Since by (1) the set { bn,km : n ∈ N } is bounded, we can choose nm ∈ Nm such

that

1

|bnm ,km | ≥ 2 |bj,km | for j > nm

(3)

|bnm ,km | > mkm

We can choose also (nm ) strictly increasing, for if they were bounded we would get

1

|bnm ,km rkm | < C for some r > 0 and C by (2). But ( m )km ’ 0.

We pass now to the subsequence (bnm ) which we denote again by (bm ). We put

«

1 1

tj bj,km ·

(4) tm := sign .

4m

bm,km j<m

Assume now that b∞ = m tm bm converges weakly with respect to S to a holomor-

phic germ. Then its Taylor series is b∞ (z) = k≥0 b∞,k z k , where the coe¬cients

are given by b∞,k = m≥0 tm bm,k . But we may compute as follows, using (3) and

(4) :

|b∞,km | ≥ tj bj,km ’ |tj bj,km |

j>m

j¤m

8.10

9.1 9. Real analytic curves 97

tj bj,km + |tm bm,km |

= (same sign)

j<m

’ |tj bj,km | ≥

j>m

«

≥ 0 + |bm,km | · |tm | ’ 2 |tj |

j>m

mkm

1

= |bm,km | · ≥ .

3 · 4m 3 · 4m

So |b∞,km |1/km goes to ∞, hence b∞ cannot have a positive radius of convergence,

a contradiction. So the theorem follows for the space H({t}, C).

Let us consider now an arbitrary closed smooth submanifold A ⊆ C. By (8.8) the

projective cone H(N ⊇ A, C) ’ {H(N ⊇ {z}, C), z ∈ A} generates the bornology.

Hence, the result follows from the case where A = {0} by (5.25).

9. Real Analytic Curves

9.1. As for smoothness and holomorphy we would like to obtain cartesian closed-

ness for real analytic mappings. Thus, one should have at least the following:

f : R2 ’ R is real analytic in the classical sense if and only if f ∨ : R ’ C ω (R, R)

is real analytic in some appropriate sense.

The following example shows that there are some subtleties involved.

(s, t) ’ (st)1 +1 ∈ R is real analytic, whereas

Example. The mapping f : R2 2

there is no reasonable topology on C ω (R, R), such that the mapping f ∨ : R ’

C ω (R, R) is locally given by its convergent Taylor series.

Proof. For a topology on C ω (R, R) to be reasonable we require only that all eval-

uations evt : C ω (R, R) ’ R are bounded linear functionals. Now suppose that

∞

f ∨ (s) = k=0 fk sk converges in C ω (R, R) for small s, where fk ∈ C ω (R, R). Then

the series converges even bornologically, see (9.5) below, so f (s, t) = evt (f ∨ (s)) =

∞

fk (t) sk for all t and small s. On the other hand f (s, t) = k=0 (’1)k (st)2k for

|s| < 1/|t|. So for all t we have fk (t) = (’1)m tk for k = 2m, and 0 otherwise, since

for ¬xed t we have a real analytic function in one variable. Moreover, the series

fk z k (t) = (’1)k t2k z 2k has to converge in C ω (R, R) — C for |z| ¤ δ and all

√

t, see (9.5). This is not the case: use z = ’1 δ, t = 1/δ.

There is, however, another notion of real analytic curves.

Example. Let f : R ’ R be a real analytic function with ¬nite radius of conver-

gence at 0. Now consider the curve c : R ’ RN de¬ned by c(t) := (f (k · t))k∈N .

Clearly, the composite of c with any continuous linear functional is real analytic,

since these functionals depend only on ¬nitely many coordinates. But the Taylor

9.1

98 Chapter II. Calculus of holomorphic and real analytic mappings 9.3

series of c at 0 does not converge on any neighborhood of 0, since the radii of con-

vergence of the coordinate functions go to 0. For an even more natural example see

(11.8).

ak tk with real coe¬cients the fol-

9.2. Lemma. For a formal power series k≥0

lowing conditions are equivalent.

(1) The series has positive radius of convergence.

ak rk converges absolutely for all sequences (rk ) with rk tk ’ 0 for all

(2)

t > 0.

(3) The sequence (ak rk ) is bounded for all (rk ) with rk tk ’ 0 for all t > 0.

(4) For each sequence (rk ) satisfying rk > 0, rk r ≥ rk+ , and rk tk ’ 0 for all

t > 0 there exists an µ > 0 such that (ak rk µk ) is bounded.

This bornological description of real analytic curves will be rather important for

the theory presented here, since condition (3) and (4) are linear conditions on the

coe¬cients of a formal power series enforcing local convergence.

(ak tk )(rk t’k ) converges absolutely for

Proof. (1) ’ (2) The series ak rk =

some small t.

(2) ’ (3) ’ (4) is clear.

1

|ak | ( n2 )k = ∞ for all

(4) ’ (1) If the series has radius of convergence 0, then k

∞ with

n. There are kn

kn ’1

|ak | ( n2 )k ≥ 1.

1

k=kn’1

1 1

We put rk := ( n )k for kn’1 ¤ k < kn , then k |ak | rk ( n )k = ∞ for all n, so

1 t

(ak rk ( 2n )k )k is not bounded for any n, but rk tk , which equals ( n )k for kn’1 ¤ k <

kn , converges to 0 for all t > 0, and the sequence (rk ) is subadditive as required.

9.3. Theorem (Description of real analytic functions). For a smooth func-

tion c : R ’ R the following statements are equivalent.

(1) The function f is real analytic.

(2) For each sequence (rk ) with rk tk ’ 0 for all t > 0, and each compact set

1

K in R, the set { k! c(k) (a) rk : a ∈ K, k ∈ N} is bounded.

(3) For each sequence (rk ) satisfying rk > 0, rk r ≥ rk+ , and rk tk ’ 0 for

all t > 0, and each compact set K in R, there exists an µ > 0 such that

1

{ k! c(k) (a) rk µk : a ∈ K, k ∈ N} is bounded.

(4) For each compact set K ‚ R there exist constants M, ρ > 0 with the property

1

that | k! c(k) (a)| < M ρk for all k ∈ N and a ∈ K.

Proof. (1) ’ (4) Clearly, c is smooth. Since the Taylor series of c converges at a

there are constants Ma , ρa satisfying the claimed inequality for ¬xed a. For a with

9.3

9.5 9. Real analytic curves 99

()

c (a)

1

|a ’ a | ¤ 2ρa we obtain by di¬erentiating c(a ) = ! (a ’ a) with respect

≥0

to a the estimate

c(k) (a ) k

k1 ‚ 1

¤ M a ρa ,

k! ‚tk t= 1 1 ’ t

k!

2

hence the condition is satis¬ed locally with some new constants Ma , ρa incorporat-

1

ing the estimates for the coe¬cients of 1’t . Since K is compact the claim follows.

1

(4) ’ (2) We have | k! c(k) (a) rk | ¤ M rk (ρ)k which is bounded since rk ρk ’ 0, as

required.

(2) ’ (3) follows by choosing µ = 1.

1

(3) ’ (1) Let ak := supa∈K | k! c(k) (a)|. Using (9.2).(4’1) these are the coe¬cients

of a power series with positive radius ρ of convergence. Hence, the remainder

1 (k+1)

(a + θ(a ’ a))(a ’ a)k+1 of the Taylor series goes locally to zero.

(k+1)! c

9.4. Corollary. Real analytic curves. For a curve c : R ’ E in a convenient

vector space E are equivalent:

—¦ c : R ’ R is real analytic for all in some family of bounded linear

(1)

functionals, which generates the bornology of E.

(2) —¦ c : R ’ R is real analytic for all ∈ E

A curve satisfying these equivalent conditions will be called real analytic.

Proof. The non-trivial implication is (1 ’ 2). So assume (1). By (2.14.6) the

curve c is smooth and hence —¦ c is smooth for all bounded linear : E ’ R and

satis¬es ( —¦ c)(k) (t) = (c(k) (t)). In order to show that —¦ c is real analytic, we have

to prove boundedness of

1 (k) 1

( —¦ c)(k) (a)rk : a ∈ K, k ∈ N

c (a)rk : a ∈ K, k ∈ N =

k! k!

for all compact K ‚ R and appropriate rk , by (9.3). Since is bounded it su¬ces

1

to show that { k! c(k) (a)rk : a ∈ K, k ∈ N} is bounded, we follows since its image

under all mentioned in (1) is bounded, again by (9.3).

9.5. Lemma. Let E be a convenient vector space and let c : R ’ E be a curve.

Then the following conditions are equivalent.

(1) The curve c is locally given by a power series converging with respect to the

locally convex topology.

(2) The curve c factors locally over a topologically real analytic curve into EB

for some bounded absolutely convex set B ⊆ E.

(3) The curve c extends to a holomorphic curve from some open neighborhood

U of R in C into the complexi¬cation (EC , EC ).

Where a curve satisfying condition (1) will be called topologically real analytic. One

that satis¬es condition (2) will be called bornologically real analytic.

Proof. (1) ’ (3) For every t ∈ R one has for some δ > 0 and all |s| < δ a

∞ k

converging power series representation c(t + s) = k=1 xk s . For any complex

9.5

100 Chapter II. Calculus of holomorphic and real analytic mappings 9.7

number z with |z| < δ the series converges for z = s in EC , hence c can be locally

extended to a holomorphic curve into EC . By the 1-dimensional uniqueness theorem

for holomorphic maps, these local extensions ¬t together to give a holomorphic

extension as required.

(3) ’ (2) A holomorphic curve factors locally over (EC )B by (7.6), where B can

√

be chosen of the form B — ’1B. Hence, the restriction of this factorization to R

is real analytic into EB .

(2) ’ (1) Let c be bornologically real analytic, i.e. c is locally real analytic into some

EB , which we may assume to be complete. Hence, c is locally even topologically

real analytic in EB by (9.6) and so also in E.

Although topological real analyticity is a strictly stronger than real analyticity,

cf. (9.4), sometimes the converse is true as the following slight generalization of

[Bochnak, Siciak, 1971, Lemma 7.1] shows.

9.6. Theorem. Let E be a convenient vector space and assume that a Baire

vector space topology on E exists for which the point evaluations evx for x ∈ E are

continuous. Then any real analytic curve c : R ’ E is locally given by its Mackey

convergent Taylor series, and hence is bornologically real analytic and topologically

real analytic for every locally convex topology compatible with the bornology.

Proof. Since c is real analytic, it is smooth and all derivatives exist in E, since E

is convenient, by (2.14.6).

1

Let us ¬x t0 ∈ R, let an := n! c(n) (t0 ). It su¬ces to ¬nd some r > 0 for which

{rn an : n ∈ N0 } is bounded; because then tn an is Mackey-convergent for |t| < r,

and its limit is c(t0 + t) since we can test this with functionals.

Consider the sets Ar := {» ∈ E : |»(an )| ¤ rn for all n ∈ N}. These Ar are closed

in the Baire topology, since the point evaluations at an are continuous. Since c

is real analytic, r>0 Ar = E , and by the Baire property there is an r > 0 such

that the interior U of Ar is not empty. Let »0 ∈ U , then for all » in the open

neighborhood U ’ »0 of 0 we have |»(an )| ¤ |(» + »0 )(an )| + |»0 (an )| ¤ 2rn . The

set U ’ »0 is absorbing, thus for every » ∈ E some multiple µ» is in U ’ »0 and so

»(an ) ¤ 2 rn as required.

µ

9.7. Theorem. Linear real analytic mappings. Let E and F be convenient

vector spaces. For any linear mapping » : E ’ F the following assertions are

equivalent.

(1) » is bounded.

(2) » —¦ c : R ’ F is real analytic for all real analytic c : R ’ E.

(3) »—¦c : R ’ F is bornologically real analytic for all bornologically real analytic

curves c : R ’ E

(4) » —¦ c : R ’ F is real analytic for all bornologically real analytic curves

c:R’E

This will be generalized in (10.4) to non-linear mappings.

9.7

10.1 10. Real analytic mappings 101

Proof. (1) ’ (3) ’ (4), and (2) ’ (4) are obvious.

(4) ’ (1) Let » satisfy (4) and suppose that » is unbounded. By composing with

an ∈ E we may assume that » : E ’ R and there is a bounded sequence

(xk ) such that »(xk ) is unbounded. By passing to a subsequence we may suppose

that |»(xk )| > k 2k . Let ak := k ’k xk , then (rk ak ) is bounded and (rk »(ak )) is

∞

unbounded for all r > 0. Hence, the curve c(t) := k=0 tk ak is given by a Mackey

convergent power series. So » —¦ c is real analytic and near 0 we have »(c(t)) =

∞ N

k k N k’N

k=0 bk t for some bk ∈ R. But »(c(t)) = k=0 »(ak )t + t »( k>N ak t )

and t ’ k>N ak tk’N is still a Mackey converging power series in E. Comparing

coe¬cients we see that bk = »(ak ) and consequently »(ak )rk is bounded for some

r > 0, a contradiction.

Proof of (1) ’ (2) Let c : R ’ E be real analytic. By theorem (9.3) the set

1

{ k! c(k) (a) rk : a ∈ K, k ∈ N} is bounded for all compact sets K ‚ R and for all

sequences (rk ) with rk tk ’ 0 for all t > 0. Since c is smooth and bounded linear

mappings are smooth by (2.11), the function » —¦ c is smooth and (» —¦ c)(k) (a) =

»(c(k) (a)). By applying (9.3) we obtain that » —¦ c is real analytic.

9.8. Corollary. For two convenient vector space structures on a vector space E

the following statements are equivalent:

(1) They have the same bounded sets.

(2) They have the same smooth curves.

(3) They have the same real analytic curves.

Proof. (1) ” (2) was shown in (2.11). The implication (1) ’ (3) follows from (9.3),

which shows that real analyticity is a bornological concept, whereas the implication

(1) ⇐ (3) follows from (9.7).

9.9. Corollary. If a cone of linear maps T± : E ’ E± between convenient vector

spaces generates the bornology on E, then a curve c : R ’ E is C ω resp. C ∞

provided all the composites T± —¦ c : R ’ E± are.

Proof. The statement on the smooth curves is shown in (3.8). That on the real

analytic curves follows again from the bornological condition of (9.3).

10. Real Analytic Mappings

10.1. Theorem (Real analytic functions on Fr´chet spaces). Let U ⊆ E

e

be open in a real Fr´chet space E. The following statements on f : U ’ R are

e

equivalent:

(1) f is smooth and is real analytic along topologically real analytic curves.

(2) f is smooth and is real analytic along a¬ne lines.

(3) f is smooth and is locally given by its pointwise converging Taylor series.

(4) f is smooth and is locally given by its uniformly and absolutely converging

Taylor series.

10.1

102 Chapter II. Calculus of holomorphic and real analytic mappings 10.4

(5) f is locally given by a uniformly and absolutely converging power series.

˜˜ ˜

(6) f extends to a holomorphic mapping f : U ’ C for an open subset U in the

˜

complexi¬cation EC with U © E = U .

Proof. (1) ’ (2) is obvious. The implication (2) ’ (3) follows from (7.14), (1)

’ (2), whereas (3) ’ (4) follows from (7.14),(2) ’ (3), and (4) ’ (5) is obvious.

Proof of (5) ’ (6) Locally we can extend converging power series into the complex-

˜

i¬cation by (7.14). Then we take the union U of their domains of de¬nition and

˜

use uniqueness to glue f which is holomorphic by (7.24).

Proof of (6) ’ (1) Obviously, f is smooth. Any topologically real analytic curve c

in E can locally be extended to a holomorphic curve in EC by (9.5). So f —¦ c is real

analytic.

10.2. The assumptions ˜f is smooth™ cannot be dropped in (10.1.1) even in ¬nite

dimensions, as shown by the following example, due to [Boman, 1967].

n+2

xy

Example. The mapping f : R2 ’ R, de¬ned by f (x, y) := x2 +y2 is real analytic

along real analytic curves, is n-times continuous di¬erentiable but is not smooth

and hence not real analytic.

Proof. Take a real analytic curve t ’ (x(t), y(t)) into R2 . The components can be

factored as x(t) = tk u(t), y(t) = tk v(t) for some k and real analytic curves u, v with

uv n+2

u(0)2 +v(0)2 = 0. The composite f —¦(x, y) is then the function t ’ tk(n+1) u2 +v2 (t),

which is obviously real analytic near 0. The mapping f is n-times continuous

di¬erentiable, since it is real analytic on R2 \{0} and the directional derivatives of

order i are (n + 1 ’ i)-homogeneous, hence continuously extendable to R2 . But f

cannot be (n + 1)-times continuous di¬erentiable, otherwise the derivative of order

n + 1 would be constant, and hence f would be a polynomial.

10.3. De¬nition (Real analytic mappings). Let E be a convenient vector

space. Let us denote by C ω (R, E) the space of all real analytic curves.

Let U ⊆ E be c∞ -open, and let F be a second convenient vector space. A mapping

f : U ’ F will be called real analytic or C ω for short, if f is real analytic along

real analytic curves and is smooth (i.e. is smooth along smooth curves); so f —¦

c ∈ C ω (R, F ) for all c ∈ C ω (R, E) with c(R) ⊆ U and f —¦ c ∈ C ∞ (R, F ) for all

c ∈ C ∞ (R, E) with c(R) ⊆ U . Let us denote by C ω (U, F ) the space of all real

analytic mappings from U to F .

10.4. Analogue of Hartogs™ Theorem for real analytic mappings. Let E

and F be convenient vector spaces, let U ⊆ E be c∞ -open, and let f : U ’ F .

Then f is real analytic if and only if f is smooth and » —¦ f is real analytic along

each a¬ne line in E, for all » ∈ F .

Proof. One direction is clear, and by de¬nition (10.3) we may assume that F = R.

Let c : R ’ U be real analytic. We show that f —¦ c is real analytic by using theorem

(9.3). So let (rk ) be a sequence such that rk r ≥ rk+ and rk tk ’ 0 for all t > 0

10.4

10.4 10. Real analytic mappings 103

and let K ‚ R be compact. We have to show, that there is an µ > 0 such that the

set { 1! (f —¦ c)( ) (a) rl ( 2 ) : a ∈ K, ∈ N} is bounded.

µ

1

By theorem (9.3) the set { n! c(n) (a) rn : n ≥ 1, a ∈ K} is contained in some bounded

absolutely convex subset B ⊆ E, such that EB is a Banach space. Clearly, for the

inclusion iB : EB ’ E the function f —¦ iB is smooth and real analytic along a¬ne

lines. Since EB is a Banach space, by (10.1.2) ’ (10.1.4) f —¦ iB is locally given

by its uniformly and absolutely converging Taylor series. Then for each a ∈ K by

1

(7.14.2) ’ (7.14.4) there is an µ > 0 such that the set { k! dk f (c(a))(x1 , . . . , xk ) :

k ∈ N, xj ∈ µB} is bounded. For each y ∈ 1 µB termwise di¬erentiation gives

2

1

p k

d f (c(a) + y)(x1 , . . . , xp ) = k≥p (k’p)! d f (c(a))(x1 , . . . , xp , y, . . . , y), so we may

assume that {dk f (c(a))(x1 , . . . , xk )/k! : k ∈ N, xj ∈ µB, a ∈ K} is contained in

[’C, C] for some C > 0 and some uniform µ > 0.

The Taylor series of f —¦ c at a is given by

k!