exact:

q

ι

0 ’ F ’ E ’ RN ’ 0.

’’

Moreover, ι— : E ’ F is a quotient mapping between the strong duals. Every

bounded linear mapping s : RN ’ E the composite q —¦s factors over prN : RN ’ RN

for some N ∈ N, and so the sequence does not split.

Proof. The mapping q : E ’ RN is a quotient mapping by the open mapping

theorem (52.11) & (52.12), since both spaces are Fr´chet and q is surjective by

e

Borel™s theorem (15.4). The inclusion ι is an embedding of Fr´chet spaces, so the

e

—

adjoint ι is a quotient mapping for the strong duals (52.28). Note that these duals

are bornological by (52.29).

Let s : RN ’ E be an arbitrary bounded linear mapping. Since RN is bornological

s has to be continuous. The set U := {g ∈ E : |g(t)| ¤ 1 for |t| ¤ 1} is a 0-

neighborhood in the locally convex topology of E. So there has to exist an N ∈ N

1

such that s(V ) ⊆ U with V := {x ∈ RN : |xn | < N for all n ¤ N }. We show that

q —¦ s factors over RN . So let x ∈ RN with xn = 0 for all n ¤ N . Then k · x ∈ V

21.5

224 Chapter V. Extensions and liftings of mappings 21.6

1

for all k ∈ N, hence k · s(x) ∈ U , i.e. |s(x)(t)| ¤ k for all |t| ¤ 1 and k ∈ N. Hence

s(x)(t) = 0 for |t| ¤ 1 and therefore q(s(x)) = 0.

Suppose now that there exists a bounded linear mapping ρ : E ’ F with ρ—¦ι = IdF .

De¬ne s(q(x)) := x ’ ιρx. This de¬nition makes sense, since q is surjective and

q(x) = q(x ) implies x ’ x ∈ F and thus x ’ x = ρ(x ’ x ). Moreover s is a

bounded linear mapping, since q is a quotient map, as surjective continuous map

between Fr´chet spaces; and (q —¦ s)(q(x)) = q(x) ’ q(ι(ρ(x))) = q(x) ’ 0.

e

21.6. Proposition. [Fr¨licher, Kriegl, 1988], 7.1.5 Let ι— : E ’ F the quotient

o

map of (21.5). The curve c : R ’ F de¬ned by c(t) := evt for t ≥ 0 and c(t) = 0

for t < 0 is smooth but has no smooth lifting locally around 0. In contrast, bounded

sets and Mackey convergent sequences are liftable.

Proof. By the uniform boundedness principle (5.18) c is smooth provided evf —¦c :

R ’ R is smooth for all f ∈ F . Since (evf —¦c)(t) = f (t) for t ≥ 0 an (evf —¦c)(t) = 0

for t ¤ 0 this obviously holds.

Assume ¬rst that there exists a global smooth lifting of c, i.e. a smooth curve

e : R ’ E with ι— —¦ e = c. By exchanging the variables, c corresponds to a

bounded linear mapping c : F ’ E and e corresponds to a bounded linear mapping

˜

e : E ’ E with e —¦ ι = c. The curve c was chosen in such a way that c(f )(t) = f (t)

˜ ˜ ˜ ˜

for t ≥ 0 and c(f )(t) = 0 for t ¤ 0.

˜

We show now that such an extension e of c cannot exist. In (16.8) we have shown the

˜˜

existence of a retraction s to the embedding of the subspace F+ := {f ∈ F : f (t) = 0

for t ¤ 0} of E. For f ∈ F one has s(˜(f )) = s(˜(f )) = c(f ) since c(E) ⊆ F+ .

e c ˜ ˜

Now let Ψ : E ’ E, Ψ(f )(t) := f (’t) be the re¬‚ection at 0. Then Ψ(F ) ⊆ F and

f = c(f ) + Ψ(˜(Ψ(f ))) for f ∈ F . We claim that ρ := s —¦ e + Ψ —¦ s —¦ e —¦ Ψ : E ’ F

˜ c ˜ ˜

is a retraction to the inclusion, and this is a contradiction with (21.5). In fact

ρ(f ) = (s —¦ e)(f ) + (Ψ —¦ s —¦ e —¦ Ψ)(f ) = c(f ) + Ψ(˜(Ψ(f ))) = f

˜ ˜ ˜ c

for all f ∈ F . So we have proved that c has no global smooth lifting.

Assume now that c|I has a smooth lifting e0 : U ’ E for some open neighborhood

I of 0. Trivially c|R {0} has a smooth lifting e1 de¬ned by the same formula as c.

Take now a smooth partition {f0 , f1 } of the unity subordinated to the open covering

(’µ, µ), R {0} of R, i.e. f0 + f1 = 1 with supp(f0 ) ⊆ (’µ, µ) and 0 ∈ supp(f1 ).

/

Then f0 e0 + f1 e1 gives a global smooth lifting of c, in contradiction with the case

treated above.

Let now B ⊆ F be bounded. Without loss of generality we may assume that

B = U o for some 0-neighborhood U in F . Since F is a subspace of the Fr´chet

e

space E, the set U can be written as U = F © V for some 0-neighborhood V in

E. Then the bounded set V o ⊆ E is mapped onto B = U o by the Hahn-Banach

theorem.

21.6

21.9 21. Extension and lifting properties 225

21.7. Proposition. [Fr¨licher, Kriegl, 1988], 7.1.7 Let ι : F ’ E be as in (21.5).

o

The function • : F ’ R de¬ned by •(f ) := f (f (1)) for f (1) ≥ 0 and •(f ) := 0 for

f (1) < 0 is smooth but has no smooth extension to E and not even to a neighborhood

of F in E.

Proof. We ¬rst show that • is smooth. Using the bounded linear c : F ’ E

˜

associated to the smooth curve c : R ’ F of (21.6) we can write • as the composite

ev —¦(˜, ev1 ) of smooth maps.

c

Assume now that a smooth global extension ψ : E ’ R of • exists. Using a ¬xed

smooth function h : R ’ [0, 1] with h(t) = 0 for t ¤ 0 and h(t) = 1 for t ≥ 1, we

then de¬ne a map σ : E ’ E as follows:

(σg)(t) := ψ g + t ’ g(1) h ’ t ’ g(1) h(t).

Obviously σg ∈ E for any g ∈ E, and using cartesian closedness (3.12) one easily

veri¬es that σ is a smooth map. For f ∈ F one has, using that f + t’f (1) h (1) =

t, the equations

(σf )(t) = f + t ’ f (1) h (t) ’ t ’ f (1) h(t) = f (t)

for t ≥ 0 and (σf )(t) = 0 ’ (t ’ f (1))h(t) = 0 for t ¤ 0. This means σf = cf for

˜

f ∈ F . So one has c = σ—¦ι with σ smooth. Di¬erentiation gives c = c (0) = σ (0)—¦ι,

˜ ˜˜

and σ (0) is a bounded linear mapping E ’ E. But in the proof of (21.6) it was

shown that such an extension of c does not exist.

˜

Let us now assume that a local extension to some neighborhood of F in E exists.

This extension could then be multiplied with a smooth function E ’ R being 1

on F and having support inside the neighborhood (E as nuclear Fr´chet space has

e

smooth partitions of unity see (16.10)) to obtain a global extension.

21.8. Remark. As a corollary it is shown in [Fr¨licher, Kriegl, 1988, 7.1.6] that

o

the category of smooth spaces is not locally cartesian closed, since pullbacks do not

commute with coequalizers.

Furthermore, this examples shows that the structure curves of a quotient of a

Fr¨licher space need not be liftable as structure curves and the structure functions

o

on a subspace of a Fr¨licher space need not be extendable as structure functions.

o

In fact, since Mackey-convergent sequences are liftable in the example, one can show

that every f : F ’ R is smooth, provided f —¦ ι— is smooth, see [Fr¨licher,Kriegl,

o

1988, 7.1.8].

21.9. Example. In [Jarchow, 1981, 11.6.4] a Fr´chet Montel space is given, which

e

has 1 as quotient. The standard basis in 1 cannot have a bounded lift, since in

a Montel space every bounded set is by de¬nition relatively compact, hence the

standard basis would be relatively compact.

21.9

226 Chapter V. Extensions and liftings of mappings 22.1

21.10. Result. [Jarchow, 1981, remark after 9.4.5]. Let q : E ’ F be a quotient

map between Fr´chet spaces. Then (Mackey) convergent sequences lift along q.

e

This is not true for general spaces. In [Fr¨licher, Kriegl, 1988, 7.2.10] it is shown

o

that the quotient map dens A=0 RA ’ E := {x ∈ RN : dens(carr(x)) = 0} does not

lift Mackey-converging sequences. Note, however, that this space is not convenient.

We do not know whether smooth curves can be lifted over quotient mappings, even

in the case of Banach spaces.

ι

21.11. Example. There exists a short exact sequence 2 ’ E ’ 2 , which does

’

not split, see (13.18.6). The square of the norm on the subspace 2 does not extend

to a smooth function on E.

Proof. Assume indirectly that a smooth extension of the square of the norm exists.

Let 2b be the second derivative of this extension at 0, then b(x, y) = x, y for all

x, y ∈ 2 , and hence the following diagram commutes

y wE

ι

2

∼

u u

∨

= b

uu ι—

2—

E—

()

giving a retraction to ι.

22. Whitney™s Extension Theorem Revisited

Whitney™s extension theorem [Whitney, 1934] concerns extensions of jets and not

of functions. In particular it says, that a real-valued function f from a closed

subset A ⊆ R has a smooth extension if and only if there exists a (not uniquely

determined) sequence fn : A ’ R, such that the formal Taylor series satis¬es the

appropriate remainder conditions, see (22.1). Following [Fr¨licher, Kriegl, 1993],

o

we will characterize in terms of a simple boundedness condition on the di¬erence

quotients those functions f : A ’ R on an arbitrary subset A ⊆ R which admit a

˜

smooth extension f : R ’ R as well as those which admit an m-times di¬erentiable

˜

extension f having locally Lipschitzian derivatives.

We shall use Whitney™s extension theorem in the formulation given in [Stein, 1970].

In order to formulate it we recall some de¬nitions.

22.1. Notation on jets. An m-jet on A is a family F = (F k )k¤m of continuous

functions on A. With J m (A, R) one denotes the vector space of all m-jets on A.

The canonical map j m : C ∞ (R, R) ’ J m (A, R) is given by f ’ (f (k) |A )k¤m .

For k ¤ m one has the ˜di¬erentiation operator™ Dk : J m (A, R) ’ J m’k (A, R)

given by Dk : (F i )i¤m ’ (F i+k )i¤m’k .

22.1

22.3 22. Whitney™s extension theorem revisited 227

For a ∈ A the Taylor-expansion operator Ta : J m (A, R) ’ C ∞ (R, R) of order m

m

k

at a is de¬ned by Ta ((F i )i¤m ) : x ’ k¤m (x’a) F k (a).

m

k!

Finally the remainder operator Ra : J m (A, R) ’ J m (A, R) at a of order m is given

m

by F ’ F ’ j m (Ta F ).

m

In [Stein, 1970, p.176] the space Lip(m + 1, A) denotes all m-jets on A for which

there exists a constant M > 0 such that

|F j (a)| ¤ M and (Ra F )j (b) ¤ M |a ’ b|m+1’j

m

for all a, b ∈ A and all j ¤ m.

The smallest constant M de¬nes a norm on Lip(m + 1, A).

22.2. Whitney™s Extension. The construction of Whitney for ¬nite order m

goes as follows, see [Malgrange, 1966], [Tougeron, 1972] or [Stein, 1970]:

First one picks a special partition of unity ¦ for Rn \ A satisfying in particular

diam(supp •) ¤ 2 d(supp •, A) for • ∈ ¦. For every • ∈ ¦ one chooses a nearest

˜

point a• ∈ A, i.e. a point a• with d(supp •, A) = d(supp •, a• ). The extension F

of the jet F is then de¬ned by

F 0 (x) for x ∈ A

˜

F (x) := m

•(x)Ta• F (x) otherwise,

•∈¦

where the set ¦ consists of all • ∈ ¦ such that d(supp •, A) ¤ 1.

The version of [Stein, 1970, theorem 4, p. 177] of Whitney™s extension theorem is:

Whitney™s Extension Theorem. Let m be an integer and A a compact subset

˜

of R. Then the assignment F ’ F de¬nes a bounded linear mapping E m : Lip(m +

1, A) ’ Lip(m + 1, R) such that E m (F )|A = F 0 .

In order that E m makes sense, one has to identify Lip(m + 1, R) with a space of

functions (and not jets), namely those functions on R which are m-times di¬eren-

tiable on R and the m-th derivative is Lipschitzian. In this way Lip(m + 1, R) is

identi¬ed with the space Lipm (R, R) in (1.2) (see also (12.10)).

Remark. The original condition of [Whitney, 1934] which guarantees a C m -exten-

sion is:

(Ra F )k (b) = o(|a ’ b|m’k ) for a, b ∈ A with |a ’ b| ’ 0 and k ¤ m.

m

In the following A will be an arbitrary subset of R.

22.3. Di¬erence Quotients. The de¬nition of di¬erence quotients δ k f given in

(12.4) works also for functions f : A ’ R de¬ned on arbitrary subsets A ⊆ R. The

natural domain of de¬nition of δ k f is the subset A<k> of Ak+1 of pairwise distinct

points, i.e.

A<k> := (t0 , . . . , tk ) ∈ Ak+1 : ti = tj for all i = j .

The following product rule can be found for example in [Verde-Star, 1988] or

[Fr¨licher, Kriegl, 1993, 3.3].

o

22.3

228 Chapter V. Extensions and liftings of mappings 22.6

22.4. The Leibniz product rule for di¬erence quotients.

k

ki

δ k (f · g) (t0 , . . . , tk ) = δ f (t0 , . . . , ti ) · δ k’i g(ti , . . . , tk )

i

i=0

Proof. This is easily proved by induction on k.

We will make strong use of interpolation polynomials as they have been already

used in the proof of lemma (12.4). The following descriptions are valid for them:

22.5. Lemma. Interpolation polynomial. Let f : A ’ E be a function with

values in a vector space E and let (t0 , . . . , tm ) ∈ A<m> . Then there exists a unique

m

polynomial P(t0 ,...,tm ) f of degree at most m which takes the values f (tj ) on tj for

all j = 0, . . . , m. It can be written in the following ways:

m k’1

1k

m

P(t0 ,...,tm ) f : t ’ (t ’ tj )

δ f (t0 , . . . , tk ) (N ewton)

k! j=0

k=0

m

t ’ tj

t’ f (tk ) (Lagrange).

tk ’ tj

k=0 j=k

See, for example, [Fr¨licher, Kriegl, 1988, 1.3.7] for a proof of the ¬rst description.

o

The second one is obvious.

22.6. Lemma. For pairwise distinct points a, b, t1 , . . . , tm and k ¤ m one has:

(k)

m m

’

P(a,t1 ,...,tm ) f P(b,t1 ,...,tm ) f (t) =

= (a ’ b) (m+1)! δ m+1 f (a, b, t1 , . . . , tm )·

1

· k! (t ’ t1 ) · . . . · (t ’ ti1 ) · . . . · (t ’ tik ) · . . . · (t ’ tm ).

i1 <···<ik

Proof. For the interpolation polynomial we have

m m

P(a,t1 ,...,tm ) f (t) = P(t1 ,...,tm ,a) f (t) =

= f (t1 ) + · · · + (t ’ t1 ) · . . . · (t ’ tm’1 ) (m’1)! δ m’1 f (t1 , . . . , tm )

1

+ (t ’ t1 ) · . . . · (t ’ tm ) m! δ m f (t1 , . . . , tm , a).

1

Thus we obtain

m m

P(a,t1 ,...,tm ) f (t) ’ P(b,t1 ,...,tm ) f (t) =

= 0 + · · · + 0 + (t ’ t1 ) · . . . · (t ’ tm ) m! δ m f (t1 , . . . , tm , a)

1

’ (t ’ t1 ) · . . . · (t ’ tm ) m! δ m f (t1 , . . . , tm , b)

1

= (t ’ t1 ) · . . . · (t ’ tm ) m! m+1 δ m+1 f (t1 , . . . , tm , a, b)

1 a’b

= (a ’ b) · (t ’ t1 ) · . . . · (t ’ tm ) (m+1)! δ m+1 f (a, b, t1 , . . . , tm ).

1

Di¬erentiation using the product rule (22.4) gives the result.

22.6

22.8 22. Whitney™s extension theorem revisited 229

22.7. Proposition. Let f : A ’ R be a function, whose di¬erence quotient of

order m + 1 is bounded on A<m+1> . Then the approximation polynomial Pa f m

m

converges to some polynomial denoted by Px f of degree at most m if the point

a ∈ A<m> converges to x ∈ Am+1 .

Proof. We claim that Pa f is a Cauchy net for A<m>

m m

a ’ x. Since Pa f is

symmetric in the entries of a we may assume without loss of generality that the

entries xj of x satisfy x0 ¤ x1 ¤ · · · ¤ xm . For a point a ∈ A<m> which is close

to x and any two coordinates i and j with xi < xj we have ai < aj . Let a and b

be two points close to x. Let J be a set of indices on which x is constant. If the

set {aj : j ∈ J} di¬ers from the set {bj : j ∈ J}, then we may order them as in the

proof of lemma (12.4) in such a way that ai = bj for i ¤ j in J. If the two sets

are equal we order both strictly increasing and thus have ai < aj = bj for i < j

in J. Since x is constant on J the distance |ai ’ bj | ¤ |ai ’ xi | + |xj ’ bj | goes to

zero as a and b approach x. Altogether we obtained that ai = bj for all i < j and

applying now (22.6) for k = 0 inductively one obtains:

m m

P(a0 ,...,am ) f (t) ’ P(b0 ,...,bm ) f (t) =

m

m m

P(a0 ,...,aj’1 ,bj ,...,bm ) f (t) ’ P(a0 ,...,aj ,bj+1 ,...,bm ) f (t)

=

j=0

m

(aj ’ bj )(t ’ a0 ) . . . (t ’ aj’1 )(t ’ bj+1 ) . . . (t ’ bm )·

=

j=0

m+1

1

· (m+1)! δ f (a0 , . . . , aj , bj , . . . , bm ).

Where those summands with aj = bj have to be de¬ned as 0. Since aj ’ bj ’ 0

m

the claim is proved and thus also the convergence of Pa f .

22.8. De¬nition of Lipk function spaces. Let E be a convenient vector space,

let A be a subset of R and k a natural number or 0. Then we denote with

Lipk (A, E) the vector space of all maps f : A ’ E for which the di¬erence

ext

quotient of order k + 1 is bounded on bounded subsets of A<k> . As in (12.10) “

but now for arbitrary subsets A ⊆ R “ we put on this space the initial locally convex

topology induced by f ’ δ j f ∈ ∞ (A j , E) for 0 ¤ j ¤ k + 1, where the spaces

∞

carry the topology of uniform convergence on bounded subsets of A j ⊆ Rj+1 .

In case where A = R the elements of Lipk (A, R) are exactly the k-times di¬er-

ext

entiable functions on R having a locally Lipschitzian derivative of order k + 1 and

the locally convex space Lipk (A, R) coincides with the convenient vector space

ext

k

Lip (R, R) studied in section (12).

If k is in¬nite, then Lip∞ (A, E) or alternatively Cext (A, E) denotes the intersection

∞

ext

of Lipj (A, E) for all ¬nite j.

ext

∞

If A = R then the elements of Cext (R, R) are exactly the smooth functions on R

and the space Cext (R, R) coincides with the usual Fr´chet space C ∞ (R, R) of all

∞

e

smooth functions.

22.8

230 Chapter V. Extensions and liftings of mappings 22.10

22.9. Proposition. Uniform boundedness principle for Lipk . For any ext

k

¬nite or in¬nite k and any convenient vector space E the space Lipext (A, E) is also

convenient. It carries the initial structure with respect to

: Lipk (A, E) ’ Lipk (A, R) for ∈E.

— ext ext

Moreover, it satis¬es the {evx : x ∈ A}-uniform boundedness principle. If E is

Fr´chet then so is Lipk (A, E).

e ext

Proof. We consider the commutative diagram

w Lip

—

Lipm (A, E) m

ext (A, R)

ext

u u

δj δj

w

—

∞ ∞

(A<j> , E) (A<j> , R)

Obviously the bornology is initial with respect to the bottom arrows for ∈ E

and by de¬nition also with respect to the vertical arrows for j ¤ k + 1. Thus also

the top arrows form an initial source. By (2.15) the spaces in the bottom row are

c∞ -complete and are metrizable if E is metrizable. Since the boundedness of the

di¬erence quotient of order k + 1 implies that of order j ¤ k + 1, also Lipm (A, E)

ext

is convenient, and it is Fr´chet provided E is. The uniform boundedness principle

e

follows also from this diagram, using the stability property (5.25) and that the

Fr´chet and hence webbed space ∞ (A<j> , R) has it by (5.24).

e

22.10. Proposition. For a convenient vector space E the following operators are

well-de¬ned bounded linear mappings:

(1) The restriction operator Lipm (A1 , E) ’ Lipm (A2 , E) de¬ned by f ’ f |A2

ext ext

for A2 ⊆ A1 .

(2) For g ∈ Lipm (A, R) the multiplication operator

ext

Lipm (A, E) ’ Lipm (A, E)

ext ext

f ’ g · f.

(3) The gluing operator

Lipm (A1 , E) —A1 ©A2 Lipm (A2 , E) ’ Lipm (A, E)

ext ext ext

de¬ned by (f1 , f2 ) ’ f1 ∪ f2 for any covering of A by relatively open subsets

A1 ⊆ A and A2 ⊆ A.

The ¬bered product (pull back) Lipm (A1 , E)—A1 ©A2 Lipm (A2 , E) ’ Lipm (A, E)

ext ext ext

m m

is the subspace of Lipext (A1 , E) — Lipext (A2 , E) formed by all (f1 , f2 ) with f1 = f2

on A0 := A1 © A2 .

Proof. It is enough to consider the particular case where E = R. The general case

follows easily by composing with — for each ∈ E .

22.10

22.12 22. Whitney™s extension theorem revisited 231

(1) is obvious.

(2) follows from the Leibniz formula (22.4).

(3) First we show that the gluing operator has values in Lipm (A, R). Suppose the

ext

j

di¬erence quotient δ f is not bounded for some j ¤ m + 1, which we assume to

be minimal. So there exists a bounded sequence xn ∈ A<j> such that (δ j f )(xn )

converges towards in¬nity. Since A is compact we may assume that xn converges to

some point x∞ ∈ A(j+1) . If x∞ does not lie on the diagonal, there are two indices

i1 = i2 and some δ > 0, such that |xn i1 ’ xn i2 | ≥ δ. But then

δ j f (xn )(xn i1 ’ xn i2 ) = δ j’1 f (. . . , xn i2 , . . . ) ’ δ j’1 f (. . . , xn i1 , . . . ) .

1

j

Which is a contradiction to the boundedness of δ j’1 f and hence the minimality of

j. So x∞ = (x∞ , . . . , x∞ ) and since the covering {A1 , A2 } of A is open x∞ lies in

Ai for i = 1 or i = 2. Thus we have that xn ∈ Ai <j> for almost all n, and hence

δ j f (xn ) = δ j fi (xn ), which is bounded by assumption on fi .

Because of the uniform boundedness principle (22.9) it only remains to show that

(f1 , f2 ) ’ f (a) is bounded, which is obvious since f (a) = fi (a) for some i depending

on the location of a.

˜

22.11. Remark. If A is ¬nite, we de¬ne an extension f : R ’ E of the given

function f : A ’ E as the interpolation polynomial of f at all points in A. For

in¬nite compact sets A ‚ R we will use Whitney™s extension theorem (22.2), where

we will replace the Taylor polynomial in the de¬nition (22.2) of the extension by

the interpolation polynomial at appropriately chosen points near a• . For this we

associate to each point a ∈ A a sequence a = (a0 , a1 , . . . ) of points in A starting

from the given point a0 = a.

22.12. De¬nition of a ’ a. Let A be a closed in¬nite subset of R, and let a ∈ A.

Our aim is to de¬ne a sequence a = (a0 , a1 , a2 , . . . ) in a certain sense close to a.

The construction is by induction and goes as follows: a0 := a. For the induction

step we choose for every non-empty ¬nite subset F ‚ A a point aF in the closure of

A \ F having minimal distance to F . In case F does not contain an accumulation

point the set A \ F is closed and hence aF ∈ F , otherwise the distance of A \ F to F

/

is 0 and aF is an accumulation point in F . In both cases we have for the distances

d(aF , F ) = d(A \ F, F ). Now suppose (a0 , . . . , aj’1 ) is already constructed. Then

let F := {a0 , . . . , aj’1 } and de¬ne aj := aF .

Lemma. Let a = (a0 , . . . ) and b = (b0 , . . . ) be constructed as above.

If {a0 , . . . , ak } = {b0 , . . . , bk } then we have for all i, j ¤ k the estimates

|ai ’ bj | ¤ (i + j + 1) |a0 ’ b0 |

|ai ’ aj | ¤ max{i, j} |a0 ’ b0 |

|bi ’ bj | ¤ max{i, j} |a0 ’ b0 |.

Proof. First remark that if {a0 , . . . , ai } = {b0 , . . . , bi } for some i, then the same

is true for all larger i, since the construction of ai+1 depends only on the set

22.12

232 Chapter V. Extensions and liftings of mappings 22.12

{a0 , . . . , ai }. Furthermore the set {a0 , . . . , ai } contains at most one accumulation

point, since for an accumulation point aj with minimal index j we have by con-

struction that aj = aj+1 = · · · = ai .

We now show by induction on i ∈ {1, . . . , k} that

d(ai+1 , {a0 , . . . , ai }) ¤ |a0 ’ b0 |,

d(bi+1 , {b0 , . . . , bi }) ¤ |a0 ’ b0 |.

We proof this statement for ai+1 , it then follows for bi+1 by symmetry.

In case where {a0 , . . . , ai } ⊇ {b0 , . . . , bi } we have that {a0 , . . . , ai } ⊃ {b0 , . . . , bi } by

assumption. Thus some of the elements of {b0 , . . . , bi } have to be equal and hence

are accumulation points. So {a0 , . . . , ai } contains an accumulation point, and hence

ai+1 ∈ {a0 , . . . , ai } and the claimed inequality is trivially satis¬ed.

In the other case there exist some j ¤ i such that bj ∈ {a0 , . . . , ai }. We choose the

/

minimal j with this property and obtain

d(ai+1 , {a0 , . . . , ai }) := d(A \ {a0 , . . . , ai }, {a0 , . . . , ai }) ¤ d(bj , {a0 , . . . , ai }).

If j = 0, then this can be further estimated as follows

d(bj , {a0 , . . . , ai }) ¤ |a0 ’ b0 |.

Otherwise {b0 , . . . , bj’1 } ⊆ {a0 , . . . , aj } and hence we have

d(bj , {a0 , . . . , ai }) ¤ d(bj , {b0 , . . . , bj’1 }) ¤ |a0 ’ b0 |

by induction hypothesis. Thus the induction is completed.

From the proven inequalities we deduce by induction on k := max{i, j} that

|ai ’ aj | ¤ max{i, j} |a0 ’ b0 |

and similarly for |bj ’ bi |:

For k = 0 this is trivial. Now for k > 0. We may assume that i > j. Let i < i be

such that |ai ’ai | = d(ai , {a0 , . . . , ai’1 }) ¤ |a0 ’b0 |. Thus by induction hypothesis

|ai ’ aj | ¤ (k ’ 1) |a0 ’ b0 | and hence

|ai ’ aj | ¤ |ai ’ ai | + |ai ’ aj | ¤ k |a0 ’ b0 |.

By the triangle inequality we ¬nally obtain

|ai ’ bj | ¤ |ai ’ a0 | + |a0 ’ b0 | + |b0 ’ bj | ¤ (i + 1 + j) |a0 ’ b0 |.

22.12

22.13 22. Whitney™s extension theorem revisited 233

22.13. Finite Order Extension Theorem. Let E be a convenient vector space,

A a subset of R and m be a natural number or 0. A function f : A ’ E admits

an extension to R which is m-times di¬erentiable with locally Lipschitzian m-th

derivative if and only if its di¬erence quotient of order m + 1 is bounded on bounded

sets.

Proof. Without loss of generality we may assume that A is in¬nite. We consider

¬rst the case that A is compact and E = R.

So let f : A ’ R be in Lipm . We want to apply Whitney™s extension theorem

ext

(22.2). So we have to ¬nd an m-jet F on A. For this we de¬ne

F k (a) := (Pa f )(k) (a),

m

where a denotes the sequence obtained by this construction starting with the point

m

aand where Pa f denotes the interpolation polynomial of f at the ¬rst m + 1

points of a if these are all di¬erent; if not, at least one of these m + 1 points is an

m

accumulation point of A and then Pa f is taken as limit of interpolation polynomials

according to (22.7).

Let ¦ be the partition of unity mentioned in (22.2) and ¦ the subset speci¬ed there.

˜

Then we de¬ne f analogously to (22.2) where a• denotes the sequence obtained by

construction (22.12) starting with the point a• chosen in (22.2):

for x ∈ A

f (x)

˜

f (x) := m

•(x)Pa• f (x) otherwise.

•∈¦

In order to verify that F belongs to Lip(m + 1, A) we need the Taylor polynomial

m m

(x ’ a)k k (x ’ a)k m (k)

m m

Ta F (x) := F (a) = (Pa f ) (a) = Pa f (x),

k! k!

k=0 k=0

m

where the last equation holds since Pa f is a polynomial of degree at most m. This

˜ ˜

shows that our extension f coincides with the classical extension F given in (22.2)

of the m-jet F constructed from f .

The remainder term Ra F := F ’ j m (Ta F ) is given by:

m m

(Ra F )k (b) = F k (b) ’ (Ta F )(k) (b) = (Pb f )(k) (b) ’ (Pa f )(k) (b)

m m m m

We have to show that for some constant M one has (Ra F )k (b) ¤ M |a ’ b|m+1’k

m

for all a, b ∈ A and all k ¤ m.

In order to estimate this di¬erence we write it as a telescoping sum of terms which

can written by (22.6) as

(k)

m m

’

P(a0 ,...,ai’1 ,bi ,bi+1 ,...,bm ) f P(a0 ,...,ai’1 ,ai ,bi+1 ,...,bm ) f (t) =

k!

δ m+1 f (a0 , . . . , ai , bi , . . . , bm )·

=

(m + 1)!

· (bi ’ ai ) (t ’ a0 ) . . . (t ’ ai1 ) . . . (t ’ bik ) . . . (t ’ bm ).

i1 <···<ik

22.13

234 Chapter V. Extensions and liftings of mappings 22.13

Note that this formula remains valid also in case where the points are not pairwise

di¬erent. This follows immediately by passing to the limit with the help of (22.7).

We have estimates for the distance of points in {a0 , . . . , am ; b0 , . . . , bm } by (22.12)

and so we obtain the required constant M as follows

m

k!

|(Ra F )k (b)|

m

(2i + 1) |b ’ a|m+1’k

¤

(m + 1)! i=0

1 · 2 · . . . · (1 + i1 ) . . . ik . . . · m·

i1 <···<ik

· max{|δ m+1 f ({a0 , . . . , am , b0 , . . . , bm }<m+1> )|}.

˜

In case, where E is an arbitrary convenient vector space we de¬ne an extension f

for f ∈ Lipm (A, E) by the same formula as before. Since ¦ is locally ¬nite, this

ext

˜ ˜

de¬nes a function f : R ’ E. In order to show that f ∈ Lipm (R, E) we compose

˜

with an arbitrary ∈ E . Then —¦ f is just the extension of —¦ f given above, thus

belongs to Lipm (R, R).

Let now A be a closed subset of R. Then let the compact subsets An ‚ R be

de¬ned by A1 := A © [’2, 2] and An := [’n + 1, n ’ 1] ∪ (A © [’n ’ 1, n + 1]) for

n > 1. We de¬ne recursively functions fn ∈ Lipm (An , E) as follows: Let f1 be

ext

a Lip -extension of f |A1 . Let fn : An ’ R be a Lipm -extension of the function

m

which equals fn’1 on [’n + 1, n ’ 1] and which equals f on A © [’n ’ 1, n + 1].

This de¬nition makes sense, since the two sets

An \ [’n + 1, n ’ 1] = A © [’n ’ 1, n + 1] \ [’n + 1, n ’ 1] ,

An \ [’n ’ 1, ’n] ∪ [n, n + 1] = [’n + 1, n ’ 1] ∪ A © [’n, n]

form an open cover of An , and their intersection is contained in the set A © [’n, n]

on which f and fn’1 coincide. Now we apply (22.10). The sequence fn converges

˜

uniformly on bounded subsets of R to a function f : R ’ E, since fj = fn on

˜ ˜

[’n, n] for all j > n. Since each fn is Lipm , so is f . Furthermore, f is an extension

˜

of f , since f = fn on [’n, n] and hence on A © [’n + 1, n ’ 1] equal to f .

¯

Finally the case, where A ⊆ R is completely arbitrary. Let A denote the closure of

A in R. Since the ¬rst di¬erence quotient is bounded on bounded subsets of A one

concludes that f is Lipschitzian and hence uniformly continuous on bounded subsets

of A, moreover, the values f (a) form a Mackey Cauchy net for A a ’ a ∈ R. Thus ¯

˜ ˜a

¯

f has a unique continuous extension f to A, since the limit f (¯) := lima’¯ f (a) a

exists in E, because E is convenient. Boundedness of the di¬erence quotients of

˜

order j of f can be tested by composition with linear continuous functionals, so we

¯

˜ ˜

may assume E = R. Its value at (t0 , . . . , tj ) ∈ A<j> is the limit of δ j f (t0 , . . . , tj ),

˜ ˜

where A<j> (t0 , . . . , tj ) converges to (t0 , . . . , tj ), since in the explicit formula for

˜˜

δ j the factors f (ti ) converge to f (ti ). Now we may apply the result for closed A to

obtain the required extension.

22.13

22.16 22. Whitney™s extension theorem revisited 235

22.14. Extension Operator Theorem. Let E be a convenient vector space and

let m be ¬nite. Then the space Lipm (A, E) of functions having an extension in

ext

the sense of (22.13) is a convenient vector space and there exists a bounded linear

extension operator from Lipm (A, E) to Lipm (R, E).

ext

Proof. This follows from (21.2).

Explicitly the proof runs as follows: For any convenient vector space E we have

to construct a bounded linear operator

T : Lipm (A, E) ’ Lipm (R, E)

ext

satisfying T (f )|A = f for all f ∈ Lipm (A, E). Since Lipm (A, E) is a convenient

ext ext

vector space, this is by (12.12) via a ¬‚ip of variables equivalent to the existence of

a Lipm -curve

˜

T : R ’ L(Lipm (A, E), E)

ext

˜ ˜

satisfying T (a)(f ) = T (f )(a) = f (a). Thus T should be a Lipm -extension of the

map e : A ’ L(Lipm (A, E), E) de¬ned by e(a)(f ) := f (a) = eva (f ).

ext

By the vector valued ¬nite order extension theorem (22.13) it su¬ces to show that

this map e belongs to Lipm (A, L(Lipm (A, E), E)). So consider the di¬erence

ext ext

m+1

quotient δ e of e. Since, by the linear uniform boundedness principle (5.18),

boundedness in L(F, E) can be tested pointwise, we consider

δ m+1 e(a0 , . . . , am+1 )(f ) = δ m+1 (evf —¦e)(a0 , . . . , am+1 )

= δ m+1 f (a0 , . . . , am+1 ).

This expression is bounded for (a0 , . . . , am+1 ) varying in bounded sets, since f ∈

Lipm (A, E).

ext

In order to obtain a extension theorem for smooth mappings, we use a modi¬cation

of the original construction of [Whitney, 1934]. In particular we need the following

result.

22.15. Result. [Malgrange, 1966, lemma 4.2], also [Tougeron, 1972, lemme 3.3].

There exist constants ck , such that for any compact set K ‚ R and any δ > 0 there

exists a smooth function hδ on R which satis¬es

(1) hδ = 1 locally around K and hδ (x) = 0 for d(x, K) ≥ δ;

(k) c

(2) for all x ∈ R and k ≥ 0 one has: hδ (x) ¤ δk . k

22.16. Lemma. Let A be compact and Aacc be the compact set of accumulation

∞

points of A. We denote by CA (R, R) the set of smooth functions on R which vanish

on A. For ¬nite m we denote by CA (R, R) the set of C m -functions on R, which

m

vanish on A, are m-¬‚at on Aacc and are smooth on the complement of Aacc . Then

m+1

∞

CA (R, R) is dense in CA (R, R) with respect to the structure of C m (R, R).

m+1

Proof. Let µ > 0 and let g ∈ CA (R, R) be the function which we want to

approximate. By Taylor™s theorem we have for f ∈ C m+1 (R, R) the equation

k

f (i) (a) (k+1)

k+1 f (ξ)

i

f (x) ’ (x ’ a) = (x ’ a)

i! (k + 1)!

i=0

22.16

236 Chapter V. Extensions and liftings of mappings 22.17

for some ξ between a and x. If we apply this equation for j ¤ m and k = m ’ j to

g (j) for some point a ∈ Aacc we obtain

g (m+1)

(j) m+1’j

(x) ’ 0 ¤ |x ’ a|

g (ξ)

(m + 1 ’ j)!

Taking the in¬mum over all a ∈ Aacc we obtain a constant

g (m+1)

(ξ) : d(ξ, Aacc ) ¤ 1

K := sup

(m + 1 ’ j)!

g (j) (x) ¤ K · d(x, Aacc )m+1’j

satisfying

for all x with d(x, A) ¤ 1.

We choose 0 < δ < 1 depending on µ such that δ · max{ci : i ¤ m} · K · 2m ¤ µ, and

let hδ be the function given in (22.15) for K := Aacc . The function (1 ’ hδ ) · g is

smooth, since on R \ Aacc both factors are smooth and on a neighborhood of Aacc

one has hδ = 1. The function (1 ’ hδ ) · g equals g on {x : d(x, Aacc ) ≥ δ}, since hδ

vanishes on this set. So it remains to show that the derivatives of hδ · g up to order

m are bounded by µ on {x : d(x, Aacc ) ¤ δ}. By the Leibniz rule we have:

j

j (i)

(hδ · g)(j) = hδ g (j’i) .

i

i=0

The i-th summand can be estimated as follows:

ci

(i)

hδ (x)g (j’i) (x) ¤ K d(x, Aacc )m+1+i’j ¤ ci K δ m+1’j

i

δ

An estimate for the derivative now is

j

j

(hδ · g)(j) (x) ¤ ci K δ m+1’j

i

i=0

¤ 2 K δ m+1’j max{ci : 0 ¤ i ¤ j} ¤ µ.

j

22.17. Smooth Extension Theorem. Let E be a Fr´chet space (or, slightly

e

more general, a convenient vector space satisfying Mackey™s countability condition)

A function f : A ’ E admits a smooth extension to R if and only if each of its

di¬erence quotients is bounded on bounded sets.

A convenient vector space is said to satisfy Mackey™s countability condition if for

every sequence of bounded sets Bn ⊆ E there exists a sequence »n > 0 such that

n∈N »n Bn is bounded in E.

˜

Proof. We consider ¬rst the case, where E = R. For k ≥ 0 let f k be a Lipk -

˜ ˜

extension of f according to (22.13). The di¬erence f k+1 ’ f k is an element of

CA (R, R): It is by construction C k and on R\A smooth. At an accumulation point a

k

22.17

22.17 22. Whitney™s extension theorem revisited 237

˜

of A the Taylor expansion of f k of order j ¤ k is just the approximation polynomial

j ˜ ˜

P(a,...,a) f by (22.13). Thus the derivatives up to order k of f k+1 and f k are equal

in a, and hence the di¬erence is k-¬‚at at a. Locally around any isolated point of A,

˜

i.e. a point a ∈ A \ Aacc , the extension f k is just the approximation polynomial Pak

and hence smooth. In order to see this, use that for x with |x ’ a| < 1 d(a, A \ {a})

4

the point a• has as ¬rst entry a for every • with x ∈ supp •: Let b ∈ A \ {a} and

y ∈ supp • be arbitrary, then

|b ’ x| ≥ |b ’ a| ’ |a ’ x| ≥ d(a, A \ {a}) ’ |a ’ x| > (4 ’ 1) |a ’ x|

|b ’ y| ≥ |b ’ x| ’ |x ’ y| > 3 |a ’ x| ’ diam(supp •)

≥ 3 d(a, supp •) ’ 2 d(a, supp •) = d(a, supp •)

’ d(b, supp •) > d(a, supp •) ’ a• = a.

∞

By lemma (22.16) there exists an hk ∈ CA (R, R) such that

1

˜ ˜

(f k+1 ’ f k ’ hk )(j) (x) ¤ k for all j ¤ k ’ 1.

2

˜ ˜ ˜ ˜

Now we consider the function f := f 0 + k≥0 (f k+1 ’ f k ’ hk ). It is the required

˜ ˜

smooth extension of f , since the summands f k+1 ’ f k ’ hk vanish on A, and since

˜˜ ˜ ˜

for any n it can be rewritten as f = f n + (f k+1 ’ f k ’ hk ), where

hk +

k<n k≥n

∞

n

the ¬rst summand is C , the ¬rst sum is C , and the derivatives up to order n ’ 1

of the terms of the second sum are uniformly summable.

Now we prove the vector valued case, where E satis¬es Mackey™s countability con-

dition. It is enough to show the result for compact subsets A ‚ R, since the

generalization arguments given in the proof of (22.13) can be applied equally in

the smooth case. First one has to give a vector valued version of (22.16): Let a

function g ∈ Lipm (R, E) with compact support be given, which vanishes on A, is

m-¬‚at on Aacc and smooth on the complement of Aacc . Then for every µ > 0 there

exists a h ∈ C ∞ (R, R), which equals 1 on a neighborhood of Aacc and such that

δ m (h · g)(Rm+1 ) is contained in µ times the absolutely convex hull of the image of

δ m+1 g.

The proof of this assertion is along the lines of that of (22.16). One only has to

de¬ne K as the absolutely convex hull of the image of δ m+1 g and choose 0 < δ < 1

such that δ · max{ci : i ¤ m} · 2m ¤ µ.

˜

Now one proceeds as in scalar valued part: Let f k be the Lipk -extension of f accord-

˜ ˜

ing to (22.13). Then gk := f k+1 ’ f k satis¬es the assumption of the vector valued

version of (22.16). Let Kk be the absolutely convex hull of the bounded image of

δ k+1 gk . By assumption on E there exist »n > 0 such that K := k∈N »k · Kk is

∞

bounded. Hence we may choose an hk ∈ CA (R, R) such that δ k (hk · gk )(R k+1 ) ⊆

˜

»k

Kk . Now the extension f is given by

2k

˜˜ ˜

f = f0 + hk · gk = f n + (1 ’ hk ) · gk + hk · gk

k≥0 k<n k≥n

and the result follows as above using convergence in the Banach space EK .

22.17

238 Chapter V. Extensions and liftings of mappings 23.1

22.18. Remark. The restriction operator Lipm (R, E) ’ Lipm (A, E) is a quo-

ext

tient mapping. We constructed a section for it, which is bounded and linear in the

¬nite order case. It is unclear, whether it is possible to obtain a bounded linear

section also in the smooth case, even if E = R.

If the smooth extension theorem were true for any arbitrary convenient vector space

E, then it would also give the extension operator theorem for the smooth case. Thus

in order to obtain a counter-example to the latter one, the ¬rst step might be to

¬nd a counter-example to the vector valued extension theorem. In the particular

cases, where the values lie in a Fr´chet space E the vector valued smooth extension

e

theorem is however true.

22.19. Proposition. Let A be the image of a strictly monotone bounded sequence

{an : n ∈ N}. Then a map f : A ’ R has a Lipm -extension to R if and only

if the sequence δ k f (an , an+1 , . . . , an+k ) is bounded for k = m + 1 if m is ¬nite,

respectively for all k if m = ∞.

Proof. By [Fr¨licher, Kriegl, 1988, 1.3.10], the di¬erence quotient δ k f (ai0 , . . . , aik )

o

is an element of the convex hull of the di¬erence quotients δ k f (an , . . . , an+k ) for

all min{i0 , . . . , ik } ¤ n ¤ n + k ¤ max{i0 , . . . , ik }. So the result follows from the

extension theorems (22.13) and (22.17).

For explicit descriptions of the boundedness condition for Lipk -mappings de¬ned

on certain sequences and low k see [Fr¨licher, Kriegl, 1993, Sect. 6].

o

23. Fr¨licher Spaces and Free Convenient Vector Spaces

o

The central theme of this book is ˜in¬nite dimensional manifolds™. But many natural

examples suggest that this is a quite restricted notion, and it will be very helpful to

have at hand a much more general and also easily useable concept, namely smooth

spaces as they were introduced by [Fr¨licher, 1980, 1981]. We follow his line of

o

development, replacing technical arguments by simple use of cartesian closedness

of smooth calculus on convenient vector spaces, and we call them Fr¨licher spaces.

o

23.1. The category of Fr¨licher spaces.

o

A Fr¨licher space or a space with smooth structure is a triple (X, CX , FX ) consisting

o

of a set X, a subset CX of the set of all mappings R ’ X, and a subset FX of the

set of all functions X ’ R, with the following two properties:

(1) A function f : X ’ R belongs to FX if and only if f —¦ c ∈ C ∞ (R, R) for all

c ∈ CX .

(2) A curve c : R ’ X belongs to CX if and only if f —¦ c ∈ C ∞ (R, R) for all

f ∈ FX .

Note that a set X together with any subset F of the set of functions X ’ R

23.1

23.2 23. Fr¨licher spaces and free convenient vector spaces

o 239

generates a unique Fr¨licher space (X, CX , FX ), where we put in turn:

o

CX := {c : R ’ X : f —¦ c ∈ C ∞ (R, R) for all f ∈ F},

FX := {f : X ’ R : f —¦ c ∈ C ∞ (R, R) for all c ∈ CX },

so that F ⊆ FX . The set F will be called a generating set of functions for the

Fr¨licher space. A locally convex space is convenient if and only if it is a Fr¨licher

o o

space with the smooth curves and smooth functions from section (1) by (2.14).

Furthermore, c∞ -open subsets U of convenient vector spaces E are Fr¨licher spaces,

o

where CU = C ∞ (R, U ) and FU = C ∞ (U, R). Here we can use as generating set F of

functions the restrictions of any set of bounded linear functionals which generates

the bornology of E, see (2.14.4).

A mapping • : X ’ Y between two Fr¨licher spaces is called smooth if the following

o

three equivalent conditions hold

(3) For each c ∈ CX the composite • —¦ c is in CY .

(4) For each f ∈ FY the composite f —¦ • is in FX .

(5) For each c ∈ CX and for each f ∈ FY the composite f —¦ • —¦ c is in C ∞ (R, R).

Note that FY can be replaced by any generating set of functions. The set of all

smooth mappings from X to Y will be denoted by C ∞ (X, Y ). Then we have

C ∞ (R, X) = CX and C ∞ (X, R) = FX . Fr¨licher spaces and smooth mappings

o

form a category.

23.2. Theorem. The category of Fr¨licher spaces and smooth mappings has the

o

following properties:

(1) Complete, i.e., arbitrary limits exist. The underlying set is formed as in the