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