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