<< . .

. 34
( : 97)

. . >>

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 .


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

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

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

for all n ∈ N, a contradiction.

20.2. Proposition. [Kriegl, Nel, 1990, 2.3] for A-paracompact, [Bistr¨m, Bjon,
Lindstr¨m, 1993, Prop.2]. If X is A-realcompact then every A-bounding subset of
X is relatively compact in XA .

Proof. Consider the diagram
y w Hom(A) y w


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

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

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

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 .

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∞ .

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).


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
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
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
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-
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:
RR i

R i f˜

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
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.

21.2 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.
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
— ∞ ∞
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 =
˜ ˜

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:

RR i

R 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

R p
R j
RR ii
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
¡¡¡ R
¡ p
evz evz

C ∞ (Z, Y )
h eg
u h

∞ ∞
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

21.5 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
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.

<< . .

. 34
( : 97)

. . >>