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