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.

Remains to show that this bijection is a bornological isomorphism. In order to

show that the linear mapping C ∞ (X, G) ’ L(»X, G) is bounded we can reformu-

late this equivalently using (3.12), the universal property of »X and the uniform

boundedness principle (5.18) in turn:

C ∞ (X, G) ’ L(»X, G) is L

⇐’ »X ’ L(C ∞ (X, G), G) is L

⇐’ X ’ L(C ∞ (X, G), G) is C ∞

evf

⇐’ X ’ L(C ∞ (X, G), G) ’ G is C ∞

and since the composition is just f we are done.

Conversely we have to show that L(»X, G) ’ C ∞ (X, G) belongs to L. Composed

with evx : C ∞ (X, G) ’ G this yields the bounded linear map evδ(x) : L(»X, G) ’

G. Thus this follows from the uniform boundedness principle (5.26).

That δX is initial follows immediately from the fact that the structure of X is initial

with respect to family {f = evo —¦δX : f ∈ C ∞ (X, R)}.

Remark. The corresponding result with the analogous proof is true for holomor-

phic Fr¨licher spaces, Lipk -spaces, and ∞ -spaces. For the ¬rst see [Siegl, 1997] for

o

the last two see [Fr¨licher, Kriegl, 1988].

o

23.7. Corollary. Let X be a Fr¨licher space such that the functions in C ∞ (X, R)

o

separate points on X. Then X is di¬eomorphic as Fr¨licher space to a subspace

o

of the convenient vector space »(X) ⊆ C ∞ (X, R) with the initial smooth structure

(generated by the restrictions of linear bounded functionals, among other possibili-

ties).

We have constructed the free convenient vector space »X as the c∞ -closure of the

linear subspace generated by the point evaluations in C ∞ (X, R) . This is not very

constructive, in particular since adding Mackey-limits of sequences (or even nets)

of a subspace does not always give its Mackey-closure. In important cases (like

when X is a ¬nite dimensional smooth manifold) one can show however that not

only »X = C ∞ (X, R) , but even that every element of »X is the Mackey-limit of

a sequence of linear combinations of point evaluations, and that C ∞ (X, R) is the

space of distributions of compact support.

23.8. Proposition. Let E be a convenient vector space and X a ¬nite dimensional

smooth separable manifold. Then for every ∈ C ∞ (X, E) there exists a compact

set K ⊆ X such that (f ) = 0 for all f ∈ C ∞ (X, E) with f |K = 0.

Proof. Since X is separable its compact bornology has a countable basis {Kn :

n ∈ N} of compact sets. Assume now that no compact set has the claimed property.

Then for every n ∈ N there has to exist a function fn ∈ C ∞ (X, E) with fn |Kn = 0

23.8

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

o 243

n

but (fn ) = 0. By multiplying fn with (fn ) we may assume that (fn ) = n. Since

every compact subset of X is contained in some Kn one has that {fn : n ∈ N} is

bounded in C ∞ (X, E), but ({fn : n ∈ N} is not; this contradicts the assumption

that is bounded.

23.9. Remark. The proposition above remains true if X is a ¬nite dimensional

smooth paracompact manifold with non-measurably many components. In order

to show this generalization one uses that for the partition {Xj : j ∈ J} by the

non-measurably many components one has C ∞ (X, E) ∼ j∈J C ∞ (Xj , E), and the

=

fact that an belongs to the dual of such a product if it is a ¬nite sum of elements

of the duals of the factors. Now the result follows from (23.8) since the components

of a paracompact manifold are paracompact and hence separable.

For such manifolds X the dual C ∞ (X, R) is the space of distributions with compact

support. In fact, in case X is connected, C ∞ (X, R) is the space of all linear func-

tionals which are continuous for the classically considered topology on C ∞ (X, R)

by (6.1); and in case of an arbitrary X this result follows using the isomorphism

C ∞ (X, R) ∼ j C ∞ (Xj , R) where the Xj denote the connected components of X.

=

23.10. Theorem. [Fr¨licher, Kriegl, 1988], 5.1.7 Let E be a convenient vector

o

space and X a ¬nite dimensional separable smooth manifold. Then the Mackey-

adherence of the linear subspace generated by { —¦evx : x ∈ X, ∈ E } is C ∞ (X, E) .

Proof. The proof is in several steps.

(Step 1) There exist gn ∈ C ∞ (R, R) with supp(gn ) ⊆ [’ n , n ] such that for every

22

f ∈ C ∞ (R, E) the set {n· f ’ k∈Z f (rn,k )gn,k : n ∈ N} is bounded in C ∞ (R, E),

where rn,k := 2k and gn,k (t) := gn (t ’ rn,k ).

n

We choose a smooth h : R ’ [0, 1] with supp(h) ⊆ [’1, 1] and k∈Z h(t ’ k) = 1

for all t ∈ R and we de¬ne Qn : C ∞ (R, E) ’ C ∞ (R, E) by setting

Qn (f )(t) := k

f ( n )h(tn ’ k).

k

Let K ⊆ R be compact. Then

n(Qn (f ) ’ f )(t) = k 1

(f ( n ) ’ f (t)) · n · h(tn ’ k) ∈ B1 (f, K + supp(h))

n

k

for t ∈ K, where Bn (f, K1 ) denotes the absolutely convex hull of the bounded set

n

δ n f (K1 ).

To get similar estimates for the derivatives we use convolution. Let h1 : R ’ R be

a smooth function with support in [’1, 1] and R h1 (s)ds = 1. Then for t ∈ K one

has

(f — h1 )(t) := f (t ’ s)h1 (s)ds ∈ B0 (f, K + supp(h1 )) · h1 1 ,

R

where h1 1 := R |h1 (s)|ds. For smooth functions f, h : R ’ R one has (f —h)(k) =

f —h(k) ; one immediately deduces that the same holds for smooth functions f : R ’

23.10

244 Chapter V. Extensions and liftings of mappings 23.10

E and one obtains (f — h1 )(t) ’ f (t) = R (f (t ’ s) ’ f (t))h1 (s)ds ∈ diam(supp(h1 )) ·

h1 1 · B1 (f, K + supp(h1 )) for t ∈ K, where diam(S) := sup{|s| : s ∈ S}. Using

now hn (t) := n · h1 (nt) we obtain for t ∈ K:

(Qm (f ) — hn ’ f )(k) (t) = (Qm (f ) — h(k) ’ f — h(k) )(t) + (f (k) — hn ’ f (k) )(t)

n n

= (Qm (f ) ’ f ) — h(k) (t) + (f (k) — hn ’ f (k) )(t)

n

∈ B0 (Qm (f ) ’ f, K + supp(hn )) · h(k) 1 +

n

+ B1 (f (k) , K + supp(hn )) · diam(supp(hn )) · hn 1

(k)

1k 1

⊆ · B1 (f, K + supp(hn ) + supp(h)) · h1

mn 1

m

+ n · B1 (f (k) , K + supp(hn )) · hn 1 .

Let now m := 2n and P n (f ) := Qm (f ) — hn . Then

1

(k) (k)

(t) ∈ nk+1 2’n · B1 (f, K + ( n +

n · P n (f ) ’ f 1

)[’1, 1]) h1 1

2n

+ B1 (f (k) , K + n [’1, 1]) h1 1

1

for t ∈ K and the right hand side is uniformly bounded for n ∈ N.

h(s2n ’ k)hn (t + k2’n ’ s)ds = h(s2n )hn (t ’ s)ds we obtain

With gn (t) := R R

n

f (k2’n )h(t2n ’ k) — hn

P n (f )(t) = (Q2 (f ) — hn )(t) =

k

f (k2’n ) h(s2n ’ k)hn (t ’ s)ds

=

R

k

f (k2’n )gn (t ’ k2’n ).

=

k

Thus rn,k := k2’n and the gn have all the claimed properties.

(Step 2) For every m ∈ N and every f ∈ C ∞ (Rm , E) the set

n· f ’ f (rn;k1 ,...,km )gn;k1 ,...,km : n ∈ N

k1 ∈Z,...,km ∈Z

is bounded in C ∞ (Rm , E), where rn;k1 ,...,km := (rn,k1 , . . . , rn,km ) and

gn;k1 ,...,km (x1 , . . . , xm ) := gn,k1 (x1 ) · . . . · gn,km (xm ).

We prove this statement by induction on m. For m = 1 it was shown in step 1.

Now assume that it holds for m and C ∞ (R, E) instead of E. Then by induction

hypothesis applied to f ∨ : C ∞ (Rm , C ∞ (R, E)) we conclude that

n· f ’ )gn;k1 ,...,km : n ∈ N

f (rn;k1 ,...,km ,

k1 ∈Z,...,km ∈Z

23.10

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

o 245

is bounded in C ∞ (Rm+1 , E). Thus it remains to show that

f (rn;k1 ,...,km , rkm+1 )gn,km+1 : n ∈ N

n gn;k1 ,...,km f (rn;k1 ,...,km , )’

k1 ,...,km km+1

is bounded in C ∞ (Rm+1 , E). Since the support of the gn;k1 ,...,km is locally ¬nite

only ¬nitely many summands of the outer sum are non-zero on a given compact set.

Thus it is enough to consider each summand separately. By step (1) we know that

the linear operators h ’ n h ’ k h(rn,k )gn,k , n ∈ N, are pointwise bounded.

So they are bounded on bounded sets, by the linear uniform boundedness principle

(5.18). Hence

n · f (rn;k1 ,...,km , )’ f (rn;k1 ,...,km , rkm+1 )gn,km+1 : n ∈ N

km+1

is bounded in C ∞ (Rm+1 , E). Using that the multiplication R — E ’ E is bounded

one concludes immediately that also the multiplication with a map g ∈ C ∞ (X, R)

is bounded from C ∞ (X, E) ’ C ∞ (X, E) for any Fr¨licher space X. Thus the proof

o

of step (2) is complete.

(Step 3) For every ∈ C ∞ (X, E) there exist xn,k ∈ X and n,k ∈ E such that

{n( ’ k n,k —¦ evxn,k ) : n ∈ N} is bounded in C ∞ (X, E) , where in the sum only

¬nitely many terms are non-zero. In particular the subspace generated by E —¦ evx

for E ∈ E and x ∈ X is c∞ -dense.

By (23.8) there exists a compact set K with f |K = 0 implying (f ) = 0. One

can cover K by ¬nitely many relatively compact Uj ∼ Rm (j = 1 . . . N ). Let

=

{hj : j = 0 . . . N } be a partition of unity subordinated to {X K, U1 , . . . , UN }.

N

Then (f ) = j=1 (hj · f ) for every f . By step (2) the set

n(hj f ’ hj f (rn,k1 ,...,km )gn,k1 ,...,km : n ∈ N

is bounded in C ∞ (Uj , E). Since supp(hj ) is compact in Uj this is even bounded in

C ∞ (X, E) and for ¬xed n only ¬nitely many rn,k1 ,...,km belong to supp(hj ). Thus

the above sum is actually ¬nite and the supports of all functions in the bounded

subset of C ∞ (Uj , E) are included in a common compact subset. Applying to this

subset yields that n ( (hj f ) ’ n,k1 ,...,km —¦ ev(rn,k1 ,...,km ) : n ∈ N is bounded

in R, where n,k1 ,...,km (x) := hj (rn,k1 ,...,km )gn;k1 ,...,km · x .

To complete the proof one only has to take as xn,k all the rn,k1 ,...,km for the ¬nitely

many charts Uj ∼ Rm and as n,k the corresponding functionals n,k1 ,...,km ∈ E .

=

23.11. Corollary. [Fr¨licher, Kriegl, 1988], 5.1.8 Let X be a ¬nite dimensional

o

separable smooth manifold. Then the free convenient vector space »X over X is

equal to C ∞ (X, R) .

23.12. Remark. In [Kriegl, Nel, 1990] it was shown that the free convenient

vector space over the long line L is not C ∞ (L, R) and the same for the space E of

points with countable support in an uncountable product of R.

23.12

246 Chapter V. Extensions and liftings of mappings 23.13

In [Adam, 1995, 2.2.6] it is shown that the isomorphism δ — : L(C ∞ (X, R) , G) ∼ =

C ∞ (X, G) is even a topological isomorphism for (the) natural topologies on all

spaces under consideration provided X is a ¬nite dimensional separable smooth

manifold. Furthermore, the corresponding statement holds for holomorphic map-

pings, provided X is a separable complex manifold modeled on polycylinders. For

Riemannian surfaces X it is shown in [Siegl, 1997, 2.11] that the free convenient

vector space for holomorphic mappings is the Mackey adherence of the linear sub-

space of H(X, C) generated by the point evaluations evx for x ∈ X. In [Siegl, 1997,

2.52] the same is shown for pseudo-convex subsets of X ⊆ Cn . Re¬‚exivity of the

space of scalar valued functions implies that the linear space generated by the point

evaluations is dense in the dual of the function space with respect to its bornolo-

gical topology by [Siegl, 1997, 3.3]. And conversely if Λ(X) is this dual, then the

function space is re¬‚exive. Thus Λ(E) = C ∞ (E, R) for non-re¬‚exive convenient

vector spaces E. Partial positive results for in¬nite dimensional spaces have been

obtained in [Siegl, 1997, section 3].

23.13. Remark. On can de¬ne convenient co-algebras dually to convenient alge-

bras, as a convenient vector space E together with a compatible co-algebra struc-

ture, i.e. two bounded linear mappings

µ : E ’ E —β E, called co-multiplication, into the c∞ -completion (4.29) of

˜

the bornological tensor product (5.9);

and µ : E ’ R, called co-unit,

such that one has the following commutative diagrams:

w w E—˜ (E—˜ E)

˜ ∼

µ—β Id =

˜ ˜ ˜

E —β E (E —β E)—β E β β

u u

˜

µ Id —β µ

w E—˜ E

µ

E β

ee

˜

E —β E

j

h

h e˜ Id

g

e

h µ—

µ

h

w R—˜ E

∼

=

E β

In words, the co-multiplication has to be co-associative and µ has to be a co-unit

with respect to µ.

If, in addition, the following diagram commutes

w E—˜ E

∼

U

R

=

RR

˜

E —β E

j

h

β

h

R hh

µ µ

E

then the co-algebra is called co-commutative.

23.13

24.1 24. Smooth mappings on non-open domains 247

Morphisms g : E ’ F between convenient co-algebras E and F are bounded linear

mappings for which the following diagrams commute:

w F —˜u F w Ru

u

˜

g —g

u

Id

˜

E —β E R

β

µE µF

µE µF

wF wF

g g

E E

A co-idempotent in a convenient co-algebra E, is an element x ∈ E satisfying

µ(x) = 1 and µ(x) = x — x. They correspond bijectively to convenient co-algebra

morphisms R ’ E, see [Fr¨licher, Kriegl, 1988, 5.2.7].

o

In [Fr¨licher, Kriegl, 1988, 5.2.4] it was shown that »(X — Y ) ∼ »(X)—»(Y ) using

˜

o =

only the universal property of the free convenient vector space. Thus »(∆) : »(X) ’

»(X — X) ∼ »(X)—»(X) of the diagonal mapping ∆ : X ’ X — X de¬nes a co-

˜

=

multiplication on »(X) with co-unit »(const) : »(X) ’ »({—}) ∼ R. In this way »

=

becomes a functor from the category of Fr¨licher spaces into that of convenient co-

o

algebras, see [Fr¨licher, Kriegl, 1988, 5.2.5]. In fact this functor is left-adjoint to the

o

functor I, which associates to each convenient co-algebra the Fr¨licher space of co-

o

idempotents with the initial structure inherited from the co-algebra, see [Fr¨licher,

o

Kriegl, 1988, 5.2.9].

Furthermore, it was shown in [Fr¨licher, Kriegl, 1988, 5.2.18] that any co-idempo-

o

eve

tent element e of »(X) de¬nes an algebra-homomorphism C ∞ (X, R) ∼ »(X) ’ ’ ’

=

R. Thus the equality I(»(X)) = X, i.e. every co-idempotent e ∈ »(X) is given by

evx for some x ∈ X, is thus satis¬ed for smoothly realcompact spaces X, as they

are treated in chapter IV.

24. Smooth Mappings on Non-Open Domains

In this section we will discuss smooth maps f : E ⊇ X ’ F , where E and F

are convenient vector spaces and X are certain not necessarily open subsets of E.

We consider arbitrary subsets X ⊆ E as Fr¨licher spaces with the initial smooth

o

structure induced by the inclusion into E, i.e., a map f : E ⊇ X ’ F is smooth if

and only if for all smooth curves c : R ’ X ⊆ E the composite f —¦ c : R ’ F is a

smooth curve.

24.1. Lemma. Convex sets with non-void interior.

Let K ⊆ E be a convex set with non-void c∞ -interior K o . Then the segment

(x, y] := {x + t(y ’ x) : 0 < t ¤ 1} is contained in K o for every x ∈ K and y ∈ K 0 .

The interior K o is convex and open even in the locally convex topology. And K is

closed if and only if it is c∞ -closed.

Proof. Let y0 := x + t0 (y ’ x) be an arbitrary point on the segment (x, y], i.e.,

0 < t0 ¤ 1. Then x+t0 (K o ’x) is an c∞ -open neighborhood of y0 , since homotheties

are c∞ -continuous. It is contained in K, since K is convex.

24.1

248 Chapter V. Extensions and liftings of mappings 24.2

In particular, the c∞ -interior K o is convex, hence it is not only c∞ -open but open

in the locally convex topology (4.5).

Without loss of generality we now assume that 0 ∈ K o . We claim that the closure of

K is the set {x : tx ∈ K o for 0 < t < 1}. This implies the statement on closedness.

Let U := K o and consider the Minkowski-functional pU (x) := inf{t > 0 : x ∈ tU }.

Since U is convex, the function pU is convex, see (52.2). Using that U is c∞ -open

it can easily be shown that U = {x : pU (x) < 1}. From (13.2) we conclude that pU

is c∞ -continuous, and furthermore that it is even continuous for the locally convex

topology. Hence, the set {x : tx ∈ K o for 0 < t < 1} = {x : pU (x) ¤ 1} = {x :

pK (x) ¤ 1} is the closure of K in the locally convex topology by (52.3).

24.2. Theorem. Derivative of smooth maps.

Let K ⊆ E be a convex subset with non-void interior K o , and let f : K ’ R be a

smooth map. Then f |K o : K o ’ F is smooth, and its derivative (f |K o ) extends

(uniquely) to a smooth map K ’ L(E, F ).

Proof. Only the extension property is to be shown. Let us ¬rst try to ¬nd a

candidate for f (x)(v) for x ∈ K and v ∈ E with x + v ∈ K o . By convexity the

smooth curve cx,v : t ’ x + t2 v has for 0 < |t| < 1 values in K o and cx,v (0) =

x ∈ K, hence f —¦ cx,v is smooth. In the special case where x ∈ K o we have by

the chain rule that (f —¦ cx,v ) (t) = f (x)(cx,v (t))(cx,v (t)), hence (f —¦ cx,v ) (t) =

f (cx,v (t))(cx,v (t), cx,v (t)) + f (cx,v (t))(cx,v (t)), and for t = 0 in particular (f —¦

cx,v ) (0) = 2 f (x)(v). Thus we de¬ne

2 f (x)(v) := (f —¦ cx,v ) (0) for x ∈ K and v ∈ K o ’ x.

√

Note that for 0 < µ < 1 we have f (x)(µ v) = µ f (x)(v), since cx,µ v (t) = cx,v ( µ t).

Let us show next that f ( )(v) : {x ∈ K : x + v ∈ K o } ’ R is smooth. So let

s ’ x(s) be a smooth curve in K, and let v ∈ K 0 ’ x(0). Then x(s) + v ∈ K o for

all su¬ciently small s. And thus the map (s, t) ’ cx(s),v (t) is smooth from some

neighborhood of (0, 0) into K. Hence (s, t) ’ f (cx(s),v (t)) is smooth and also its

second derivative s ’ (f —¦ cx(s),v ) (0) = 2 f (x(s))(v).

In particular, let x0 ∈ K and v0 ∈ K o ’ x0 and x(s) := x0 + s2 v0 . Then

2f (x0 )(v) := (f —¦ cx0 ,v ) (0) = lim (f —¦ cx(s),v ) (0) = lim 2 f (x(s))(v),

s’0 s’0

with x(s) ∈ K o for 0 < |s| < 1. Obviously this shows that the given de¬nition of

f (x0 )(v) is the only possible smooth extension of f ( )(v) to {x0 } ∪ K o .

Now let v ∈ E be arbitrary. Choose a v0 ∈ K o ’ x0 . Since the set K o ’ x0 ’ v0 is

a c∞ -open neighborhood of 0, hence absorbing, there exists some µ > 0 such that

v0 + µv ∈ K o ’ x0 . Thus

f (x)(v) = 1 f (x)(µv) = 1

f (x)(v0 + µv) ’ f (x)(v0 )

µ µ

for all x ∈ K 0 . By what we have shown above the right side extends smoothly

to {x0 } ∪ K o , hence the same is true for the left side. I.e. we de¬ne f (x0 )(v) :=

24.2

24.3 24. Smooth mappings on non-open domains 249

lims’0 f (x(s))(v) for some smooth curve x : (’1, 1) ’ K with x(s) ∈ K o for

0 < |s| < 1. Then f (x) is linear as pointwise limit of f (x(s)) ∈ L(E, R) and is

bounded by the Banach-Steinhaus theorem (applied to EB ). This shows at the

same time, that the de¬nition does not depend on the smooth curve x, since for

v ∈ x0 + K o it is the unique extension.

In order to show that f : K ’ L(E, F ) is smooth it is by (5.18) enough to show

that

f evx

x

s ’ f (x(s))(v), R ’ K ’ L(E, F ) ’ F

is smooth for all v ∈ E and all smooth curves x : R ’ K. For v ∈ x0 + K o

this was shown above. For general v ∈ E, this follows since f (x(s))(v) is a linear

combination of f (x(s))(v0 ) for two v0 ∈ x0 + K o not depending on s locally.

By (24.2) the following lemma applies in particular to smooth maps.

24.3. Lemma. Chain rule. Let K ⊆ E be a convex subset with non-void interior

K o , let f : K ’ R be smooth on K o and let f : K ’ L(E, F ) be an extension of

(f |K o ) , which is continuous for the c∞ -topology of K, and let c : R ’ K ⊆ E be a

smooth curve. Then (f —¦ c) (t) = f (c(t))(c (t)).

Proof.

Claim Let g : K ’ L(E, F ) be continuous along smooth curves in K, then g : ˆ

K — E ’ F is also continuous along smooth curves in K — E.

In order to show this let t ’ (x(t), v(t)) be a smooth curve in K — E. Then

g —¦ x : R ’ L(E, F ) is by assumption continuous (for the bornological topology on

L(E, F )) and v — : L(E, F ) ’ C ∞ (R, F ) is bounded and linear (3.13) and (3.17).

Hence, the composite v — —¦ g —¦ x : R ’ C ∞ (R, F ) ’ C(R, F ) is continuous. Thus,

(v — —¦ g —¦ x)§ : R2 ’ F is continuous, and in particular when restricted to the

diagonal in R2 . But this restriction is just g —¦ (x, v).

Now choose a y ∈ K o . And let cs (t) := c(t) + s2 (y ’ c(t)). Then cs (t) ∈ K o for 0 <

|s| ¤ 1 and c0 = c. Furthermore, (s, t) ’ cs (t) is smooth and cs (t) = (1 ’ s2 )c (t).

And for s = 0

1 1

f (cs (t)) ’ f (cs (0)) 2

(f —¦ cs ) (t„ )d„ = (1 ’ s )

= f (cs (t„ ))(c (t„ ))d„ .

t 0 0

Now consider the speci¬c case where c(t) := x + tv with x, x + v ∈ K. Since

f is continuous along (t, s) ’ cs (t), the left side of the above equation converges

to f (c(t))’f (c(0)) for s ’ 0. And since f (·)(v) is continuous along (t, „, s) ’

t

cs (t„ ) we have that f (cs (t„ ))(v) converges to f (c(t„ ))(v) uniformly with respect

to 0 ¤ „ ¤ 1 for s ’ 0. Thus, the right side of the above equation converges to

1

f (c(t„ ))(v)d„ . Hence, we have

0

1 1

f (c(t)) ’ f (c(0))

f (c(t„ ))(v)d„ ’

= f (c(0))(v)d„ = f (c(0))(c (0))

t 0 0

24.3

250 Chapter V. Extensions and liftings of mappings 24.5

for t ’ 0.

Now let c : R ’ K be an arbitrary smooth curve. Then (s, t) ’ c(0)+s(c(t)’c(0))

is smooth and has values in K for 0 ¤ s ¤ 1. By the above consideration we have

for x = c(0) and v = (c(t) ’ c(0))/t that

1

f (c(t)) ’ f (c(0)) c(t) ’ c(0)

f c(0) + „ (c(t) ’ c(0))

=

t t

0

which converges to f (c(0))(c (0)) for t ’ 0, since f is continuous along smooth

curves in K and thus f (c(0) + „ (c(t) ’ c(0))) ’ f (c(0)) uniformly on the bounded

set { c(t)’c(0) : t near 0}. Thus, f —¦ c is di¬erentiable with derivative (f —¦ c) (t) =

t

f (c(t))(c (t)).

Since f can be considered as a map df : E — E ⊇ K — E ’ F it is important to

study sets A — B ⊆ E — F . Clearly, A — B is convex provided A ⊆ E and B ⊆ F

are. Remains to consider the openness condition. In the locally convex topology

(A — B)o = Ao — B o , which would be enough to know in our situation. However,

we are also interested in the corresponding statement for the c∞ -topology. This

topology on E — F is in general not the product topology c∞ E — c∞ F . Thus, we

cannot conclude that A — B has non-void interior with respect to the c∞ -topology

on E — F , even if A ⊆ E and B ⊆ F have it. However, in case where B = F

everything is ¬ne.

24.4. Lemma. Interior of a product.

Let X ⊆ E. Then the interior (X — F )o of X — F with respect to the c∞ -topology

on E — F is just X o — F , where X o denotes the interior of X with respect to the

c∞ -topology on E.

Proof. Let W be the saturated hull of (X — F )o with respect to the projection

pr1 : E — F ’ E, i.e. the c∞ -open set (X — F )o + {0} — F ⊆ X — F . Its projection

to E is c∞ -open, since it agrees with the intersection with E — {0}. Hence, it is

contained in X o , and (X — F )o ⊆ X o — F . The converse inclusion is obvious since

pr1 is continuous.

24.5. Theorem. Smooth maps on convex sets.

Let K ⊆ E be a convex subset with non-void interior K o , and let f : K ’ F be

a map. Then f is smooth if and only if f is smooth on K o and all derivatives

(f |K o )(n) extend continuously to K with respect to the c∞ -topology of K.

Proof. (’) It follows by induction using (24.2) that f (n) has a smooth extension

K ’ Ln (E; F ).

(⇐) By (24.3) we conclude that for every c : R ’ K the composite f —¦ c : R ’ F

is di¬erentiable with derivative (f —¦ c) (t) = f (c(t))(c (t)) =: df (c(t), c (t)).

The map df is smooth on the interior K o — E, linear in the second variable, and

its derivatives (df )(p) (x, w)(y1 , w1 ; . . . , yp , wp ) are universal linear combinations of

f (p+1) (x)(y1 , . . . , yp ; w) and of f (k+1) (x)(yi1 , . . . , yik ; wi0 ) for k ¤ p.

24.5

24.6 24. Smooth mappings on non-open domains 251

These summands have unique extensions to K — E. The ¬rst one is continuous

along smooth curves in K — E, because for such a curve (t ’ (x(t), w(t)) the

extension f (k+1) : K ’ L(E k , L(E, F )) is continuous along the smooth curve x,

and w— : L(E, F ) ’ C ∞ (R, F ) is continuous and linear, so the mapping t ’

(s ’ f (k+1) (x(t))(yi1 , . . . , yik ; w(s))) is continuous from R ’ C ∞ (R, F ) and thus

as map from R2 ’ F it is continuous, and in particular if restricted to the diagonal.

And the other summands only depend on x, hence have a continuous extension by

assumption.

So we can apply (24.3) inductively using (24.4), to conclude that f —¦ c : R ’ F is

smooth.

In view of the preceding theorem (24.5) it is important to know the c∞ -topology

c∞ X of X, i.e. the ¬nal topology generated by all the smooth curves c : R ’

X ⊆ E. So the ¬rst question is whether this is the trace topology c∞ E|X of the

c∞ -topology of E.

24.6. Lemma. The c∞ -topology is the trace topology.

In the following cases of subsets X ⊆ E the trace topology c∞ E|X equals the topol-

ogy c∞ X:

(1) X is c∞ E-open.

(2) X is convex and locally c∞ -closed.

(3) The topology c∞ E is sequential and X ⊆ E is convex and has non-void

interior.

(3) applies in particular to the case where E is metrizable, see (4.11). A topology

is called sequential if and only if the closure of any subset equals its adherence,

i.e. the set of all accumulation points of sequences in it. By (2.13) and (2.8) the

adherence of a set X with respect to the c∞ -topology, is formed by the limits of all

Mackey-converging sequences in X.

Proof. Note that the inclusion X ’ E is by de¬nition smooth, hence the identity

c∞ X ’ c∞ E|X is always continuous.

(1) Let U ⊆ X be c∞ X-open and let c : R ’ E be a smooth curve with c(0) ∈ U .

Since X is c∞ E-open, c(t) ∈ X for all small t. By composing with a smooth

map h : R ’ R which satis¬es h(t) = t for all small t, we obtain a smooth curve

c —¦ h : R ’ X, which coincides with c locally around 0. Since U is c∞ X-open we

conclude that c(t) = (c —¦ h)(t) ∈ U for small t. Thus, U is c∞ E-open.

¯

(2) Let A ⊆ X be c∞ X-closed. And let A be the c∞ E-closure of A. We have to

¯ ¯

show that A © X ⊆ A. So let x ∈ A © X. Since X is locally c∞ E-closed, there

exists a c∞ E-neighborhood U of x ∈ X with U © X c∞ -closed in U . For every

c∞ E-neighborhood U of x we have that x is in the closure of A © U in U with

respect to the c∞ E-topology (otherwise some open neighborhood of x in U does

not meet A © U , hence also not A). Let an ∈ A © U be Mackey converging to a ∈ U .

Then an ∈ X © U which is closed in U thus a ∈ X. Since X is convex the in¬nite

polygon through the an lies in X and can be smoothly parameterized by the special

24.6

252 Chapter V. Extensions and liftings of mappings 24.8

curve lemma (2.8). Using that A is c∞ X-closed, we conclude that a ∈ A. Thus,

A © U is c∞ U -closed and x ∈ A.

¯

(3) Let A ⊆ X be c∞ X-closed. And let A denote the closure of A in c∞ E. We

¯ ¯

have to show that A © X ⊆ A. So let x ∈ A © X. Since c∞ E is sequential there

is a Mackey converging sequence A an ’ x. By the special curve lemma (2.8)

the in¬nite polygon through the an can be smoothly parameterized. Since X is

convex this curve gives a smooth curve c : R ’ X and thus c(0) = x ∈ A, since A

is c∞ X-closed.

24.7. Example. The c∞ -topology is not trace topology.

Let A ⊆ E be such that the c∞ -adherence Adh(A) of A is not the whole c∞ -closure

¯ ¯

A of A. So let a ∈ A \ Adh(A). Then consider the convex subset K ⊆ E — R de¬ned

by K := {(x, t) ∈ E — R : t ≥ 0 and (t = 0 ’ x ∈ A ∪ {a})} which has non-empty

interior E — R+ . However, the topology c∞ K is not the trace topology of c∞ (E — R)

which equals c∞ (E) — R by (4.15).

Note that this situation occurs quite often, see (4.13) and (4.36) where A is even a

linear subspace.

Proof. Consider A = A — {0} ⊆ K. This set is closed in c∞ K, since E © K is

closed in c∞ K and the only point in (K © E) \ A is a, which cannot be reached by

a Mackey converging sequence in A, since a ∈ Adh(A).

/

It is however not the trace of a closed subset in c∞ (E) — R. Since such a set has to

¯

contain A and hence A a.

24.8. Theorem. Smooth maps on subsets with collar.

Let M ⊆ E have a smooth collar, i.e., the boundary ‚M of M is a smooth sub-

manifold of E and there exists a neighborhood U of ‚M and a di¬eomorphism

ψ : ‚M — R ’ U which is the identity on ‚M and such that ψ(M — {t ∈ R :

t ≥ 0}) = M © U . Then every smooth map f : M ’ F extends to a smooth map

˜

f : M ∪ U ’ F . Moreover, one can choose a bounded linear extension operator

˜

C ∞ (M, F ) ’ C ∞ (M ∪ U, F ), f ’ f .

Proof. By (16.8) there is a continuous linear right inverse S to the restriction

map C ∞ (R, R) ’ C ∞ (I, R), where I := {t ∈ R : t ≥ 0}. Now let x ∈ U and

(px , tx ) := ψ ’1 (x). Then f (ψ(px , ·)) : I ’ F is smooth, since ψ(px , t) ∈ M

for t ≥ 0. Thus, we have a smooth map S(f (ψ(px , ·))) : R ’ F and we de¬ne

˜ ˜

f (x) := S(f (ψ(px , ·)))(tx ). Then f (x) = f (x) for all x ∈ M © U , since for such

˜

an x we have tx ≥ 0. Now we extend the de¬nition by f (x) = f (x) for x ∈ M o .

˜

Remains to show that f is smooth (on U ). So let s ’ x(s) be a smooth curve

in U . Then s ’ (ps , ts ) := ψ ’1 (x(s)) is smooth. Hence, s ’ (t ’ f (ψ(ps , t))

is a smooth curve R ’ C ∞ (I, F ). Since S is continuous and linear the composite

s ’ (t ’ S(f ψ(ps , ·))(t)) is a smooth curve R ’ C ∞ (R, F ) and thus the associated

˜

map R2 ’ F is smooth, and also the composite f (xs ) of it with s ’ (s, ts ).

The existence of a bounded linear extension operator follows now from (21.2).

In particular, the previous theorem applies to the following convex sets:

24.8

24.10 24. Smooth mappings on non-open domains 253

24.9. Proposition. Convex sets with smooth boundary have a collar.

Let K ⊆ E be a closed convex subset with non-empty interior and smooth boundary

‚K. Then K has a smooth collar as de¬ned in (24.8).

Proof. Without loss of generality let 0 ∈ K o .

In order to show that the set U := {x ∈ E : tx ∈ K for some t > 0} is c∞ -open let

/

s ’ x(s) be a smooth curve R ’ E and assume that t0 x(0) ∈ K for some t0 > 0.

/

Since K is closed we have that t0 x(s) ∈ K for all small |s|.

/

1

For x ∈ U let r(x) := sup{t ≥ 0 : tx ∈ K o } > 0, i.e. r = pK o as de¬ned in the

proof of (24.1) and r(x)x is the unique intersection point of ‚K © (0, +∞)x. We

claim that r : U ’ R+ is smooth. So let s ’ x(s) be a smooth curve in U and

x0 := r(x(0))x(0) ∈ ‚K. Choose a local di¬eomorphism ψ : (E, x0 ) ’ (E, 0) which

maps ‚K locally to some closed hyperplane F ⊆ E. Any such hyperplane is the

kernel of a continuous linear functional : E ’ R, hence E ∼ F — R.

=

We claim that v := ψ (x0 )(x0 ) ∈ F . If this were not the case, then we consider the

/

smooth curve c : R ’ ‚K de¬ned by c(t) = ψ ’1 (’tv). Since ψ (x0 ) is injective its

derivative is c (0) = ’x0 and c(0) = x0 . Since 0 ∈ K o , we have that x0 + c(t)’c(0) ∈

t

c(t)’c(0)

o o

K for all small |t|. By convexity c(t) = x0 + t ∈ K for small t > 0, a

t

contradiction.

So we may assume that (ψ (x)(x)) = 0 for all x in a neighborhood of x0 .

For s small r(x(s)) is given by the implicit equation (ψ(r(x(s))x(s))) = 0. So let

g : R2 ’ R be the locally de¬ned smooth map g(t, s) := (ψ(tx(s))). For t = 0

its ¬rst partial derivative is ‚1 g(t, s) = (ψ (tx(s))(x(s))) = 0. So by the classical

implicit function theorem the solution s ’ r(x(s)) is smooth.

Now let Ψ : U —R ’ U be the smooth map de¬ned by (x, t) ’ e’t r(x)x. Restricted

to ‚K — R ’ U is injective, since tx = t x with x,x ∈ ‚K and t, t > 0 implies

x = x and hence t = t . Furthermore, it is surjective, since the inverse mapping is

1

given by x ’ (r(x)x, ln(r(x))). Use that r(»x) = » r(x). Since this inverse is also

smooth, we have the required di¬eomorphism Ψ. In fact, Ψ(x, t) ∈ K if and only if

e’t r(x) ¤ r(x), i.e. t ¤ 0.

That (24.8) is far from being best possible shows the

24.10. Proposition. Let K ⊆ Rn be the quadrant K := {x = (x1 , . . . , xn ) ∈

Rn : x1 ≥ 0, . . . , xn ≥ 0}. Then there exists a bounded linear extension operator

C ∞ (K, F ) ’ C ∞ (Rn , F ) for each convenient vector space F .

This can be used to obtain the same result for submanifolds with convex corners

sitting in smooth ¬nite dimensional manifolds.

Proof. Since K = (R+ )n ⊆ Rn and the inclusion is the product of inclusions

ι : R+ ’ R we can use the exponential law (23.2.3) to obtain C ∞ (K, F ) ∼ =

∞ ∞

n’1

C ((R+ ) , C (R+ , F )). By Seeley™s theorem (16.8) we have a bounded lin-

ear extension operator S : C ∞ (R+ , F ) ’ C ∞ (R, F ). We now proceed by induction

24.10

254 Chapter V. Extensions and liftings of mappings 25.1

on n. So we have an extension operator Sn’1 : C ∞ ((R+ )n’1 , G) ’ C ∞ (Rn’1 , G)

for the convenient vector space G := C ∞ (R, F ) by induction hypothesis. The

composite gives up to natural isomorphisms the required extension operator

S—

C ∞ (K, F ) ∼ C ∞ ((R+ )n’1 , C ∞ (R+ , F )) ’’ C ∞ ((R+ )n’1 , C ∞ (R, F )) ’

=

Sn’1

’ ’ C ∞ (Rn’1 , C ∞ (R, F )) ∼ C ∞ (Rn , F ).

’’ =

25. Real Analytic Mappings on Non-Open Domains

In this section we will consider real analytic mappings de¬ned on the same type of

convex subsets as in the previous section.

25.1. Theorem. Power series in Fr´chet spaces. Let E be a Fr´chet space and

e e

(F, F ) be a dual pair. Assume that a Baire vector space topology on E exists for

which the point evaluations are continuous. Let fk be k-linear symmetric bounded

functionals from E to F , for each k ∈ N. Assume that for every ∈ F and every x

∞

in some open subset W ⊆ E the power series k=0 (fk (xk ))tk has positive radius of

convergence. Then there exists a 0-neighborhood U in E, such that {fk (x1 , . . . , xk ) :

∞

k ∈ N, xj ∈ U } is bounded and thus the power series x ’ k=0 fk (xk ) converges

Mackey on some 0-neighborhood in E.

Proof. Choose a ¬xed but arbitrary ∈ F . Then —¦ fk satisfy the assumptions

of (7.14) for an absorbing subset in a closed cone C with non-empty interior. Since

this cone is also complete metrizable we can proceed with the proof as in (7.14)

to obtain a set AK,r ⊆ C whose interior in C is non-void. But this interior has

to contain a non-void open set of E and as in the proof of (7.14) there exists

some ρ > 0 such that for the ball Uρ in E with radius ρ and center 0 the set

{ (fk (x1 , . . . , xk )) : k ∈ N, xj ∈ Uρ } is bounded.

Now let similarly to (9.6)

{ ∈ F : | (fk (x1 , . . . , xk ))| ¤ Krk }

AK,r,ρ :=

k∈N x1 ,...xn ∈Uρ

for K, r, ρ > 0. These sets AK,r,ρ are closed in the Baire topology, since evaluation

at fk (x1 , . . . , xk ) is assumed to be continuous.

By the ¬rst part of the proof the union of these sets is F . So by the Baire property,

there exist K, r, ρ > 0 such that the interior U of AK,r,ρ is non-empty. As in the

proof of (9.6) we choose an 0 ∈ U . Then for every ∈ F there exists some µ > 0

such that µ := µ ∈ U ’ 0 . So | (y)| ¤ 1 (| µ (y) + 0 (y)| + | 0 (y)|) ¤ 2 Krn for

µ µ

every y = fk (x1 , . . . , xk ) with xi ∈ Uρ . Thus, {fk (x1 , . . . , xk ) : k ∈ N, xi ∈ U ρ } is

r

bounded.

On every smaller ball we have therefore that the power series with terms fk con-

verges Mackey.

Note that if the vector spaces are real and the assumption above hold, then the

conclusion is even true for the complexi¬ed terms by (7.14).

25.1

25.2 25. Real analytic mappings on non-open domains 255

25.2. Theorem. Real analytic maps I ’ R are germs.

Let f : I := {t ∈ R : t ≥ 0} ’ R be a map. Suppose t ’ f (t2 ) is real analytic

˜˜ ˜

R ’ R. Then f extends to a real analytic map f : I ’ R, where I is an open

neighborhood of I in R.

Proof. We show ¬rst that f is smooth. Consider g(t) := f (t2 ). Since g : R ’ R

is assumed to be real analytic it is smooth and clearly even. We claim that there

exists a smooth map h : R ’ R with g(t) = h(t2 ) (this is due to [Whitney, 1943]).

In fact, by h(t2 ) := g(t) a continuosu map h : {t :∈ R : t ≥ 0} ’ R is uniquely

determined. Obviously, h|{t∈R:t>0} is smooth. Di¬erentiating for t = 0 the de¬ning

equation gives h (t2 ) = g 2t =: g1 (t). Since g is smooth and even, g is smooth and

(t)

odd, so g (0) = 0. Thus

1

g (t) ’ g (0) 1

t ’ g1 (t) = = g (ts) ds

2t 2 0

is smooth. Hence, we may de¬ne h on {t ∈ R : t ≥ 0} by the equation h (t2 ) = g1 (t)

with even smooth g1 . By induction we obtain continuous extensions of h(n) : {t ∈

R : t > 0} ’ R to {t ∈ R : t ≥ 0}, and hence h is smooth on {t ∈ R : t ≥ 0} and so

can be extended to a smooth map h : R ’ R.

From this we get f (t2 ) = g(t) = h(t2 ) for all t. Thus, h : R ’ R is a smooth

extension of f .

Composing with the exponential map exp : R ’ R+ shows that f is real analytic

on {t : t > 0}, and has derivatives f (n) which extend by (24.5) continuously to

1

maps I ’ R. It is enough to show that an := n! f (n) (0) are the coe¬cients of a

power series p with positive radius of convergence and for t ∈ I this map p coincides

with f .

Claim. We show that a smooth map f : I ’ R, which has a real analytic composite

with t ’ t2 , is the germ of a real analytic mapping.

Consider the real analytic curve c : R ’ I de¬ned by c(t) = t2 . Thus, f —¦ c is

real analytic. By the chain rule the derivative (f —¦ c)(p) (t) is for t = 0 a universal

linear combination of terms f (k) (c(t))c(p1 ) (t) · · · c(pk ) (t), where 1 ¤ k ¤ p and

p1 + . . . + pk = p. Taking the limit for t ’ 0 and using that c(n) (0) = 0 for

all n = 2 and c (0) = 2 shows that there is a universal constant cp satisfying

(f —¦ c)(2p) (0) = cp · f (p) (0). Take as f (x) = xp to conclude that (2p)! = cp · p!.

∞ 1

Now we use (9.2) to show that the power series k=0 k! f (k) (0)tk converges locally.

So choose a sequence (rk ) with rk tk ’ 0 for all t > 0. De¬ne a sequence (¯k ) by r

¯ ¯¯ ¯¯ ¯

r2n = r2n+1 := rn and let t > 0. Then rk tk = rn tn for 2n = k and rk tk = rn tn t

¯ ¯

¯

for 2n + 1 = k, where t := t2 > 0, hence (¯k ) satis¬es the same assumptions as (rk )

r

1

and thus by (9.3) (1 ’ 3) the sequence k! (f —¦ c)(k) (0)¯k is bounded. In particular,

r

this is true for the subsequence

cp

—¦ c)(2p) (0)¯2p = (p) 1 (p)

1

(2p)! (f r (2p)! f (0)rp = p! f (0)rp .

1 (p)

Thus, by (9.3) (1 ⇐ 3) the power series with coe¬cients p! f (0) converges locally

˜

to a real analytic function f .

25.2

256 Chapter V. Extensions and liftings of mappings 25.5

˜ ˜

Remains to show that f = f on J. But since f —¦ c and f —¦ c are both real analytic

near 0, and have the same Taylor series at 0, they have to coincide locally, i.e.

˜

f (t2 ) = f (t2 ) for small t.

Note however that the more straight forward attempt of a proof of the ¬rst step,

namely to show that f —¦ c is smooth for all c : R ’ {t ∈ R : t ≥ 0} by showing that

for such c there is a smooth map h : R ’ R, satisfying c(t) = h(t)2 , is doomed to

fail as the following example shows.

25.3. Example. A smooth function without smooth square root.

Let c : R ’ {t ∈ R : t ≥ 0} be de¬ned by the general curve lemma (12.2) using

pieces of parabolas cn : t ’ 2n t2 + 41 . Then there is no smooth square root of c.

2n n

Proof. The curve c constructed in (12.2) has the property that there exists a

converging sequence tn such that c(t + tn ) = cn (t) for small t. Assume there were

a smooth map h : R ’ R satisfying c(t) = h(t)2 for all t. At points where c(t) = 0

we have in turn:

c (t) = 2h(t)h (t)

c (t) = 2h(t)h (t) + 2h (t)2

2c(t)c (t) = 4h(t)3 h (t) + c (t)2 .

Choosing tn for t in the last equation gives h (tn ) = 2n, which is unbounded in n.

Thus h cannot be C 2 .

25.4. De¬nition. (Real analytic maps I ’ F )

Let I ⊆ R be a non-trivial interval. Then a map f : I ’ F is called real analytic

if and only if the composites —¦ f —¦ c : R ’ R are real analytic for all real analytic

c : R ’ I ⊆ R and all ∈ F . If I is an open interval then this de¬nition coincides

with (10.3).

25.5. Lemma. Bornological description of real analyticity.

Let I ⊆ R be a compact interval. A curve c : I ’ E is real analytic if and only if c

1

is smooth and the set { k! c(k) (a) rk : a ∈ I, k ∈ N} is bounded for all sequences (rk )

with rk tk ’ 0 for all t > 0.

Proof. We use (9.3). Since both sides can be tested with ∈ E we may assume

that E = R.

˜

(’) By (25.2) we may assume that c : I ’ R is real analytic for some open

˜

neighborhood I of I. Thus, the required boundedness condition follows from (9.3).

(⇐) By (25.2) we only have to show that f : t ’ c(t2 ) is real analytic. For

this we use again (9.3). So let K ⊆ R be compact. Then the Taylor series of f

is obtained by that of c composed with t2 . Thus, the composite f satis¬es the

required boundedness condition, and hence is real analytic.

This characterization of real analyticity can not be weakened by assuming the

2

boundedness conditions only for single pointed K as the map c(t) := e’1/t for

25.5

25.9 25. Real analytic mappings on non-open domains 257

t = 0 and c(0) = 0 shows. It is real analytic on R \ {0} thus the condition is

satis¬ed at all points there, and at 0 the power series has all coe¬cients equal to

0, hence the condition is satis¬ed there as well.

25.6. Corollary. Real analytic maps into inductive limits.

Let T± : E ’ E± be a family of bounded linear maps that generates the bornology

on E. Then a map c : I ’ F is real analytic if and only if all the composites

T± —¦ c : I ’ F± are real analytic.

Proof. This follows either directly from (25.5) or from (25.2) by using the corre-

sponding statement for maps R ’ E, see (9.9).

25.7. De¬nition. (Real analytic maps K ’ F )

For an arbitrary subset K ⊆ E let us call a map f : E ⊇ K ’ F real analytic if

and only if » —¦ f —¦ c : I ’ R is a real analytic (resp. smooth) for all » ∈ F and

all real analytic (resp. smooth) maps c : I ’ K, where I ‚ R is some compact

non-trivial interval. Note however that it is enough to use all real analytic (resp.

smooth) curves c : R ’ K by (25.2).

With C ω (K, F ) we denote the vector space of all real analytic maps K ’ F . And we

topologize this space with the initial structure induced by the cone c— : C ω (K, F ) ’

C ω (R, F ) (for all real analytic c : R ’ K) and the cone c— : C ω (K, F ) ’ C ∞ (R, F )

(for all smooth c : R ’ K). The space C ω (R, F ) should carry the structure of

(11.2) and the space C ∞ (R, F ) that of (3.6).

For an open K ⊆ E the de¬nition for C ω (K, F ) given here coincides with that of

(10.3).

25.8. Proposition. C ω (K, F ) is convenient. Let K ⊆ E and F be arbitrary.

Then the space C ω (K, F ) is a convenient vector space and satis¬es the S-uniform

boundedness principle (5.22), where S := {evx : x ∈ K}.

Proof. Since both spaces C ω (R, R) and C ∞ (R, R) are c∞ -complete and satisfy the

uniform boundedness principle for the set of point evaluations the same is true for

C ω (K, F ), by (5.25).

25.9. Theorem. Real analytic maps K ’ F are often germs.

Let K ⊆ E be a convex subset with non-empty interior of a Fr´chet space and let

e

(F, F ) be a complete dual pair for which a Baire topology on F exists, as required

in (25.1). Let f : K ’ F be a real analytic map. Then there exists an open

˜ ˜

neighborhood U ⊆ EC of K and a holomorphic map f : U ’ FC such that f |K = f .

Proof. By (24.5) the map f : K ’ F is smooth, i.e. the derivatives f (k) exist on

the interior K 0 and extend continuously (with respect to the c∞ -topology of K)

to the whole of K. So let x ∈ K be arbitrary and consider the power series with

1

coe¬cients fk = k! f (k) (x). This power series has the required properties of (25.1),

since for every ∈ F and v ∈ K o ’ x the series k (fk (v k ))tk has positive radius

of convergence. In fact, (f (x + tv)) is by assumption a real analytic germ I ’ R,

25.9

258 Chapter V. Extensions and liftings of mappings 25.10

by (24.8) hence locally around any point in I it is represented by its converging

Taylor series at that point. Since (x, v ’ x] ⊆ K o and f is smooth on this set,

d

( dt )k ( (f (x + tv)) = (f (k) (x + tv)(v k ) for t > 0. Now take the limit for t ’ 0

to conclude that the Taylor coe¬cients of t ’ (f (x + tv)) at t = 0 are exactly

k! (fk ). Thus, by (25.1) the power series converges locally and hence represents a

holomorphic map in a neighborhood of x. Let y ∈ K o be an arbitrary point in this

neighborhood. Then t ’ (f (x + t(y ’ x))) is real analytic I ’ R and hence the

series converges at y ’ x towards f (y). So the restriction of the power series to the

interior of K coincides with f .

˜

We have to show that the extensions fx of f : K © Ux ’ FC to star shaped

˜˜

˜

neighborhoods Ux of x in EC ¬t together to give an extension f : U ’ FC . So let

˜ ˜

Ux be such a domain for the extension and let Ux := Ux © E.

For this we claim that we may assume that Ux has the following additional property:

y ∈ Ux ’ [0, 1]y ⊆ K o ∪ Ux . In fact, let U0 := {y ∈ Ux : [0, 1]y ⊆ K o ∪ Ux }. Then

U0 is open, since f : (t, s) ’ ty(s) being smooth, and f (t, 0) ∈ K o ∪ Ux for

t ∈ [0, 1], implies that a δ > 0 exists such that f (t, s) ∈ K o ∪ Ux for all |s| < δ

and ’δ < t < 1 + δ. The set U0 is star shaped, since y ∈ U0 and s ∈ [0, 1] implies

that t(x + s(y ’ x)) ∈ [x, t y] for some t ∈ [0, 1], hence lies in K o ∪ Ux . The set U0

contains x, since [0, 1]x = {x} ∪ [0, 1)x ⊆ {x} ∪ K o . Finally, U0 has the required

property, since z ∈ [0, 1]y for y ∈ U0 implies that [0, 1]z ⊆ [0, 1]y ⊆ K o ∪ Ux , i.e.

z ∈ U0 .

˜ ˜

Furthermore, we may assume that for x + iy ∈ Ux and t ∈ [0, 1] also x + ity ∈ Ux

˜ ˜

(replace Ux by {x + iy : x + ity ∈ Ux for all t ∈ [0, 1]}).

˜ ˜

Now let U1 and U2 be two such domains around x1 and x2 , with corresponding

˜ ˜

extensions f1 and f2 . Let x + iy ∈ U1 © U2 . Then x ∈ U1 © U2 and [0, 1]x ⊆ K o ∪ Ui

for i = 1, 2. If x ∈ K o we are done, so let x ∈ K o . Let t0 := inf{t > 0 : tx ∈ K o }.

/ /

Then t0 x ∈ Ui for i = 1, 2 and by taking t0 a little smaller we may assume that

x0 := t0 x ∈ K o © U1 © U2 . Thus, fi = f on [x0 , xi ] and the fi are real analytic on

[x0 , x] for i = 1, 2. Hence, f1 = f2 on [x0 , x] and thus f1 = f2 on [x, x + iy] by the

1-dimensional uniqueness theorem.

That the result corresponding to (24.8) is not true for manifolds with real analytic

boundary shows the following

25.10. Example. No real analytic extension exists.

Let I := {t ∈ R : t ≥ 0}, E := C ω (I, R), and let ev : E — R ⊇ E — I ’ R be the

real analytic map (f, t) ’ f (t). Then there is no real analytic extension of ev to a

neighborhood of E — I.

Proof. Suppose there is some open set U ⊆ E — R containing {(0, t) : t ≥ 0} and

a C ω -extension • : U ’ R. Then there exists a c∞ -open neighborhood V of 0

and some δ > 0 such that U contains V — (’δ, δ). Since V is absorbing in E, we

have for every f ∈ E that there exists some µ > 0 such that µf ∈ V and hence

1

µ •(µf, ·) : (’δ, δ) ’ R is a real analytic extension of f . This cannot be true, since

there are f ∈ E having a singularity inside (’δ, δ).

25.10

25.12 25. Real analytic mappings on non-open domains 259

The following theorem generalizes (11.17).

25.11. Theorem. Mixing of C ∞ and C ω .

Let (E, E ) be a complete dual pair, let X ⊆ E, let f : R—X ’ R be a mapping that

extends for every B locally around every point in R—(X ©EB ) to a holomorphic map

C — (EB )C ’ C, and let c ∈ C ∞ (R, X). Then c— —¦ f ∨ : R ’ C ω (X, R) ’ C ∞ (R, R)

is real analytic.

Proof. Let I ⊆ R be open and relatively compact, let t ∈ R and k ∈ N. Now

¯

choose an open and relatively compact J ⊆ R containing the closure I of I. By

(1.8) there is a bounded subset B ⊆ E such that c|J : J ’ EB is a Lipk -curve in the

Banach space EB generated by B. Let XB denote the subset X © EB of the Banach

space EB . By assumption on f there is a holomorphic extension f : V — W ’ C

¯

of f to an open set V — W ⊆ C — (EB )C containing the compact set {t} — c(I). By

cartesian closedness of the category of holomorphic mappings f ∨ : V ’ H(W, C) is

holomorphic. Now recall that the bornological structure of H(W, C) is induced by

that of C ∞ (W, C) := C ∞ (W, R2 ). Furthermore, c— : C ∞ (W, C) ’ Lipk (I, C) is a

bounded C-linear map (see tyhe proof of (11.17)). Thus, c— —¦ f ∨ : V ’ Lipk (I, C)

is holomorphic, and hence its restriction to R © V , which has values in Lipk (I, R),

is (even topologically) real analytic by (9.5). Since t ∈ R was arbitrary we conclude

that c— —¦ f ∨ : R ’ Lipk (I, R) is real analytic. But the bornology of C ∞ (R, R) is

generated by the inclusions into Lipk (I, R), by the uniform boundedness principles

(5.26) for C ∞ (R, R) and (12.9) for Lipk (R, R), and hence c— —¦ f ∨ : R ’ C ∞ (R, R)

is real analytic.

This can now be used to show cartesian closedness with the same proof as in (11.18)

for certain non-open subsets of convenient vector spaces. In particular, the previous

theorem applies to real analytic mappings f : R — X ’ R, where X ⊆ E is convex

with non-void interior. Since for such a set the intersection XB with EB has the

same property and since EB is a Banach space, the real analytic mapping is the

germ of a holomorphic mapping.

25.12. Theorem. Exponential law for real analytic germs.

Let K and L be two convex subsets with non-empty interior in convenient vector

spaces. A map f : K ’ C ω (L, F ) is real analytic if and only if the associated

ˆ

mapping f : K — L ’ F is real analytic.

Proof. (’) Let c = (c1 , c2 ) : R ’ K — L be C ± (for ± ∈ {∞, ω}) and let ∈ F .

ˆ

We have to show that —¦ f —¦ c : R ’ R is C ± . By cartesian closedness of C ± it is

ˆ

enough to show that the map —¦ f —¦ (c1 — c2 ) : R2 ’ R is C ± . This map however is

associated to — —¦ (c2 )— —¦ f —¦ c1 : R ’ K ’ C ω (L, F ) ’ C ± (R, R), hence is C ± by

assumption on f and the structure of C ω (L, F ).

(⇐) Let conversely f : K — L ’ F be real analytic. Then obviously f (x, ·) :

L ’ F is real analytic, hence f ∨ : K ’ C ω (L, F ) makes sense. Now take an

arbitrary C ± -map c1 : R ’ K. We have to show that f ∨ —¦ c1 : R ’ C ω (L, F )

is C ± . Since the structure of C ω (L, F ) is generated by C β (c1 , ) for C β -curves

25.12

260 Chapter V. Extensions and liftings of mappings 25.13

c2 : R ’ L (for β ∈ {∞, ω}) and ∈ F , it is by (9.3) enough to show that

C β (c2 , ) —¦ f ∨ —¦ c1 : R ’ C β (R, R) is C ± . For ± = β it is by cartesian closedness of

C ± maps enough to show that the associate map R2 ’ R is C ± . Since this map is

just —¦ f —¦ (c1 — c2 ), this is clear. In fact, take for γ ¤ ±, γ ∈ {∞, ω} an arbitrary

C γ -curve d = (d1 , d2 ) : R ’ R2 . Then (c1 — c2 ) —¦ (d1 , d2 ) = (c1 —¦ d1 , c2 —¦ d2 ) is C γ ,

and so the composite with —¦ f has the same property.

It remains to show the mixing case, where c1 is real analytic and c2 is smooth or

conversely. First the case c1 real analytic, c2 smooth. Then —¦ f —¦ (c1 — Id) :

R — L ’ R is real analytic, hence extends to some holomorphic map by (25.9), and

by (25.11) the map

C ∞ (c2 , ) —¦ f ∨ —¦ c1 = c— —¦ ( —¦ f —¦ (c1 — Id))∨ : R ’ C ∞ (R, R)

2

is real analytic. Now the case c1 smooth and c2 real analytic. Then —¦ f —¦ (Id —c2 ) :

˜

K — R ’ R is real analytic, so by the same reasoning as just before applied to f

˜

de¬ned by f (x, y) := f (y, x), the map

˜ ˜

C ∞ (c1 , ) —¦ (f )∨ —¦ c2 = c— —¦ ( —¦ f —¦ (Id —c2 ))∨ : R ’ C ∞ (R, R)

1

is real analytic. By (11.16) the associated mapping

˜ ˜

(c— —¦ ( —¦ f —¦ (Id —c2 ))∨ )∼ = C ω (c2 , ) —¦ f —¦ c1 : R ’ C ω (R, R)

1

is smooth.

The following example shows that theorem (25.12) does not extend to arbitrary

domains.

25.13. Example. The exponential law for general domains is false.

’2

Let X ⊆ R2 be the graph of the map h : R ’ R de¬ned by h(t) := e’t for

t = 0 and h(0) = 0. Let, furthermore, f : R — X ’ R be the mapping de¬ned by

r

f (t, s, r) := t2 +s2 for (t, s) = (0, 0) and f (0, 0, r) := 0. Then f : R — X ’ R is real

analytic, however the associated mapping f ∨ : R ’ C ω (X, R) is not.

Proof. Obviously, f is real analytic on R3 \ {(0, 0)} — R. If u ’ (t(u), s(u), r(u))

is real analytic R ’ R — X, then r(u) = h(s(u)). Suppose s is not constant and

t(0) = s(0) = 0, then we have that r(u) = h(un s0 (u)) cannot be real analytic, since

it is not constant but the Taylor series at 0 is identical 0, a contradiction. Thus,

s = 0 and r = h —¦ s = 0, therefore u ’ f (t(u), s(u), r(u)) = 0 is real analytic.

Remains to show that u ’ f (t(u), s(u), r(u)) is smooth for all smooth curves

h(s(u))

(t, s, r) : R ’ R — X. Since f (t(u), s(u), r(u)) = t(u)2 +s(u)2 it is enough to show

that • : R2 ’ R de¬ned by •(t, s) = th(s)2 is smooth. This is obviously the case,

2 +s

since each of its partial derivatives is of the form h(s) multiplied by some rational

function of t and s, hence extends continuously to {(0, 0)}.

Now we show that f ∨ : R ’ C ω (X, R) is not real analytic. Take the smooth curve

c : u ’ (u, h(u)) into X and consider c— —¦f ∨ : R ’ C ∞ (R, R), which is given by t ’

25.13

26.3 26. Holomorphic mappings on non-open domains 261

(s ’ f (t, c(s)) = th(s)2 ). Suppose it is real analytic into C([’1, +1], R). Then it has

2 +s

an tn ∈ C([’1, +1], R).

to be locally representable by a converging power series

∞

So there has to exist a δ > 0 such that an (s)z n = h(s) k=0 (’1)k ( z )2k converges

s2 s

for all |z| < δ and |s| < 1. This is impossible, since at z = si there is a pole.

26. Holomorphic Mappings on Non-Open Domains

In this section we will consider holomorphic maps de¬ned on two types of convex

subsets. First the case where the set is contained in some real part of the vector

space and has non-empty interior there. Recall that for a subset X ⊆ R ⊆ C the

space of germs of holomorphic maps X ’ C is the complexi¬cation of that of germs

of real analytic maps X ’ R, (11.2). Thus, we give the following

26.1. De¬nition. (Holomorphic maps K ’ F )

Let K ⊆ E be a convex set with non-empty interior in a real convenient vector space.

And let F be a complex convenient vector space. We call a map f : EC ⊇ K ’ F

holomorphic if and only if f : E ⊇ K ’ F is real analytic.

26.2. Lemma. Holomorphic maps can be tested by functionals.

Let K ⊆ E be a convex set with non-empty interior in a real convenient vector

space. And let F be a complex convenient vector space. Then a map f : K ’ F

is holomorphic if and only if the composites —¦ f : K ’ C are holomorphic for all

∈ LC (E, C), where LC (E, C) denotes the space of C-linear maps.

Proof. (’) Let ∈ LC (F, C). Then the real and imaginary part Re , Im ∈

LR (F, R) and since by assumption f : K ’ F is real analytic so are the composites

Re —¦ f and Im —¦ f , hence —¦ f : K ’ R2 is real analytic, i.e. —¦ f : K ’ C is

holomorphic.

(⇐) We have to show that —¦ f : K ’ R is real analytic for every ∈ LR (F, R).

So let ˜ : F ’ C be de¬ned by ˜(x) = i (x) + (ix). Then ˜ ∈ LC (F, C), since

i ˜(x) = ’ (x) + i (ix) = ˜(ix). Note that = Im —¦ ˜. By assumption, ˜—¦ f : K ’ C

is holomorphic, hence its imaginary part —¦ f : K ’ R is real analytic.

26.3. Theorem. Holomorphic maps K ’ F are often germs.

Let K ⊆ E be a convex subset with non-empty interior in a real Fr´chet space E

e

and let F be a complex convenient vector space such that F carries a Baire topology

as required in (25.1). Then a map f : EC ⊇ K ’ F is holomorphic if and only if

˜˜ ˜

it extends to a holomorphic map f : K ’ F for some neighborhood K of K in EC .

˜˜

Proof. Using (25.9) we conclude that f extends to a holomorphic map f : K ’ FC

˜

for some neighborhood K of K in EC . The map pr : FC ’ F , given by pr(x, y) =

x + iy ∈ F for (x, y) ∈ F 2 = F —R C, is C-linear and restricted to F — {0} = F it

˜˜

is the identity. Thus, pr —¦f : K ’ FC ’ F is a holomorphic extension of f .

˜˜ ˜

Conversely, let f : K ’ F be a holomorphic extension to a neighborhood K of K.

˜

So it is enough to show that the holomorphic map f is real analytic. By (7.19) it

26.3

262 Chapter V. Extensions and liftings of mappings 26.8

is smooth. So it remains to show that it is real analytic. For this it is enough to

˜

consider a topological real analytic curve in K by (10.4). Such a curve is extendable

˜˜

to a holomorphic curve c by (9.5), hence the composite f —¦ c is holomorphic and its

˜

˜

restriction f —¦ c to R is real analytic.

26.4. De¬nition. (Holomorphic maps on complex vector spaces)

Let K ⊆ E be a convex subset with non-empty interior in a complex convenient

vector space. And map f : E ⊇ K ’ F is called holomorphic i¬ it is real analytic

and the derivative f (x) is C-linear for all x ∈ K o .

26.5. Theorem. Holomorphic maps are germs.

Let K ⊆ E be a convex subset with non-empty interior in a complex convenient

vector space. Then a map f : E ⊇ K ’ F into a complex convenient vector space

F is holomorphic if and only if it extends to a holomorphic map de¬ned on some

neighborhood of K in E.

Proof. Since f : K ’ F is real analytic, it extends by (25.9) to a real analytic map

˜

f : E ⊇ U ’ F , where we may assume that U is connected with K by straight

˜

line segments. We claim that f is in fact holomorphic. For this it is enough to

show that f (x) is C-linear for all x ∈ U . So consider the real analytic mapping

g : U ’ F given by g(x) := if (x)(v) ’ f (x)(iv). Since it is zero on K o it has to

be zero everywhere by the uniqueness theorem.

26.6. Remark. (There is no de¬nition for holomorphy analogous to (25.7))

In order for a map K ’ F to be holomorphic it is not enough to assume that all

composites f —¦ c for holomorphic c : D ’ K are holomorphic, where D is the open

unit disk. Take as K the closed unit disk, then c(D) © ‚K = φ. In fact let z0 ∈ D

then c(z) = (z ’ z0 )n (cn + (z ’ z0 ) k>n ck (z ’ z0 )k’n’1 ) for z close to z0 , which

covers a neighborhood of c(z0 ). So the boundary values of such a map would be

completely arbitrary.

26.7. Lemma. Holomorphy is a bornological concept.

Let T± : E ’ E± be a family of bounded linear maps that generates the bornology

on E. Then a map c : K ’ F is holomorphic if and only if all the composites

T± —¦ c : I ’ F± are holomorphic.

Proof. It follows from (25.6) that f is real analytic. And the C-linearity of f (x)

can certainly be tested by point separating linear functionals.

26.8. Theorem. Exponential law for holomorphic maps.

Let K and L be convex subsets with non-empty interior in complex convenient vector

spaces. Then a map f : K — L ’ F is holomorphic if and only if the associated

map f ∨ : K ’ H(L, F ) is holomorphic.

Proof. This follows immediately from the real analytic result (25.12), since the

C-linearity of the involved derivatives translates to each other, since we obviously

have f (x1 , x2 )(v1 , v2 ) = evx2 ((f ∨ ) (x1 )(v1 )) + (f ∨ (x1 )) (x2 )(v2 ) for x1 ∈ K and

x2 ∈ L.

26.8

263

Chapter VI

In¬nite Dimensional Manifolds

27. Di¬erentiable Manifolds . . . . . . . . . . . . . . . . . . . . . 264

28. Tangent Vectors ........... . . . . . . . . . . . . . 276

29. Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . 287

30. Spaces of Sections of Vector Bundles .. . . . . . . . . . . . . . 293

31. Product Preserving Functors on Manifolds . . . . . . . . . . . . . 305

This chapter is devoted to the foundations of in¬nite dimensional manifolds. We

treat here only manifolds described by charts onto c∞ -open subsets of convenient

vector spaces.

Note that this limits cartesian closedness of the category of manifolds. For ¬nite

dimensional manifolds M, N, P we will show later that C ∞ (N, P ) is not locally con-

tractible (not even locally pathwise connected) for the compact-open C ∞ -topology

if N is not compact, so one has to use a ¬ner structure to make it a manifold

C∞ (N, P ), see (42.1). But then C ∞ (M, C∞ (N, P )) ∼ C ∞ (M — N, P ) if and only if

=

∞

N is compact see (42.14). Unfortunately, C (N, P ) cannot be generalized to in¬-

nite dimensional N , since this structure becomes discrete. Let us mention, however,

that there exists a theory of manifolds and vector bundles, where the structure of

charts is replaced by the set of smooth curves supplemented by other requirements,

where one gets a cartesian closed category of manifolds and has the compact-open

C ∞ -topology on C ∞ (N, P ) for ¬nite dimensional N , P , see [Seip, 1981], [Kriegl,

1980], [Michor, 1984a].

We start by treating the basic concept of manifolds, existence of smooth bump

functions and smooth partitions of unity. Then we investigate tangent vectors

seen as derivations or kinematically (via curves): these concepts di¬er, and we

show in (28.4) that even on Hilbert spaces there exist derivations which are not

tangent to any smooth curve. In particular, we have di¬erent kinds of tangent

bundles, the most important ones are the kinematic and the operational one. We

treat smooth, real analytic, and holomorphic vector bundles and spaces of sections

of vector bundles, we give them structures of convenient vector spaces; they will

become important as modeling spaces for manifolds of mappings in chapter IX.

Finally, we discuss Weil functors (certain product preserving functors of manifolds)

as generalized tangent bundles. This last section is due to [Kriegl, Michor, 1997].

264 Chapter VI. In¬nite dimensional manifolds 27.3

27. Di¬erentiable Manifolds