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

category of sets as a certain subset of the cartesian product, and the smooth

structure is generated by the smooth functions on the factors.

(2) Cocomplete, i.e., arbitrary colimits exist. The underlying set is formed as

in the category of set as a certain quotient of the disjoint union, and the

smooth functions are exactly those which induce smooth functions on the

cofactors.

(3) Cartesian closed, which means: The set C ∞ (X, Y ) carries a canonical

smooth structure described by

C ∞ (c,f ) »

C (X, Y ) ’ ’ ’ C ∞ (R, R) ’ R

∞

’’’ ’

where c ∈ C ∞ (R, X), where f is in C ∞ (Y, R) or in a generating set of

functions, and where » ∈ C ∞ (R, R) . With this structure the exponential

law holds:

C ∞ (X — Y, Z) ∼ C ∞ (X, C ∞ (Y, Z)).

=

Proof. Obviously, the limits and colimits described above have all required uni-

versal properties.

We have the following implications:

•∨ : X ’ C ∞ (Y, Z) is smooth.

23.2

240 Chapter V. Extensions and liftings of mappings 23.5

” •∨ —¦ cX : R ’ C ∞ (Y, Z) is smooth for all smooth curves cX ∈ C ∞ (R, X),

by de¬nition.

” C ∞ (cY , fZ ) —¦ •∨ —¦ cX : R ’ C ∞ (R, R) is smooth for all smooth curves

cX ∈ C ∞ (R, X), cY ∈ C ∞ (R, Y ), and smooth functions fZ ∈ C ∞ (Z, R), by

de¬nition.

” fZ —¦ • —¦ (cX — cY ) = fZ —¦ (c— —¦ •∨ —¦ cX )§ : R2 ’ R is smooth for all smooth

Y

curves cX , cY , and smooth functions fZ , by the simplest case of cartesian

closedness of smooth calculus (3.10).

’ • : X — Y ’ Z is smooth, since each curve into X — Y is of the form

(cX , cY ) = (cX — cY ) —¦ ∆, where ∆ is the diagonal mapping.

’ • —¦ (cX — cY ) : R2 ’ Z is smooth for all smooth curves cX and cY , since

the product and the composite of smooth mappings is smooth.

As in the proof of (3.13) it follows in a formal way that the exponential law is a

di¬eomorphism for the smooth structures on the mapping spaces.

23.3. Remark. By [Fr¨licher, Kriegl, 1988, 2.4.4] the convenient vector spaces are

o

exactly the linear Fr¨licher spaces for which the smooth linear functionals generate

o

the smooth structure, and which are separated and ˜complete™. On a locally convex

space which is not convenient, one has to saturate to the scalarwise smooth curves

and the associated functions in order to get a Fr¨licher space.

o

23.4 Proposition. Let X be a Fr¨licher space and E a convenient vector space.

o

Then C ∞ (X, E) is a convenient vector space with the smooth structure described

in (23.2.3).

Proof. We consider the locally convex topology on C ∞ (X, E) induced by c— :

C ∞ (X, E) ’ C ∞ (R, E) for all c ∈ C ∞ (R, X). As in (3.11) one shows that this

describes C ∞ (X, E) as inverse limit of spaces C ∞ (R, E), which are convenient by

(3.7). Thus also C ∞ (X, E) is convenient by (2.15). By (2.14.4), (3.8), (3.9) and

(3.7) its smooth curves are exactly those γ : R ’ C ∞ (X, E), for which

c—

γ f— »

R ’ C (X, E) ’ C ∞ (R, E) ’ C ∞ (R, R) ’ R

∞

’ ’ ’ ’

is smooth for all c ∈ C ∞ (R, X), for all f in the generating set E of functions, and

all » ∈ C ∞ (R, R). This is the smooth structure described in (23.2.3).

23.5. Related concepts: Holomorphic Fr¨licher spaces. They can be de-

o

¬ned in a way similar as smooth Fr¨licher spaces in (23.1), with the following

o

changes: As curves one has to take mappings from the complex unit disk. Then

the results analogous to (23.2) hold, where for the proof one has to use the holo-

morphic exponential law (7.22) instead of the smooth one (3.10), see [Siegl, 1995]

and [Siegl, 1997].

The concept of holomorphic Fr¨licher spaces is not without problems: Namely

o

¬nite dimensional complex manifolds are holomorphic Fr¨licher spaces if they are

o

Stein, and compact complex manifolds are never holomorphic Fr¨licher spaces. But

o

arbitrary subsets A of complex convenient vector spaces E are holomorphic Fr¨licher

o

23.5

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

o 241

spaces with the initial structure, again generated by the restrictions of bounded

complex linear functionals. Note that analytic subsets of complex convenient spaces,

i.e., locally zero sets of holomorphic mappings, are holomorphic spaces. But usually,

as analytic sets, holomorphic functions on them are restrictions of holomorphic

functions de¬ned on neighborhoods, whereas as holomorphic spaces they admit

more holomorphic functions, as the following example shows:

Example. Neil™s parabola P := {z1 ’ z2 = 0} ‚ C2 has the holomorphic curves

2 3

a : D ’ P ‚ C2 of the form a = (b3 , b2 ) for holomorphic b : D ’ C: If a(z) =

(z k a1 (z), z l a2 (z)) with a(0) = 0 and ai (0) = 0, then k = 3n and l = 2n for some

n > 0 and (a1 , a2 ) is still a holomorphic curve in P \ 0, so (a1 , a2 ) = (c3 , c2 ) by the

implicit function theorem, then b(z) = z n c(z) is the solution. Thus, z ’ (z 3 , z 2 )

is biholomorphic C ’ P . So z is a holomorphic function on P which cannot be

extended to a holomorphic function on a neighborhood of 0 in C2 , since this would

have in¬nite di¬erential at 0.

23.6. Theorem. Free Convenient Vector Space. [Fr¨licher, Kriegl, 1988],

o

5.1.1 For every Fr¨licher space X there exists a free convenient vector space »X,

o

i.e. a convenient vector space »X together with a smooth mapping δX : X ’ »X,

such that for every smooth mapping f : X ’ G with values in a a convenient

˜

vector space G there exists a unique linear bounded mapping f : »X ’ G with

f —¦ δX = f . Moreover δ — : L(»X, G) ∼ C ∞ (X, G) is an isomorphisms of convenient

˜ =

vector spaces and δ is an initial morphism.

Proof. In order to obtain a candidate for »X, we put G := R and thus should have

(»X) = L(»X, R) ∼ C ∞ (X, R) and hence »X should be describable as subspace of

=

(»X) ∼ C (X, R) . In fact every f ∈ C ∞ (E, R) acts as bounded linear functional

∞

=

evf : C ∞ (X, R) ’ R and if we de¬ne δX : X ’ C ∞ (X, R) to be δX : x ’ evx

then evf —¦δX = f and δX is smooth, since by the uniform boundedness principle

(5.18) it is su¬cient to check that evf —¦δX = f : X ’ C ∞ (X, R) ’ R is smooth for

˜

all f ∈ C ∞ (X, R). In order to obtain uniqueness of the extension f := evf , we have

to restrict it to the c∞ -closure of the linear span of δX (X). So let »X be this closure

and let f : X ’ G be an arbitrary smooth mapping with values in some convenient

vector space. Since δ belongs to C ∞ we have that δ — : L(»X, G) ’ C ∞ (X, G) is

well de¬ned and it is injective since the linear subspace generated by the image of

δ is c∞ -dense in »X by construction. To show surjectivity consider the following

diagram:

wy w

δ C ∞ (X)

i (2) ¡¡

X »X

i u £¡ (1)

¡

i

fi RR

i RR ev

f »—¦f

i(

(3)

RR

&

G

i& pr

Tu

R

»

u ki& δ »

4

wR

G

23.6

242 Chapter V. Extensions and liftings of mappings 23.8

Note that (2) has values in δ(G), since this is true on the evx , which generate by

de¬nition a c∞ -dense subspace of »X.