∞ ∞

(“)) 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.