Chapter 2. Analytic Spaces

65

CHAPTER 2

ANALYTIC SPACES

The analytic spaces that we are about to consider generalizes the idea of reduced analytic spaces of finite dimension.

We

shall regard them a s ringed spaces a s defined, for instance, in [FI]

.

However, as we shall limit ourselves to the case of

reduced spaces,

we concentrate on ringed spaces (X,

o,

is a topological space and germs of continuous

(I:

xo)

where X

is a subsheaf of the sheaf o f

X

- v a l u e d functions o n

.

These spaces

(cf. [ D D ] ) ; this is

are sometimes referred to a s Cartan spaces

the name that we shall use in order to distinguish them from more general ringed spaces.

xO(U) or T ( U , ,O) to denote defined o n a n open subset U of X, o, a t the point a

As usual we shall use the notation the space of sections of and

xo(a)

o,

for the fibre of

.

We recall that a morphism from a Cartan space Cartan space (Y,

is a continuous map IP

such that, for each point to

,

yO(p(a))

a

the composition

of Q

X 9

(X,

from

and each germ belongs to

xO) X CL

to a to

Y

belonging

.

,O(a)

Y is a subset of the Cartan space (X, x O ) we can define the induced Cartan space (Y, where the germs of yo are Although this the restrictions to Y of the germs of ,o to Y is not the same thing as the restriction of the sheaf o, (in sheaf-theoretical terms), the sheaf yo will also be denoted by I n the language of the theory of ringed xO,y If

.

spaces,

YO

.

is obtained by forming the quotient of

the ideal of germs of

o,

that vanirh o n

Let us thus consider a locally convex space of a map defined o n an open subset o f is clearly a local property.

Y

.

xo

by

E ; the analyticity

E , with values in

(c

,

This allows us to equip each open

CHAPTER 2

66

set

C

for

uO

E

of

with the structure of a Cartan space by taking

uo)

(U,

The spaces

-

VO)

V

Suppose now that

F

space

(U',

U,

open subset of

and that

u ~ o ) is

(V, ,O) ci

is a map from

U

(U,

,,o)

that

p

V

an

V ;

If 9

is a

to the Cartan space

V , and therefore

to

F

a from

In particular this is true for

to

.

a E F'

E,

is continuous and weakly analytic, and hence by proposi-

9

tion 1 . 6 to

U

to

Also, for each analytic map

i s analytic.

q

Thus

.

F

to

U'

We note that, for

the Cartan space induced by

then 9 is continuous from

U

&.

is an open subset of a locally convex 9

morphism from the Cartan space from

to

obtained in this way provide us with our

first examples of analytic spaces. (U,

U

the sheaf of germs of analytic functions from

9

i s analytic from

U

is analytic from and proposition

maps shows that

q

U to

1.10

to

F ;

.

F

Conversely, suppose

is continuous from

9

U

on the composition of analytic

(U,

is a morphism from

uo)

to (V,

vo) .

Therefore for these examples of analytic spaces, the notion of morphism coincides with that of analytic map. We next define the term model, or globally defined analytic space, to denote a Cartan space

(U, U L ) ) space and

U

from

X , where

on

X

U

(X,

xo)

which is induced by

i s an o p e n subset of a locally convex

is the inverse image o f

0

by a n analytic map

to a locally convex space.

In general, b y an analytic space we shall mean any Cartan space (X,

xL))

Ri

such that the induced spaces

mode 1 s

for which there exists a covering of

.

(ni,

X

b y open sets

iziO) are isomorphic to

It is clear that the reduced analytic spaces of the finite dimensional theory are analytic spaces according to the definition that we have just given. Generalizing from the case where we were dealing with open subsets o f locally convex spaces, we shall use the term of analytic map for all morphisms between analytic spaces.

With

this definition it becomes clear that the composition of two analytic maps is analytic.

67

ANALYTIC SPACES

(X, ,O)

Let

be a n analytic space and 9 a section of

X ;

defined o n an open subset of

IP

locally a s the restriction of a global section J, ,I)

where

U

xo

may then be looked upon of a sheaf

is a n open subset of a locally convex space.

,

O,

the definition of

J,

is analytic, and hence s o is

By

.

9

Conversely, if 8 is a n analytic map from a n open subset o f to

(C

,

then since

IdC

8 = Id

conclude that

,

defined o n a n open subset of

X

8

o,

Thus the sections of

CO

is a section of the sheaf is a section of

the analytic maps from that open set to

xo

-

X

we

are

C ; they form a sub-

algebra of the algebra of continuous functions from that open set to

o(X)

and we denote by

C

.

xO

sections o f

In summary, to say that a map to another analytic space

IP

(Y,

yo)

the algebra of global

,o)

from a n analytic space ( X , is analytic means that

9

is continuous and that, for every analytic function a

from an a: 9 is This i s equivalent to saying that 9 is analytic o n 9-'(U) , a IP is a continuous and that, for each germ a: of germ of O, open subset

U

of

Y

.

to

C

,

the composition

.

yo

Remark Let us consider a locally convex space

E and a vector subspace E W e can equip F with two Cartan space structures, one being that endowed o n it by FI) the sheaf of C-valued analytic maps o n F and the other being that induced by Ei) If F is closed in E then this latter is in fact a n analytic space structure which is even a model since F is the inverse image of 0 by the canonical surjection from E onto E/F which is obviously analytic (as it is continuous and linear).

F

of

.

.

It is however important to note that these structures c a n be different. Since the restriction of a n analytic function is analytic we clearly have EOIF C Fo , but these two sheaves can be distinct.

I n other words there can exist analytic func-

tions o n F which cannot be continued (even locally) to E More precisely, the existence o f such a n extension, which is

.

68

CHAPTER 2

classical in the case of linear forms (the Hahn-Banach theorem), is lacking already at the level of quadratic forms. The following proposition shows that this situation does not

F

arise when

2.1.

iez

.

E

is dense in

Proposition E

be a ':ocaZly convex space.

extended t o

a n c l y t i c s t r u c t u r e on

Every germ i n

.

O(2)

( u n i q u e ) germ in

J

L)(E)

can b e

I n other words,

the

2.

is i n d u c e d b y t h a t of t h e c o m p l e t i o n

E

Proof

Let 7

be a n analytic map defined o n a n open neighbourhood of a

point

a

E , with values in

in

lp"

ence of a n analytic map restriction to

E

continuous semi-norm

on

p

We must prove the exist-

o n a n open subset of

coincides with 7

E

2

whose

o n a neighbourhood of

.

a

we deduce the existence o f a

From the proof o f proposition 1 . 6 Bp(a,l),

.

0

such that, on the open ball

is well-defined and equal to the sum o f a series

$

Z: qn(x- a) of homogeneous polynomials which verify Ivnl 4 M.p But then p and the q n c a n be extended continuously to give

i and

Gn

on

allows u s to define

Bp(a,p)

.

This

on

q

where

0 <

<

Mpn

I

p <

,

I

.

O n these balls we have

(G,(x - a) series. We deduce from this that restriction of A

M.6"

; to do this it suffices to prove i t for all the balls

Bp(a,l)

qn

,<

B-(a.1) a s the sum of the series Q We must n o w prove the analyticity of this sum o n

.

Z pn(x- a)

IGnl

and we still have

n

i

and s o w e have normal convergence of the A

IP

is continuous and that the

to every affine line is holomorphic (since

is a polynomial)

; proposition 1 . 3

n o w implies that

is

9

analytic. The uniqueness of the germ of such a n extension follows from the fact that the closure in point

a

a

E

of a neighbourhood in

is also a neighbourhood o f

a

in

.

E

of a

.

ANALYTIC SPACES

69

Remarks

11

Using this proposition it is easy to show that every

analytic function

9

an extension to a n open subset we do not necessarily have (cf. [ DI]

and

21

E

has

However, when

U =

defined o n an open subset A

U

-

6

.

of

A

E

and

of

U

A

U can vary with

E,

9

[NO] ).

This proposition has no analogue for the case of

analytic maps with values in locally convex spaces. defined on

E

Such a map

may thus not possess a n extension (even local)

(cf. [ D l ] ) .

to

From the proposition we immediately deduce:

Corollary

In o r d e r t h a t a map f r o m an a n a l y t i c s p a c e t o a l o c a l l y c o n v e x space

E

s o a s an

2.2.

Let

b e a n a l y t i c i t i s n e c e s s a r y and s u f f i c i e n t t h a t i t be i - v a l u e d map.

Proposition E b e a l o c a l l y c o n v e x s p a c e and

I a right d i r e c t e d f a m i l y o f s em i - nor m s t h a t d e f i n e t h e t o p o l o g y of E ; f o r e a c h i n d e x i we d e n o t e b y Ei t h e q u o t i e n t s p a c e E/p;’(O) e q u i p p e d w i t h t h e q u o t i e n t norm i n d u c e d b y pi and by si t h e c a n o n i c a l s u r j e c t i o n f r o m E o n t o Ei Then, f o r e v e r y (pi)i

.

point a an i n d e x

a = B,

of i

E and e v e r y germ a and a germ f3 in

si,

i n ,O(a) O(si(a)) Ei

,, t h e r e e x i s t s such t h a t

Proof Let

9

be a n analy.tic map defined o n an open neighbourhood o f

with values in

E

.

The proof of proposition 1.6

yields a

continuous semi-norm p o n E and a decomposition @(x) = C P n ( x - a ) which is valid o n the open ball B (a,l), P

a

CHAPTER 2

70

and such that the

irnj

.

4 Mpn

satisfy a n estimate of the form

9n

suppose that

polynomial of degree Mpn(qi)"

.

=

-

:;ln(y

Z

on e a c h ball

.

n

,

pi

E

= >3

q ( x )

vn

If

3

is the germ o f

a

then we have

E

JI CI

=

I/p)

12

,

-1

; a s in the preceding

P

is analytic.

Thus, for

we have

[ si(x- a) ]

at

si(a)

6

si

=

IJ [ si(x) I

and i f

CI

,

i.e.

is the germ of

P

a s required.

a(E,E')

)

and l e t IP

be a U

. . .,fn .

]

where t h e

.

fl,. .,fn

equipped w i t h the

E - v a l u e d a n a l y t i c func-

of

E

i n a n e i g h b o u r h o o d of e a c h p o i n t of

y a r i a b Se s

is the quotient

1.

topclogy

9 [ f!,

qi

be a weak l o c a 2 2 y c o n v e x s p a c e ( i . e . ,

f o r m s and

can

by setting

r <

where

t i o n d e f i n e d on an o p e n s u b s e t 9

qn

is a homogeneous

T h i s series then converges normally

Bp(a,l)

at

Let

Qn

.

E ] O , m [

show that the

where

I n addition, if

.

q n ( x - a) =

Corollary

Mpn(pi)n

Bqi(si(a),

si(a))

=

p

where

we still have estimates of the form

Bqi(si(a),r)

Bpi(a, I / p )

ppi

si

proposition we deduce that

x

pi

is defined by the

is right directed we can

These estimates allow u s to define a function

5 o n the open ball $(y)

<

Iqnl 9 n = JJ,

be factorized as norm induced by

pi

is of the form

p

Then the estimates

/ a n /<

E

Since the topology o n

and since the family of the

.

We c a n t h e n e x p r e s s

U

i n the form

are continuous l i n e a r

9 is a n a n a l y t i c f u n c t i o n of a f i n i t e number of

Indeed, we can apply the proposition taking for the the continuous semi-norms on finite-dimensional.

E

; the spaces

E;

pi

all

are then all

ANALYTIC SPACES Corollary

71

2.

With t h e n o t a t i o n of t h e p r o p o s i t i o n ,

from a n a n a l y t i c s p a c e

xo)

(X,

i n o r d e r t h a t a map

to

be a n a l y t i c i t i s

E

n e c e s s a r y and s u f f i c i e n t t h a t f o r e a c h i n d e x

si

be a n a l y t i c from

P

X

to

i

,

t h e map

E;.

Necessity follows from the fact that the continuous, are analytic.

P

si

,

being linear and

Conversely, the condition stated in

the corollary clearly implies that

is continuous, and we

9

infer from the proposition that it is a sheaf morphism.

This corollary c a n be improved in the case of weak spaces.

In

fact, we have the following:

Proposition

2.3.

Let

(X,

space to

E

.

xo)

E

b e an a n a l y t i c s p a c e ,

f i . e . equipped w i t h Then, in o r d e r t h a t

s u f f i c i e n t that, for every f u n c t i o n from

X

to

.

E

VJ

a weak ZocaZZy c o n v e x

I

o(E,E')

and

9

a map f r o m

X

b e a n a Z y t i c i t i s n e c e s s a r y and

,

a in E '

a

9

be a n a n a l y t i c

Proof The preceding corollary says that

L

~p

9

is analytic if and only if

is analytic for each linear continuous map

to o n e of the spaces

(En

.

The particular case

that the condition is necessary.

L n= 1

E

from shows

Conversely, in the case when

L

the condition is verified, we take a continuous linear map from E to En and show that L 9 question is local we c a n suppose that

is analytic. Since the is a globally (X, xc')

defined analytic space and therefore that it is induced by a space of the form

(U,

locally convex space. 01,

a2,..., an

are elements of

.

,o)

where

U

is an open subset of a

Let us denote the components of

By hypothesis, ai P , a2 P,..., an O(X) Subject t o shrinking X and U

.

L

by 9

we

CHAPTER 2

72

can suppose t h e r e f o r e t h a t they a r e the r e s t r i c t i o n s of a n a l y t i c

2i

functions

5 to

s2,

(El,

=

an

Thus

...,

Bn)

to

U

that

it

L

.

E

Then t h e map

to

X

U 1.6).

i s a n a l y t i c (.f. p r o p o s i t i o n

appears a s the restriction

@

3 ;

from

i s c o n t i n u o u s and weakly a n a l y t i c from

and i t f o l l o w s L

map

. . . 6,

d?,

,

the analytic

of

is analytic as required.

9

Remarks

I/

Let

b e a map f r o m a n a n a l y t i c s p a c e (X, xc))

q

l o c a l l y convex space say that

shall (1.

E

.

i s weakly a n a l y t i c i f ,

@

for each

to a

1.5

Generalizing the definition

in

(1

we

,

El

is analytic.

q

The preceding p r o p o s i t i o n

and o n l y i f

shows t h a t

~p

i s weakly a n a l y t i c i f

i t i s a n a l y t i c w h e n we g i v e

E

t h e weak t o p o l o g y

o!E,E')

.

However

the r e s u l t o b t a i n e d f o r open s u b s e t s of

spaces does not generalize s h a l l s e e an example of i s not a n a l y t i c .

l o c a l l y convex

t o g e n e r a l a n a l y t i c s p a c e s a n d we

a c o n t i n u o u s w e a k l y a n a l y t i c map w h i c h

N e v e r t h e l e s s we s h a l l e x a m i n e

c e r t a i n i m p o r t a n t c a s e s where

in detail

t h i s p a t h o l o g i c a l phenomenon

cannot occur. The r e s u l t o f

2/

proposition 2.3

as saying that in the corollary 2 need n o t r e q u i r e t h a t when t h e

E i

a r e of

the family

c a n be i n t e r p r e t e d

,

of p r o p o s i t i o n 2 . 2

f i n i t e dimension.

we

be r i g h t d i r e c t e d

(pi)i E I

I n f a c t t h e problem of

g o i n g from a n a r b i t r a r y f a m i l y t o a r i g h t d i r e c t e d f a m i l y l e a d s

t o the f o l l o w i n g problem:

let

from a n a n a l y t i c s p a c e

(x,

(resp.

(a,@)

I s t h e map

F).

T h e a n s w e r i s y e s when

X

,O)

a(resp.

B ) b e a n a n a l y t i c map

t o a l o c a l l y convex space from

X

to

i s an open s u b s e t of a l o c a l l y

convex s p a c e as i s p r o v e n by t h e c h a r a c t e r i z a t i o n s tion

1.6

.

The a n s w e r i s s t i l l y e s i f

t h e maps

he e x t e n d e d l o c a l l y t o a n o p e n s u b s e t o f However,

even i f

i t may n o t e x i s t w h e n

i n proposi-

3

and

Q

can

a l o c a l l y convex space.

s u c h a n e x t e n s i o n e x i s t s when

f i n i t e dimensional,

E

analytic ?

E x F

E

E and

and F

F have

are

ANALYTIC SPACES

arbitrary dimension. that

a +

73

E = F , it is possible and B are. These

I n particular, when

is not analytic even though

ci

pathologies are due to the fact that structure considered for the analytic spaces is very weak; in particular it is weaker "espace K - fonctiS" ( K - functored space)

than the structure of introduced by

,

A. D O U A D Y (cf. [ D O ] )

for which analytic maps

to locally convex spaces can always be extended locally.

A very pathological example

We intend to construct a n example of a n analytic space (X, ci.0

and analytic maps not analytic.

X

from

R 2 such that

to

It is clear that in this case, since a + f3

are continuous and weakly analytic,

ci ci

(x,y> (a,f3)

analytic, the map

+

( x + y)

from

X

from

to

R2

and

is f3

is continuous and

weakly analytic, without nevertheless being analytic. more, since the map

+ f3

xO)

x

k 2 x 1'

Further-

to R 2 is

R 2 is not analytic

although each o f its components is. This construction, which is fairly long, will be carried out in several steps.

*

(PI

-

...,

If a 1 , a 2 , an are elements of A and 9 is the germ of a holomorphic function defined i n a neighbourhood of the point values in to

A

.

We next define o n open subset such that, if

(E

U

X

(al(a), )

az(a),.

then the germ

the sheaf

O,

.., an(a))

of

~ ( C I I , C X ~ ,..,an) .

We shall prove :

E

U

,

belongs

whose sections over an

are those continuous functions from a

(In (with

their germs at the point

a

U

to

0:

belong to A.

CHAPTER 2

74

The C a r t a n s p a c e We remark t h a t p r o p e r t y (P) algebra of

a

to

xo)

(X,

ensures that

a ; it follows that the set

a t

O,

sections of

[ O(X)]*

proposition 2 . 3

proves t h a t

t h e s e t of c h a r a c t e r s of

.

X

to

, O(X)

a

point

and

O(X) a

of

that 2,

=

cl.

d

3.

I*

defined

are continuous

contains a l l the

which are c o n s t a n t on a

A(X)

,

S

.

A(X)

X

; hence

= S

.

a(a)

a

We c l e a r l y h a v e

7 the

and d e n o t e by

.

,

X

because

It

follows that

is invertible i n

an

s e c t i o n of

the

2

be i n

ai

are zero;

3.

if

Ker

x.

is h o l o m o r p h i c

l/z

I/a

,

o(X)

CI E

X

.

belongs to

which i m p l i e s

3,

the set

We s h a l l prove t h a t the i n t e r -

i s nonempty.

T h i s is c l e a r l y t r u e i f

t h i s is not

subject t o changing the indices,

ideal

t h e germ a t t h e

A

We d e d u c e f r o m t h i s t h a t i f

. .. ,

ri(a)

o(X)

i s a nonempty c l o s e d s u b s e t o f

L e t . x j , cxi, the

[ o(X)

to

is

S = F-’(O)

separates the points of

belongs t o

I/>

and t h u s t h a t

t - ’ ( O )

X

to

We h a v e t h u s a

.

o(X)

d o e s n o t v a n i s h on

a

on a neighbourhood of

O(X)

Let

T h e r e f o r e A d e f i n e s a homeomorphism f r o m t h e

be an element of :x E

E

where

o(X) E

o n t o i t s image

X

,

,O)

from

Since

A(X) c S ; l e t us prove t h a t

If

global

from

S i n c e t h e e l e m e n t s of

continuous f u n c t i o n s from

is injective.

(S,

the algebra

functions, A is continuous. neighbourhood of

F

is analytic.

F

x W 6,

L e t A b e t h e D i r a c map

compact s e t

of

w i t h t h e weak t o p o l o g y

We c a n d e f i n e a map

globally defined analytic space

= f(x)

O(X)

of

.

Let

X

t h e l o c a l l y convex s p a c e o b t a i n e d by endowing

~ ( [ ( ~ ( Y ) ] * , O(X))

A

O(X)

has t h e s t r u c t u r e of a u n i t a r y a l g e b r a . E

the algebraic dual

6,(f)

i s a u n i t a r y sub-

A

t h e a l g e b r a o f germs o f c o n t i n u o u s f u n c t i o n s from

u s d e n o t e by

by

i s an a n a l y t i c space.

all

t h e c a s e t h e n we m a y ,

a @ Zcrl

suppose t h a t

t h e r e e x i s t s a s t r i c t l y p o s i t i v e r e a l number

E

.

Then

such t h a t the

75

ANALYTIC SPACES

w =

set

>

lal(x)l

{ X E X

}

E

is a neighbourhood of

X

Urysohn's lemma n o w guarantees the existence of a function

X

continuous from hood of

a

,

o(X)

,

a

<

5

-

.

B1

=

X(x)

Bi .

we have 1

=

za2

a1

and

a

Za,

X

for

*

0

=

each a

a(x)

=

o(X),

in

S ,

a1

V

.

C Bi

...,n

7 and ,

ai

Bi ai(x)

take

then x

d

,

w

we conclude that

vanish simultaneously.

za

3

E

CI

are zero.

and, for

= E'

1 ,

a E

is nonempty since

E

3

,

is nonempty. For each

0 ; since

X(a)

=

a(x)

X(1)

=

c1

I

Let

x

in

o(X)

be a n

,

we have

we conclude that for

X

and hence

.

6,

=

X

induces a homeomorphism from

We shall now show that i t is i n fact an isomorphism

of Cartan spaces.

X

Since the topological space

the germ of a function defined o n all of

X

.

is compact,

x

every germ of a continuous function at the point

of

X

is

Furthermore, if

# a , then this function c a n be chosen to vanish in a neighbourhood of a I t follows that, for x € X , every germ

x

.

xo(x)

belonging to Then a

G

tion c1 =

G

is the germ of a n element

defines a continuous linear form o n to

A

S

is a n element of

o(S);

E

a

of

A

o(X).

whose restric-

since we have

we conclude that every germ belonging to

is the inverse image by

of a germ belonging to

El

1.

therefore proves that the intersection c1

Thus we have proven that A onto

2,3,

E

lai(x)12

=

ai

=

an

element of this intersection.

X(a)

i

belong to =

Blcxl(x)

Biai(x)

n...nz

The compactness o f of all the

ai

Thus, if

vanishes precisely when all the Hence

on

since 9 is holomorphic o n a

i f and only if all the

x E o

Since for we have

Z

equal to

a1 and, for

The products

= 0

a(x)

;

E

a

to

property (P) ensures that

~p

real positive values. and

a

Furthermore we can define

.

Let us then set Xi Ei

3

,

O(X).

E

from

Izl

al(a)

o(X)

belongs to

=

A;

~p

on

neighbourhood o f

B;

in

is continuous and possesses a zero germ at the

and

E

outside o ; for each a

1

and therefore

a continuous function 121

which vanishes on a neighbour-

and is equal to

1;

point

[0,1]

to

.

a

,o(x)

.

CHAPTER 2

76

Conversely, let to if

x

X

be a point of

B

Since A

and

6 a germ belonging 6 A and thus, If x = a , we k n o w

is continuous, s o is

.

xo(x)

belongs to

8 is the restriction of the germ of a n analytic function

that

E

on

x

C(&,). S # a ,

.

By corollary I

be written as

q(f1,

E

linear f o r m s on

..., fn)

and IP

3

.I

that

B

ai

= A

,O(X)

(X,,O)

are continuous

.

of

En

of O(X)

.

a

.

Each form

...,gn)

where

CX~

Thus, for every

(S,so).

denotes

Property (P) n o w proves

which means that

onto

f;

and we see that the

in

x

X , the A

are exactly the inverse images by

the elements o f from

..., fn

~ ( 2 1 ,

at the point

A E ,C(a)

elements o f

]

ai

c a n be written

the germ of

f1,

is the germ of an analytic function

corresponds to a n element germ

where

[ fl(&x),...,fn(6x)

at the point

, such a germ can

to proposition 2 . 2

A

of

is a n isomorphism

(X,xd)

This proves therefore that

i s an analytic space.

Let

E, F , G

E x F

t o

.

G

be locally convex spaces and We shall say that

P

P

a polynomial from

is decomposable if there

exists a continuous polynomial Q (resp. R) from 0

, and

have

x

G

in

P(u,v)

such that, for each (u,v)

Q(u).

=

.

R(v).x

in

We shall say that

E (resp. F) to

E P

F

x

,

we

i s of

finite type if it is the sum of a finite number o f decomposable polynomials. Finally we shall say that a germ of a n analytic map from

E

x

F

G

to

i s of finite .type if in its expansion

a s a series of polynomials only polynomials of finite type from

E x F

C

t o

Let then 7

occur. be the germ of an analytic map from

,

at the point (a,b) at the point space

H

.

and l e t $

q(a,b)

of

G

,

E x F

o

G

with values in the locally convex

We have the following series expansions:

$

to

be the germ of a n analytic map

Ip(a+u,b+~)

=

Z Rp(u,v)

ANALYTIC SPACES

and

Rp

77

is defined by

R (u,v)

L: ?jn

=

P

..

where the sum is taken over all families il +

integers which verify

(c)n

the n-linear map from extension of

Qn

to

I

p i 1 (u,v>, P ~ ~ ( u , v ) , .,P;,(u,v)

.

I

G

... + i n

..., in-

and where

p

=

to

il,

of nonzero

Qn

denotes

induced by the continuous

Since the sum which gives

Rp

involves only a finite number of indices, we conclude that, if the

are o f finite type, then s o are the

Pm

II, ,

is a germ of finite type, then s o is

3. C v n b t f i u c t i v n

Rp

.

.

Thus if

Ip

t h e example

06

Let us choose a pair

(G,B)

of continuous maps from

[0,1]

to

(I which verify :

ii)

iii) the intexval

(

Vx E [ 0,l ] )

<

IG(x)l

x

and

-

lg(x)l

<

x

for every nonzero natural number the image of

] n+l , [ 1

by the map

x

(d(x),

i(x))

is

not contained in any analytic hypersurface of an open subset of C 2 .

6(x)

We could take for example

Furthermore, let us denote by

2’

space base of the space

= x

sin

(ep)p

the set (resp.

(ep,eq)

B )

a(0) = 0

from

for [ 0,1]

and

the canonical Hilbert

of square summable sequences. Since

N2 is countable we can find a sequence such that the set of the

‘TT X

(fny gn)

(p,q) € I N 2 to

(resp. B ( 0 ) = 0 )

.

for

(fn, gn) n E IN

in

k 2 x 11’

is precisely

Let us then define a map

R 2 by :

,

and for

1 n+ 1

<

x

< -n1

y

a

CHAPTER 2

78

The m a p s s o d e f i n e d v e r i f y guarantees

# a ( x ) l l \< x

.

0

their continuity a t

the point

.&

of

f i n i t e type i f

2.

the point

outlined in

then

from

(0,O)

i s of

to

L2x f2

v

Consequently, of

C2

point

to

x

belongs t o

and

if

,

C of

t h e n t h e germ o f if

(resp.

x

= a

the point i n

from

0

R2 $

x

,

11’

i s t h e germ

t2 w i t h v a l u e s i n

of

to

a+ 5 k’

$

B-’(U))

a (resp.

to

$

A

,

CY

U

6 ) a t any

is continuous,

; i n o t h e r words,

(x,,O)

I n o r d e r t o show t h a t Q

,a = 0

[ O , l ]

=

a r e germs which b e l o n g t o

f3

@

a n a l y t i c maps f r o m

polynomial

X

i s a n a n a l y t i c map f r o m a n o p e n s u b s e t

$

a-’(IJ) A

The c o n s t r u c t i o n

It follows that i f

f i n i t e type.

o a

belongs

where q r u n s

.

1

(with

which depends o n l y on t h e

E

component of

of an a n a l y t i c f u n c t i o n a t t h e p o i n t

E , then

an

in

i s t h e g e r m o f a n a n a l y t i c map a t t h e

p

(or t h e second) q

v(a,B)

.

A

We r e m a r k t h a t i f first

of germs

A

may t h u s b e c a r r i e d o u t w i t h

1.

the results

C ) t h e n t h e germ $ ( P I , P Z , . . . q n )

has t h e p r o p e r t y (P) used i n

and t h i s s e t

point

i t s components a r e ,

(ql(a),qz(a), . . . , ~ n ( a ) )

I t follows t h a t t h e s e t

&

which a r e

C )

has the property :

values in

through

and t h e f a c t

a holomorphic f u n c t i o n i n a neighbourhood

germ o f of

which

germs of a n a l y t i c f u n c t i o n s a t

and o n l y i f

.&

show t h a t

(P’ )

. B

and

(with values i n

11’

11’x

x

S i n c e a germ w i t h v a l u e s i n a f i n i t e p r o d u c t

f i n i t e type.

is of of

the s e t of

of

(0,O)

IIB(x)ll d

Their continuity a t the

5

o t h e r p o i n t s f o l l o w s from t h a t of

Let us d e n o t e by

and

and i t

B

and

are

L*.

i s n o t a n a l y t i c we i n t r o d u c e t h e E

d e f i n e d by

Q(u)

=

IN(un)2.

79

ANALYTIC SPACES Q ( a +8 )

We shall show that We have

B)

Q(a+

= Q(a) +

is not analytic.

analytic,

Q(cu,B)

G(a,B) 0

at the point

,.k

P,

belongs

to

; thus it suffices to show

Then the germ of this map

A , and therefore there exists G(a,B)

of

0

is

P,

P,

z

is homogeneous of degree j

v ; the

in

Pi,j

i

in

E

[ O,E]

[CL(x),B(x)

1

=

z

Pi,j

-1

is a n integer greater than

n

- 1< x n+ 1

c

E

’i,m-

u

Each

i

and homogeneous of E

There

such that, for each

.

we therefore have, for

n

(G,g)

iii) concerning

the

vanish except for

pij(fn,gn) G(fn,gn)

1

[ cc(x),B(x)

1

-

The hypothesis to

9

we have

a If

.

where

being also of finite type.

then exists a strictly positive number x

9(cc,8)

are of finite type.

=

04 i
degree

is the B are

and

CL

We seek a contradiction by

where the

can be decomposed as

Pi,j

6

where

Since

as a series o f polynomials, we obtain

9

L: P,(u,v)

=

.

Q

is analytic.

such that the germ at

Expanding 9(u,v)

Q(B)

and

is not analytic.

supposing that in

Q(a)

are

so

I

that

,

Q ( B ) + 2 G(cc,B)

symmetric bilinear form induced by

.

now proves that all

Pll(fn,gn)

which is equal

In this way we see that except for a finite

number o f indices we have

o(fn,gn)

=

Pll(fn,gn)

.

It follows

that G(ep,eq) - P11 (ep,eq) is zero except for a finite number of indices. We deduce from this that there exists a natural number

N

such that the bilinear form

components of index less than ij(u,v)

-

=

Pll(U,V)

.

N +

G-P11

depends only o n

W e therefore have a relation = a u v i < N ij i j j
which proves that Q is a polynomial of finite type. Taking into account the homogeneous nature of the components we arrive

CHAPTER 2

80

a t a r e l a t i o n of the

“i

t h e form

and t h e

+i

G(u.v)

i s o v e r a f i n i t e number o f u

i n t h e i n t e r s e c t i o n of

u

we h a v e

tion since

( V v E !Z2)

Q(U,U)

=

=

Z

qi(u).

Jli(v)

where

a r e l i n e a r c o n t i n u o u s f o r m s a n d t h e sum indices.

We c a n t h e n f i n d a n o n z e r o

t h e k e r n e l s of (G(u,v> = 0

(null)*

+

o

.

the

q i

; f o r such a

which y i e l d s a c o n t r a d i c -