- Email: [email protected]
Linearization of Holomorphic Mappings of Bounded Type
Progress in Functional Analysis K.D. Bierstedt, J. Bonet, J. Horvath B M. Maestre (Eds.) 0 1992 Elsevier Science Publishers B.V. All rights reserved
149
LINEARIZATION OF HOLOMORPHIC MAPPINGS OF BOUNDED TYPE Jorge Mujica
Iiistituto cle Matemitica Universidade Estadual de Campinas Caixa Postal GOG5 13 OS1 Campinas, SP, Brazil
Dedicated to Manuel Valdivia on the occasion of his sixtieth birthday W e study the relationships between preduals of spaces of holomorphic functions of bounded t y p e and preduals of spaces of bounded holomorphic functions. In this manner we obtain information about the former from the available information about the latter.
INTRODUCTION Several authors have obtaincd linearization thcorems for various classes of holomorphic mappings. It scems that the first general result of this kind is due to Mazet [13], who constructed, for cach open subset U of a locally convex space E , a complete locally convex space G ( U ) and a holomorpliic mapping 6u : U + G ( U ) such that, for each complete locally convex space F and each holornorphic mapping f : U --+ F , there is a unique continuous linear mapping T, : G ( U ) -+ F such that Tj o 6u = f . Related results of Schottenloher [19] and Ryan [lS] were not so satisfactory. In a recent paper Nachbin and the author [16] gave a new proof and several applications of the Mazet linearization theorem. In another paper the author [15] obtained a linearization theorem for bounded holomorphic mappings. IIe constiucted, for each open subset U of a Banach space E , a Baiiach space G"(U) and a bounded liolomorpliic mapping Su : U + G"(U) such that, for each Banach space F and each bounded holornorphic mapping f : U + F , there is a unique continuous linear mapping T, : G"(U) + F such that T, o 6" = f . In a more rccent papcr Galindo ct al. [7] obtained a linearization theorem for holomorphic mappings of bounded type. They constructed, for each balanced, open subsct U of a Banach space E , an (13)-space Cb(U) and a holomorphic mapping of bounded type
150
J. Mujica
-+ Gb(U) such that, for each Banach space F and each holomorphic mapping of bounded type f : U + F, there is a unique continuous linear mapping T, : Gb(U) -+ F such that Tj o 6" = f. The Banach spaces which appear in the (LB)-representation of Gb(U) obtained by Galindo et al. [7] are rather involved. In this lecture we improve their result by showing that if (Un)is a suitable sequence of open subsets of U , then Gb(U) can be represented as the inductive limit of the Banach spaces Gm(Un).This representation theorem is used to derive several properties of the space Gb(U) from the corresponding properties of the spaces Gm(Un). In this lecture we actually deal with a class of holomorphic mappings more general than the class of holomorphic mappings of bounded type. This more general class appears in a natural way in the study of G ( U ) ,and the results obtained here will be used in a subsequent paper to derive further properties of G(U). I am grateful to Klaus Floret for several helpful comments when this paper was being prepared. I wish to thank the organizers of the meeting honoring Manuel Valdivia for their kind invitation and financial support to attend and deliver this lecture.
6u : U
1. NOTATION AND TERMINOLOGY
We follow the standard terminology from topological vector spaces, as found for instance in the books of Grothcndieck [lo] or Horvith [ll]. The letters E and F represent locally convex spaces, always assumed complex and Hausdorff. If A is a convex, balanced, bounded subset of E , then EA denotes the vector subspace of E generated by A and normed by the Minkowski functional of A . L ( E ; F ) denotes the vector space of all continuous linear mappings from E into E'. If F = C then we write E' instead of L ( E ; 6'). E[ denotes the strong dual of E , whereas E: denotes the inductive d u d of E. \'e recall that E: = ind ELo, where V varies among the convex, balanced 0-neighborhoods i n E , and V o denotes the polar of V in 15'. We refer to Bierstedt [3] for the properties of the inductive dual. We follow the standard terminology from infinite dimensional complex analysis, as found for instance in the books of Dineen [GI or the author [14]. If m E No = N U (0)) then 'P(mE; F ) denotes the vector space of all continuous rn-homogeneous polynomials from E into F . If F = C,then we write 'P(mE)instead of P(mE;C). If U is an open subset of E , then X(U;F ) denotes the vector space of all holomorphic mappings from U into F , whereas X " ( U ; F ) denotes the vector subspace of all f E X(U;F ) such that f ( U ) is bounded in F . If F is a Banach space, then X " ( U ; F ) is a Banach space for the sup norm. If F = C, then we write ' H ( U ) instead of X(U;C), and X " ( U ) instead of X"(U; 6'). T~ denotes the compact opcn topology on X ( U ; F ) or on any subset of X ( U ; F ) . I f f E X(U;F ) , (I E U and nz E No,then P m f ( u ) E P ( m E ; F) denotes the mth
151
Linearization of holomophic mappings of bounded type
homogeneous polynomial in the Taylor series expansion of f at a. If f E 'FI(U; F ) , where F is a Uanach space, then for each set A I l f l l A = S U P ~I ~ l f ( Az ) l l .
cU
we write
2. THE SPACES X"(U; F ) AND G"(U) Let E and F be locally convex spaces, let U be an open subset of El and let U = (Un) be a countable, open cover of U . Let 'FI"(U; F ) denote the locally convex space
'FIm(U;F ) = { f E X(U;F ) : f(U,,) is bounded in F for every n }, equipped with the topology of uniform convergence 011 all the sets Un. If F is a Banach space, then 'FI"(U; F ) is a 1;'ldchct space. If F = C, then we write 'FI"(U) instead of
X"(U; C).
2.1. THEOREM. Let U be an open subset of a locally convex space E , and let U = ( U n ) be a countable, open cover of U . l'hen there are a complete, barrelled (DF)-space G"(U) and a mapping 6" E 'FI"(U; G w ( U ) )with the following universal property: For each complete locally convex space F and each mapping f E Z"(U; F ) , there is a unique mapping TJ E C(C"(U); F ) such that TJ o 6" = f . This property characterizes G"(U) uniquely up to a topological isoinorphistn.
Proof. For each sequence
ct
= ( c Y , ~of ) strictly positive nurnbers, let
Bf;denote the set
and let C"(U) denote tlie locally convex space
C"(U) = { t E~ X"(U)' : .ID,", is .r,-continuous for every a ) , equipped with the topology of uiiiform convergence on all the sets B,",. It is clear that
X"(1.1)
= iiid 'H"(U)B;
and each of the sets BE is .r,-compact, by the Ascoli theorem. Furthermore, each of the sets { f E 'FI"(U) : llfll,yn 5 E ) is r,-closed. Thus an application of [16, Theorem 1.11 shows that the evaluation niappiiig
J : 'FI"(U)
--$
c - ( U ) : = G'"(U)b
is a topological isomorphism. Cleitrly the space G"(U) is complete. That G"(U) is barrelled follows from a general result of Bicrstedt and Bonct "41, or directly from tlie fact that J(B,",)is an equicontinuous subset of G"(U)' for every a. Now since 'H"(U) is a Frdchet space, its strong dual
152
J. Mujica
7 P ( U ) bis a (DF)-space. As a subspace of %"(U)',, the space G"(U) has a fundamental sequence of bounded sets, and is therefore a (DF)-space. From this point on the proof of Theorem 2.1 is similar to the proof of [16, Theorem 2.11, and is therefore omitted. PROPOSITION. Let U be an open subset of a locally convex space El and let (U,,) be a countable, open cover of U . Then: (a) For each bounded set 13 c G"(U), there are R E N and p > 0 such that B c pT6u(Un), where FA denotes the closed, convex, balanced hull of A . (b) For each complete locally convex space F , the mapping 2.2.
U
=
f
E 'H"(U; F ) -+ TI E L(G"(U); F )
is a topological isomorphism, when L(G"(U); F ) is endowed with the topology of uniform convergence on the bounded subsets of G"(U).
Proof. Since the evaluation mapping
J : 'FI"(U)
--$
G"(U);,
is a topological isomorphism, we can prove (a) by imitating the proof of [16, Lemma 4.21. And (b) is clearly a direct consequence of (a).
2.3. PROPOSITION. Let E and F be locally convex spaces, with F complete. Let U be an open subset of E , let U = (U,,) be a countable, open couer of U , and let ( f i ) c IFI"(U; F ) . The.n the set U;f;(U,) is bounded in F f o r every n E lli if and only if the family ( T j , )c L ( G m ( U ) ;F ) is equicontinuous.
The proof of this proposition is similar to t,he proof of [16, Proposition 2.51, and is therefore omitted. 2.4. PROPOSITION. Let U be an open subset of a Banach space E , and let U = (Un)
be a countable, open cover of U , with each U, bounded. Then E is topologically isomorphic to a complemented subspace of G"(U).
Proof. Under the hypotheses of the proposition, the inclusion mapping U ~t E belongs to 'FI"(U; E ) . By Theorem 2.1 there exists T E L(G"(U); E ) such that T(6,) = z for every L E U . Then the proof of [15, Proposition 2.31 or [16, Proposition 2.61 applies.
Linearization of holomorphic mappings of bounded type
153
3. A N (LB)-REPRESENTATION OF G"(U) Let U be an open subset of a locally convex space E , and let U = (U,) be a countable open cover of U . Let B, denote the closed unit ball of X"(U,), and let G"(U,) denote the Banach space
G"(U,)
= {ti E
X"(U,)'
: ulB, is .r,-continuous
},
with the induced norm. As pointed out i n the proof of [15, Theorem 2.11, a theorem of Ng [17] guarantees that the evaluation mapping
J,, : X"(U,)
G'(U,)'
-+
is an isometric isomorphism. Clearly the FrCchet space X " ( U ) can be represented as the projective limit of the Banach spaces X"(U,) by incans of the restriction mappings
R, : X " ( U )
-+
X"(U,).
Consider the dual mappings
R:, : 'H"(U,,)'
-+
X"(U)'.
One can readily verify that R:,zLE G"(U) for evcry u E G"(U,), that is, the mapping R, G"(U,)). Thus if
is continuous for the topologies a ( X " ( U ) , G " ( U ) ) and a(X"(U,),
S, : G"(U,,)
-+
G"(U)
denotes the restriction of RL to G'"(U,,), then the mappings R, and S, are dual to each G " ( U ) ) and (X"(U,,), G"(U,)). We will other with respect to the dual pairs (XFI"(U), see that, undcr suitable liypotlieses, the (DF)-space G"(U) can be represented as the inductive limit of the Baiiach spaces CFI"(U,,)by means of the mappings S,,. We begin with the following proposition. 3.1. PROPOSITION. Let U be a balanced, open subset o J a Banach space E , and let U = (U,) be a sequence of balanced, bounded, open subsets o J U such that U = UF=.=,U,. Then: (a) The mapping R, : X " ( U ) -i fZm(U,,) has a c(X"(U,,), G"(U,))-dense range. (b) The mapping S, : G"(lJ,) -+ G " ( U ) is injective.
Proof. (a) Let f E X"(U,,). Then, with the notation of [15, Proposition 5.21, the sequence of Cesbro means (a7,,f)convergcs to f i n (X"(U,), T ~ ) Since, . by [15, Propositions 4.7 and 4.91, (X"(U,), T?)' = G'=(Un),w e conclude that (a,f)converges to f for the topology c('l-lM(U,,),G"(U,)). Since each amf is a polynomial, and since each iJk is bounded, wc conclude that a,f E X " ( U ) for every m E N ,thus proving (a).
154
J. Mujica
Since (b) is a direct consequence of (a), the proof of the proposition is complete. Thus, under the hypotheses of Proposition 3.1, we may regard each G w ( U n ) as a vector subspace of G"(U).
3.2. THEOREM. Let U be a balanced, open subset of a Banach space El and let U = ((in) be a sequence of balanced, bounded, open subsets of U such that U = Ur=lCJn and pnUn c Un+I, with pn > 1, f o r every n E N . Then G"(U) = ind G w ( U n ) , and this inductive limit is boundedly retractive.
The key to the proof of Theorem 3.2 is the following lemma. 3.3. LEMMA. Let U be a balanced, open subset of a Banach space E , and let U = ( U n ) be a sequence of balanced, bounded, open subsets of U such that U = UrZlUn and pnUn C Un+l, with p n > 1, for cvcry n E N . Then G"(U) and Gm(Un+l) induce the same topology and tlie same uniform structure on the closed unit ball of G w ( U n ) .
Proof. Let A k and Bk denote the closed unit balls of G w ( U k ) and y"(uk),respectively. By a lemma of Grothendieck [lo, p. 98, Lemma], to prove Lemma 3.3 it suffices to show that for each E > 0 there exists cy = ( c y k ) , with a k > 0, such that
Let f E B,,+l. Since pnUn
c U,l+l, tlie Cauchy integral
M
formulas yield the inequalities
M
and this is less than &/2 for N sufficiently large. On the other hand, since u with ck > 0, for every k E N ,we have that N-1
N- 1
m=O
for every k E N. Set cy = ( a k ) , where preceding inequalitics show that
k
C ck
un+l)
N- 1 m=O
cyk
= fC,"lAcr,
for every k E
N. Then
and (3.1) follows directly from (3.2). Proof of Theorem 3.2. Clearly U ~ = l G w ( U nc) G"(U), and the inclusion mapping ind G m ( U n ) L) G"(U) is continuous.
the
Linearization of holomoiphic mappings of bounded type
155
On the other hand, by Tllcorem 2.1 and by [15, Theorem 2.11, we have the canonical topological isomorphisms
G"(U)b = R " ( U )
= proj
' P ( U , , ) = proj Gm(Un)'.
Thus, by polarity, each bounded subset of G"(U) is included in the closure in G"(U) of a bounded subset of some Gm(Un).Since Gm(Un+l)is complete, it follows from Lemma 3.3 that and thus each bounded subset of G"(U) is included in some Gm(Un) and is bounded there. Thus G"(U) = UF!lGm(Un), and the spaces C"(U) and ind Gm(Un)induce the same topology on each bounded subset of G"(U). Since we already know from Theorem 2.1 that G"(U) is a (DF)-space, a theorem of Grothendieck [8, p. 68, Thdorkme 31 guarantees that the identity mapping G"(U) + i d G"(U,) is continuous. This completes the proof. An cxarninatioii of the proof of Lemma 3.3 yields the following result.
3.4. PROPOSITION. Let E a d F be Danach spaces, let U be a balanced, open subset of E , and let U = ( U n ) be a sequence of balanced, bounded, open subsets of U such that U = Ur=lUn and pnUn c Un+l, with p,, > 1, f o r every n E N . T h e n 'H"(U; F ) is a quasi-normable Fre'chet space.
4. THE APPROXIMATION PROPERTY
To begin with we state the following conseyucnce of [15, Theorem 5.41. 4.1. THEOREM. Let U be a balanced, bounded, open subset of a Banach space E . Then G"(U) has the approximation propeify if a n d only if E has the approximation property. With the aid of Tlieorcm 3.2 we can easily extend this result to G"(U).
THEOREM. Let U be a balanced, open subset of a Banach space E , and let (U,,) be a sequence of balanced, bounded, open subsets o f U such that U = Ur=lUn and pnUn c Un+l, with p,, > 1, f o r every n E N . T h e n G w ( U ) has the approximation property if and only i f E has the approximation property. 4.2.
U
=
Proof. By Proposition 2.4, E is topologically isomorphic to a complemented subspace of G"(U). IIence E has the approximation property if so does G"(U). If, conversely, E has tlie approximation property, then each Gm(Un)has the approximation property, by Tlieorcm 4.1. By Theorem 3.2 we may apply a result of Bierstedt and Mcisc [5, Satz 1.21 to conclude that G"(U) = ind G"(Un) has the approximation
156
J. Mujica
property. 5. HOLOMORPHIC FUNCTIONS O N QUOTIENT SPACES In 115, Corollary 4.121 we gave an explicit description of G"(U). By examining the proofs of [15, Lemma 4.6 and Theorems 4.5 and 4.111, we can sharpen the conclusion of [15, Corollary 4.121 as follows. 5.1. THEOREM. Let U be an open subset of a Banach space E . Then G"(U) consists of all linear functionals u E G"(U)' of the form 00
u =
p1 S,,,
3=1
with
( a j )E
1' and
(zj)c
U . Aloreover,
IIuII = i n f
00
C IajI,
j=1
where the infimum is taken over all such representations of u . 5.2. THEOREM. Let E and F be Banach spaces, and let H E L ( E ; F ) be surjective. Let U be an open subset of E , atid let 1f = .(U). Let S : G"(U) 4 G"(V) be the unique continuous linear mapping such that the following diagram is commutative:
u " . v
SlJ
1
Gm(U)
5
1 Sv G"(V)
Then S maps the ball { u E G"(U) : I I I L I I < 1) onto the ball { v E G"(V) : [lull < 1). Proof. Since SV o T E 'H"(U; Gm(U)), [15, Theorem 2.11 guarantees that llSll = 116~o H I I = 1. IIence llSiill < 1 for evcry u E G"(U) with IIuJI < 1. If, conversely, v E G"(V), with ((v1(< 1, thcn by Tlicorem 5.1 we can find (y,) c V and (a,) E l', with C,"=,(a31< 1 , such that
La3by,. 00
v =
3=1
Write yI = r ( z 3 )with , zl E U , for every j E If we define
r'-
N.
Linearization of holomofphic mappings of bounded type
157
then u E G " ( U ) , llull < 1 arid
5.3. THEOREM. Let E and F be Banach spaces, and let A E C ( E ; F ) be surjective. Let U be a balanced, open subset of E , and let U = (U,) be a sequence of balanced, bounded, open subsets of U such that U = uF=lU, and p,Un c Un+,, with pn > 1, for every n E N . Let V = A ( U ) , and let V = (V,), where V, = A(U,) f o r every n E N . Let S : G w ( U ) --+ G " ( V ) be the unique continuous linear mapping such that the following diagram is comnautative:
U
L
G"(U)
--%
611
1.
V
1 6v Gm(V)
Then S is surjective and open. Each bounded subset of G"(V) is the image under S of some bounded subset of G"(U).
-
Proof. For each n E N we have a commutative diagram GCO(Un) Sn 1
Gw(Vn)
-+
G"(U)
1s
G"(V)
Let B be a bounded subset of G m ( Y ) . By Theorem 3.2, B is included in some G"(Vn) and is bounded there. By Theorem 5.2, there is a boundcd sct A in Gw(Un) such that B = & ( A ) . Thus B = S ( A ) ,with A bounded in G"(U). In particular we have shown that S is surjective, and therefore open, by an open mapping theorem due t o Grothendieck (see [9, Introduction, p. 17, Thd.orL:me B ] )or [lo, p. 200, T h d o r h e 21). 6. HOLOMORPHIC FUNCTIONS ON PRODUCT SPACES
6.1. THEOREM. Let E arid F be Banach spaces, and let U and V be open sets in E and F , rcsycctively. Then: (a) X"(U x V ) is isometrically tsoniorphic to X"(U; X W ( V ) ) . (b) G"(U x V ) is isonieti~icullyisomorphic to G " ( U ) & G " ( V ) .
Proof. (a) Clearly t h e mapping
f E X"(U x V )
--+
?€
X " ( U ; %"(\'))
defined by ?(z)(g) = f ( z , y ) for all z E U arid y E V , is an isometric isomorphism. (b) O n one hand the mapping (T, y)
E IJ x V
--t
6, @ 6, E G"(U)&G"(V)
158
J. Mujica
is holomorphic and ~ ~ 6 , @ 6=, Il6,ll ~ ~ 116,ll = 1 for all z E U and y E V . By the universal property of G"(U x V ) (see [15, Theorem 2.1]), there is a continuous linear mapping
S : G"(U x V ) -+ G"(U)&G"(V), with llSll = 1, and such that hand the mapping
SS[,,,)= 6,
@
(u,U ) E G"(U) x G"(V)
6, for all -+
z E
U and y E V . On the other
u x u E G"(U x V )
defined by (ux v)(f) = u ( v o f) for every u E G"(U), u E G"(V) and f E 'H"(U x V ) , is bilinear and IIu x 011 5 llull 1 1 ~ 1 1for all u E G"(U) and v E G"(V). By the universal property of tensor products, there is a continuous linear mapping
T : C"(U)&G"(V)
+ Gm(U x
V),
with llTll 5 1 , and such that T ( u @ v )= u X U for all u E G"(U) and u E G"(V). Hence T o S(6(,, ,I) = b(,, ), and S o T(6, @ 6,) = 6, 86, for all z E U and y E V . It follows that S is an isometric isomorphism, with inverse T . 6.2. THEOREM. Let E and F be Banach spaces, and let U and V be balanced, open sets in E and F , respectively. Let U = (Un) be a sequence of balanced, bounded, open subsets of U such that U = Uz=lUn and p,Un c Un+l, with pn > 1, f o r every n E fir. Likewise, let V = (Vn) be a sequence of balanced, bounded, open subsets of V such that V = Ur=p=,Vn and anVn c Vn+,, with an > 1, f o r every n E N . If U x V denotes the sequence (U, x Vn), then: (a) 'FI"(U x V ) is topologically isomorphic to 'H"(U; 'H"(V)). (b) G"(U x V ) is topologically isomorphic t o G"(U)&G"(V).
Proof. (a) Clearly the mapping
f E R"(U x V ) -+ JE R"(U; 'H"(V)) defined by f ( z ) ( y )= f ( z , y ) for all z E U and y E V , is a topological isomorphism. (b) A glance at the diagram
Un x Vn
1
U x V
6un@6vn +
'*
G"(Un)@,G"(Vn)
1
G"(U)G3,G"(V)
shows that the mapping (z, y) E
ux v
-+
6, @ 6, E G ~ ( U ) @ , G " ( V )
belongs to X"(U x V ; G"(U)&G"(V)). By Theorem 2.1 there is a continuous linear mapping S : Gm(Ux V ) + G"(U)&G"(V)
Linearization of holomorphic mappings of bounded type
159
such that S6(,,), = 6,@ 6, for all x E U and y E V . On the other hand, since G"(U) = ind G"(Un) and G"(V) = ind Gm(Vn),the bilinear mapping ( u , v) E G"(U) x G"(V) -+ u x
2,
E
C"(U x V )
defined by (ux v)( f ) = u(v o f)for every u E C"(U), u E G"(V) and f E 'H"(U x V ) , is separatcly continuous. By a theorem of Grothendieck on (DF)-spaces (see [8, p. 66, Corollaire] or [lo, p. 226, Corollaire l]), this bilinear mapping is actually continuous. From now on the proof proceeds exactly as the proof of Theorem 6.1.
7. HOLOMORPHIC MAPPINGS OF BOUNDED TYPE
+
Let U be an open subset of a Banach space E . If U E and x E U , then dU(x) denotes tlie distance from 5 to E\U. Since ldu(z) - d u ( y ) l 5 llz - yII for all x, y E U , we see that dU is a continuous function on U . If U = E , then for convenience we define d u ( x ) = co for every 2 E U . A set A c U is said to be U-bounded if A is bounded and infIEA &(I) > 0. If F is a locally convex space, then 'Hb(U; F ) denotes the locally convex space
'Hb(U; F ) = { f E 'H(U; F ) : f ( A ) is bounded in F for each U-bounded set A } , equipped with the topology of uniform convergence on all U-bounded sets. If F = C, then we write 'Hb(U) instead of 'Hb(U; 6').The mernbers of 'Hb(U; F ) are called holomorphic mappings of bounded type. It is clear that 7&(U; F ) = 'H"(2A; F ) if U = (Un) is any fundamental sequence of open U-bounded sets. In the next proposition we give a fundamental sequence U = (Un) of open U-bounded sets which satisfics the hypotheses of all the results in the preceding sections.
7.1. PROPOSITION. Let U be an open subset of a Banach space El and let (Un) be defined b y 11, = {x E U : llzll < PZ and d ~ ( x>) 2-"}. Then: (a) (Un) is a fuiidaniental scqueiice oJ open, U-bounded sets. (b) If U is balanced, then each U,, is balanced as well and, furthermore, there exists with pn > 1, such that pnUn c U,,, for every 12 E N. The proof of this proposition is straightforward, and is left to the reader. For each
nE
N we may take pn = 1 + 1/n2".
Theorem 2.1 yields Galindo et al. [7].
ill
particular the following result, which improves a result of
160
J. Mujica
7.2. PROPOSITION. Let U be an open subset of a Banach space E . Then there are a complete, barrelled (DF)-space Gb(U) and a mapping 6u E Ftb(U; Gb(U)) with the following universal property: For each complete locally convex space F and each mapping f € ?fb(u; F ) , there is a continuous h e a r mapping T, E L(Gb(U); F ) such that T, o 6u = f . This property characterizes Gb(U) uniquely up to a topological isomorphism.
Theorem 3.2 yields in particular the following result, which improves a result of Galindo et a1 [7]. 7.3. PROPOSITION. Let U be a balanced, open subset of a Banach space E , and let (Un)be a fundamental sequence of open U-bounded sets such that each U,, is balanced and pnUn c Un+l, with pn > 1, f o r every 11 E IN. Then Gb(U) = ind Gm(Un) and this inductive limit of boundedly retinctive. It is clear too that Proposition 3.4 yielcls in particular a result of Ansemil and Ponte [2] and Isidro [12]. Theorem 4.2 yields in particular the following result.
7.4. PROPOSITION. Let U be a balanced, open subset of a Banach space E . Thcn Gb(U) has the approximation property if and only if E has the approximation property. Theorem 5.3 yields in particular the following result. 7.5. PROPOSITION. Let E and F be Banach spaces, and let R E L ( E ; F ) be surjective. Let V be a balanced, open subset o f F , let U = R-'(V), and let S : Gb(U) -+ Gb(V) be the unique continuous linear mapping such that the following diagram is commutative: U
A
V
Gb(U)
3
Gb(V)
bu
1
1 bv
Then S is surjective and open. Each bounded subset of Gb(V) is the image under S of some bounded subset ofGb(U).
Proof. Let (Un) be a fundamcntal sequence of open U-bounded sets such that each U,, is balanced and p,Un c Unfl, with p,, > 1, for every 11 E N . Let V, = .(Un) for every n E A'. Since U = .-'(V), it follows that each V-bounded set is the image under a of some U-bounded set. Whence it follows that (Vn) is a fundamental sequence of open V-bounded sets. Thus Theorem 5.3 applics and yields the desired conclusion.
Remark. There are Banach spaces E and F , there is a surjective mapping a convex, balanced, open set U in E such that, if we set v = r ( U ) , then the canonical mapping s E L(Gb(U); Gb(V))is not surjective. Indeed, if S were surjective, then S would be open, by an open mapping theorem due 7.6. R
E
L ( E ; F ) , and there is
Linearization of holomorphic mappings of bounded type
161
to Grothendieck (see [9, Introduction, p. 17, ThCorkme B] or [lo, p. 200, Thdorkme 21). Then, by a theorem of Grothendieck on (DF)-spaces (see [8, p. 76, Proposition 51 or [lo, p. 228, Proposition 4]), each bounded subset of Gb(V)would be included in the closure of the image under S of some bounded subset of Gb(U). Then the mapping g E 'Hb(v)+ g o n E 'Hb(u)would be an embedding, that is, a topological isomorphism between 'Hb(v) and a subspace of 'Hb(u).But this is not always true, as a counterexample of Ansemil et a]. [I] shows. Finally Theorem 6.2 yields in particular tlie following result.
7.7. P R O P O S I T I O N . Let E and F bc Banach spaces, and let U and V be balanced, open sets in E and F , respectively. Then: (a) 'Ftb(U x V ) is topologzcally isomorphic to ' F t b ( U ; 'Hb(V)). (b) Gb(U x V ) is topologically isomorphic to Gb(U)& Gb(V).
REFERENCES
[l] J . M. Ansemil, R. M. Aron a n d S. P o n t e , Embeddings of spaces of holomorphic functions of bounded type. Preprint, 1990.
[2] J. M. Anseniil a n d S. P o n t e , Ail example of a quasi-normable Fre'chet function space which is not a Scliwcirfz space. In: Functional Analysis, Holomorphy and Approximation Theory, edited by S. hfacliado, pp. 1-8. Lecture Notes in Mathematics, vol. 843. Springer, Berlin, 1SS1. [3]K. D. B i e r s t e d t , A n introduction to locally convex inductive limits. In: Functional Analysis and its Applications, editcd by 1-1. IIogbe-Nlend, pp. 35-133. World Scientific, Singapore, 19SS.
[4] K. D. Bierstedt a n d J. B o n e t , Bidnulity in Fre'chet and (LB)-spaces. In: Progress in Functional Analysis, edited by J. Bonet et al. North-Holland Mathematics Studies. North-Holland, Amsterdam, to appear. [ 5 ] K . D. B i e r s t e d t a n d R. Meise, Bemerkungen uber die Approximationseigenschaft
lokalkonvexer Fur~ktionenriurne.hlatli. Ann. 209 (1974), 99-107.
[6] S. D i n e e n , Coniplcx Analysis in Locally Convex Spaces. North-Holland Mathematics Studies, vol. 57. North-IIolland, Amsterdam, 1981.
[7] P. Galindo, D. G a r c i a a n d M. M a e s t r e , IIoloniorphic mappings of bounded type. J. hlath. Anal. Appl., to appear.
J. Mujica
162
[8] A. Grotliendieck, Sur les espaces ( F ) et ( D F ) . Summa Brasil. Math. 3 (1954), 57-123. [9] A. Grotliendieck, Produits tensoriels topologiques et espaces nucle'aires. Amer. Math. SOC. 16 (1955).
[lo] A. G r o t h e n d i e c k , Espaces Paulo, 1964.
Mem.
Vectoriels Topologiques, 33 ediq8o. Universidade de SiLo
[ll] J. H o r v a t h , Topological Vector Spaces and Distributions, vol. I. Addison-Wesley, Reading, Massachusetts, 1966. [12] J. M. Isidro, O n the distinguished character of the function spaces of holomorphic mappings of bounded type. J. Funct. Anal. 3s (19SO), 139-145. [13] P. M a z e t , Analytic Sets in Locally Convex Spaces. North-Holland Mathematics Studies, vol. S9. North-IIolland, Amsterdam, 1954. [14] J . Mujica, Complex Analysis i.1~Banach Spaces. Studies, vol. 120. North-Holland, Amsterdam, 19%.
North-Holland Mathematics
[15] J . M u j i c a , Linearization of bounded holomorphic mappings on Banach spaces. Trans. Amer. Math. SOC.324 (1991), 867-887. [16] J. M u j i c a a n d L. Nachbin, Linearization of holomorphic mappings on locally convex spaces. J. hilath. Pures Appl., to appear.
[17]K . F. Ng, On a theorem of Dixmier. Math. Scand. 29 (1971), 279-280. [18] R. A. Ryan, Applications of topological tensor products to iiifiiiite dimensional holomorphy. Ph. D. thesis, Trinity College Dublin, 19SO. [19] M. S c h o t t e n l o h e r , E-products and continuation of analytic mappings. In : Analyse Fonctionnelle et Applications, edit6 par L. Nachbin, pp. 261-270. Hermann, Paris, 1975.
149
LINEARIZATION OF HOLOMORPHIC MAPPINGS OF BOUNDED TYPE Jorge Mujica
Iiistituto cle Matemitica Universidade Estadual de Campinas Caixa Postal GOG5 13 OS1 Campinas, SP, Brazil
Dedicated to Manuel Valdivia on the occasion of his sixtieth birthday W e study the relationships between preduals of spaces of holomorphic functions of bounded t y p e and preduals of spaces of bounded holomorphic functions. In this manner we obtain information about the former from the available information about the latter.
INTRODUCTION Several authors have obtaincd linearization thcorems for various classes of holomorphic mappings. It scems that the first general result of this kind is due to Mazet [13], who constructed, for cach open subset U of a locally convex space E , a complete locally convex space G ( U ) and a holomorpliic mapping 6u : U + G ( U ) such that, for each complete locally convex space F and each holornorphic mapping f : U --+ F , there is a unique continuous linear mapping T, : G ( U ) -+ F such that Tj o 6u = f . Related results of Schottenloher [19] and Ryan [lS] were not so satisfactory. In a recent paper Nachbin and the author [16] gave a new proof and several applications of the Mazet linearization theorem. In another paper the author [15] obtained a linearization theorem for bounded holomorphic mappings. IIe constiucted, for each open subset U of a Banach space E , a Baiiach space G"(U) and a bounded liolomorpliic mapping Su : U + G"(U) such that, for each Banach space F and each bounded holornorphic mapping f : U + F , there is a unique continuous linear mapping T, : G"(U) + F such that T, o 6" = f . In a more rccent papcr Galindo ct al. [7] obtained a linearization theorem for holomorphic mappings of bounded type. They constructed, for each balanced, open subsct U of a Banach space E , an (13)-space Cb(U) and a holomorphic mapping of bounded type
150
J. Mujica
-+ Gb(U) such that, for each Banach space F and each holomorphic mapping of bounded type f : U + F, there is a unique continuous linear mapping T, : Gb(U) -+ F such that Tj o 6" = f. The Banach spaces which appear in the (LB)-representation of Gb(U) obtained by Galindo et al. [7] are rather involved. In this lecture we improve their result by showing that if (Un)is a suitable sequence of open subsets of U , then Gb(U) can be represented as the inductive limit of the Banach spaces Gm(Un).This representation theorem is used to derive several properties of the space Gb(U) from the corresponding properties of the spaces Gm(Un). In this lecture we actually deal with a class of holomorphic mappings more general than the class of holomorphic mappings of bounded type. This more general class appears in a natural way in the study of G ( U ) ,and the results obtained here will be used in a subsequent paper to derive further properties of G(U). I am grateful to Klaus Floret for several helpful comments when this paper was being prepared. I wish to thank the organizers of the meeting honoring Manuel Valdivia for their kind invitation and financial support to attend and deliver this lecture.
6u : U
1. NOTATION AND TERMINOLOGY
We follow the standard terminology from topological vector spaces, as found for instance in the books of Grothcndieck [lo] or Horvith [ll]. The letters E and F represent locally convex spaces, always assumed complex and Hausdorff. If A is a convex, balanced, bounded subset of E , then EA denotes the vector subspace of E generated by A and normed by the Minkowski functional of A . L ( E ; F ) denotes the vector space of all continuous linear mappings from E into E'. If F = C then we write E' instead of L ( E ; 6'). E[ denotes the strong dual of E , whereas E: denotes the inductive d u d of E. \'e recall that E: = ind ELo, where V varies among the convex, balanced 0-neighborhoods i n E , and V o denotes the polar of V in 15'. We refer to Bierstedt [3] for the properties of the inductive dual. We follow the standard terminology from infinite dimensional complex analysis, as found for instance in the books of Dineen [GI or the author [14]. If m E No = N U (0)) then 'P(mE; F ) denotes the vector space of all continuous rn-homogeneous polynomials from E into F . If F = C,then we write 'P(mE)instead of P(mE;C). If U is an open subset of E , then X(U;F ) denotes the vector space of all holomorphic mappings from U into F , whereas X " ( U ; F ) denotes the vector subspace of all f E X(U;F ) such that f ( U ) is bounded in F . If F is a Banach space, then X " ( U ; F ) is a Banach space for the sup norm. If F = C, then we write ' H ( U ) instead of X(U;C), and X " ( U ) instead of X"(U; 6'). T~ denotes the compact opcn topology on X ( U ; F ) or on any subset of X ( U ; F ) . I f f E X(U;F ) , (I E U and nz E No,then P m f ( u ) E P ( m E ; F) denotes the mth
151
Linearization of holomophic mappings of bounded type
homogeneous polynomial in the Taylor series expansion of f at a. If f E 'FI(U; F ) , where F is a Uanach space, then for each set A I l f l l A = S U P ~I ~ l f ( Az ) l l .
cU
we write
2. THE SPACES X"(U; F ) AND G"(U) Let E and F be locally convex spaces, let U be an open subset of El and let U = (Un) be a countable, open cover of U . Let 'FI"(U; F ) denote the locally convex space
'FIm(U;F ) = { f E X(U;F ) : f(U,,) is bounded in F for every n }, equipped with the topology of uniform convergence 011 all the sets Un. If F is a Banach space, then 'FI"(U; F ) is a 1;'ldchct space. If F = C, then we write 'FI"(U) instead of
X"(U; C).
2.1. THEOREM. Let U be an open subset of a locally convex space E , and let U = ( U n ) be a countable, open cover of U . l'hen there are a complete, barrelled (DF)-space G"(U) and a mapping 6" E 'FI"(U; G w ( U ) )with the following universal property: For each complete locally convex space F and each mapping f E Z"(U; F ) , there is a unique mapping TJ E C(C"(U); F ) such that TJ o 6" = f . This property characterizes G"(U) uniquely up to a topological isoinorphistn.
Proof. For each sequence
ct
= ( c Y , ~of ) strictly positive nurnbers, let
Bf;denote the set
and let C"(U) denote tlie locally convex space
C"(U) = { t E~ X"(U)' : .ID,", is .r,-continuous for every a ) , equipped with the topology of uiiiform convergence on all the sets B,",. It is clear that
X"(1.1)
= iiid 'H"(U)B;
and each of the sets BE is .r,-compact, by the Ascoli theorem. Furthermore, each of the sets { f E 'FI"(U) : llfll,yn 5 E ) is r,-closed. Thus an application of [16, Theorem 1.11 shows that the evaluation niappiiig
J : 'FI"(U)
--$
c - ( U ) : = G'"(U)b
is a topological isomorphism. Cleitrly the space G"(U) is complete. That G"(U) is barrelled follows from a general result of Bicrstedt and Bonct "41, or directly from tlie fact that J(B,",)is an equicontinuous subset of G"(U)' for every a. Now since 'H"(U) is a Frdchet space, its strong dual
152
J. Mujica
7 P ( U ) bis a (DF)-space. As a subspace of %"(U)',, the space G"(U) has a fundamental sequence of bounded sets, and is therefore a (DF)-space. From this point on the proof of Theorem 2.1 is similar to the proof of [16, Theorem 2.11, and is therefore omitted. PROPOSITION. Let U be an open subset of a locally convex space El and let (U,,) be a countable, open cover of U . Then: (a) For each bounded set 13 c G"(U), there are R E N and p > 0 such that B c pT6u(Un), where FA denotes the closed, convex, balanced hull of A . (b) For each complete locally convex space F , the mapping 2.2.
U
=
f
E 'H"(U; F ) -+ TI E L(G"(U); F )
is a topological isomorphism, when L(G"(U); F ) is endowed with the topology of uniform convergence on the bounded subsets of G"(U).
Proof. Since the evaluation mapping
J : 'FI"(U)
--$
G"(U);,
is a topological isomorphism, we can prove (a) by imitating the proof of [16, Lemma 4.21. And (b) is clearly a direct consequence of (a).
2.3. PROPOSITION. Let E and F be locally convex spaces, with F complete. Let U be an open subset of E , let U = (U,,) be a countable, open couer of U , and let ( f i ) c IFI"(U; F ) . The.n the set U;f;(U,) is bounded in F f o r every n E lli if and only if the family ( T j , )c L ( G m ( U ) ;F ) is equicontinuous.
The proof of this proposition is similar to t,he proof of [16, Proposition 2.51, and is therefore omitted. 2.4. PROPOSITION. Let U be an open subset of a Banach space E , and let U = (Un)
be a countable, open cover of U , with each U, bounded. Then E is topologically isomorphic to a complemented subspace of G"(U).
Proof. Under the hypotheses of the proposition, the inclusion mapping U ~t E belongs to 'FI"(U; E ) . By Theorem 2.1 there exists T E L(G"(U); E ) such that T(6,) = z for every L E U . Then the proof of [15, Proposition 2.31 or [16, Proposition 2.61 applies.
Linearization of holomorphic mappings of bounded type
153
3. A N (LB)-REPRESENTATION OF G"(U) Let U be an open subset of a locally convex space E , and let U = (U,) be a countable open cover of U . Let B, denote the closed unit ball of X"(U,), and let G"(U,) denote the Banach space
G"(U,)
= {ti E
X"(U,)'
: ulB, is .r,-continuous
},
with the induced norm. As pointed out i n the proof of [15, Theorem 2.11, a theorem of Ng [17] guarantees that the evaluation mapping
J,, : X"(U,)
G'(U,)'
-+
is an isometric isomorphism. Clearly the FrCchet space X " ( U ) can be represented as the projective limit of the Banach spaces X"(U,) by incans of the restriction mappings
R, : X " ( U )
-+
X"(U,).
Consider the dual mappings
R:, : 'H"(U,,)'
-+
X"(U)'.
One can readily verify that R:,zLE G"(U) for evcry u E G"(U,), that is, the mapping R, G"(U,)). Thus if
is continuous for the topologies a ( X " ( U ) , G " ( U ) ) and a(X"(U,),
S, : G"(U,,)
-+
G"(U)
denotes the restriction of RL to G'"(U,,), then the mappings R, and S, are dual to each G " ( U ) ) and (X"(U,,), G"(U,)). We will other with respect to the dual pairs (XFI"(U), see that, undcr suitable liypotlieses, the (DF)-space G"(U) can be represented as the inductive limit of the Baiiach spaces CFI"(U,,)by means of the mappings S,,. We begin with the following proposition. 3.1. PROPOSITION. Let U be a balanced, open subset o J a Banach space E , and let U = (U,) be a sequence of balanced, bounded, open subsets o J U such that U = UF=.=,U,. Then: (a) The mapping R, : X " ( U ) -i fZm(U,,) has a c(X"(U,,), G"(U,))-dense range. (b) The mapping S, : G"(lJ,) -+ G " ( U ) is injective.
Proof. (a) Let f E X"(U,,). Then, with the notation of [15, Proposition 5.21, the sequence of Cesbro means (a7,,f)convergcs to f i n (X"(U,), T ~ ) Since, . by [15, Propositions 4.7 and 4.91, (X"(U,), T?)' = G'=(Un),w e conclude that (a,f)converges to f for the topology c('l-lM(U,,),G"(U,)). Since each amf is a polynomial, and since each iJk is bounded, wc conclude that a,f E X " ( U ) for every m E N ,thus proving (a).
154
J. Mujica
Since (b) is a direct consequence of (a), the proof of the proposition is complete. Thus, under the hypotheses of Proposition 3.1, we may regard each G w ( U n ) as a vector subspace of G"(U).
3.2. THEOREM. Let U be a balanced, open subset of a Banach space El and let U = ((in) be a sequence of balanced, bounded, open subsets of U such that U = Ur=lCJn and pnUn c Un+I, with pn > 1, f o r every n E N . Then G"(U) = ind G w ( U n ) , and this inductive limit is boundedly retractive.
The key to the proof of Theorem 3.2 is the following lemma. 3.3. LEMMA. Let U be a balanced, open subset of a Banach space E , and let U = ( U n ) be a sequence of balanced, bounded, open subsets of U such that U = UrZlUn and pnUn C Un+l, with p n > 1, for cvcry n E N . Then G"(U) and Gm(Un+l) induce the same topology and tlie same uniform structure on the closed unit ball of G w ( U n ) .
Proof. Let A k and Bk denote the closed unit balls of G w ( U k ) and y"(uk),respectively. By a lemma of Grothendieck [lo, p. 98, Lemma], to prove Lemma 3.3 it suffices to show that for each E > 0 there exists cy = ( c y k ) , with a k > 0, such that
Let f E B,,+l. Since pnUn
c U,l+l, tlie Cauchy integral
M
formulas yield the inequalities
M
and this is less than &/2 for N sufficiently large. On the other hand, since u with ck > 0, for every k E N ,we have that N-1
N- 1
m=O
for every k E N. Set cy = ( a k ) , where preceding inequalitics show that
k
C ck
un+l)
N- 1 m=O
cyk
= fC,"lAcr,
for every k E
N. Then
and (3.1) follows directly from (3.2). Proof of Theorem 3.2. Clearly U ~ = l G w ( U nc) G"(U), and the inclusion mapping ind G m ( U n ) L) G"(U) is continuous.
the
Linearization of holomoiphic mappings of bounded type
155
On the other hand, by Tllcorem 2.1 and by [15, Theorem 2.11, we have the canonical topological isomorphisms
G"(U)b = R " ( U )
= proj
' P ( U , , ) = proj Gm(Un)'.
Thus, by polarity, each bounded subset of G"(U) is included in the closure in G"(U) of a bounded subset of some Gm(Un).Since Gm(Un+l)is complete, it follows from Lemma 3.3 that and thus each bounded subset of G"(U) is included in some Gm(Un) and is bounded there. Thus G"(U) = UF!lGm(Un), and the spaces C"(U) and ind Gm(Un)induce the same topology on each bounded subset of G"(U). Since we already know from Theorem 2.1 that G"(U) is a (DF)-space, a theorem of Grothendieck [8, p. 68, Thdorkme 31 guarantees that the identity mapping G"(U) + i d G"(U,) is continuous. This completes the proof. An cxarninatioii of the proof of Lemma 3.3 yields the following result.
3.4. PROPOSITION. Let E a d F be Danach spaces, let U be a balanced, open subset of E , and let U = ( U n ) be a sequence of balanced, bounded, open subsets of U such that U = Ur=lUn and pnUn c Un+l, with p,, > 1, f o r every n E N . T h e n 'H"(U; F ) is a quasi-normable Fre'chet space.
4. THE APPROXIMATION PROPERTY
To begin with we state the following conseyucnce of [15, Theorem 5.41. 4.1. THEOREM. Let U be a balanced, bounded, open subset of a Banach space E . Then G"(U) has the approximation propeify if a n d only if E has the approximation property. With the aid of Tlieorcm 3.2 we can easily extend this result to G"(U).
THEOREM. Let U be a balanced, open subset of a Banach space E , and let (U,,) be a sequence of balanced, bounded, open subsets o f U such that U = Ur=lUn and pnUn c Un+l, with p,, > 1, f o r every n E N . T h e n G w ( U ) has the approximation property if and only i f E has the approximation property. 4.2.
U
=
Proof. By Proposition 2.4, E is topologically isomorphic to a complemented subspace of G"(U). IIence E has the approximation property if so does G"(U). If, conversely, E has tlie approximation property, then each Gm(Un)has the approximation property, by Tlieorcm 4.1. By Theorem 3.2 we may apply a result of Bierstedt and Mcisc [5, Satz 1.21 to conclude that G"(U) = ind G"(Un) has the approximation
156
J. Mujica
property. 5. HOLOMORPHIC FUNCTIONS O N QUOTIENT SPACES In 115, Corollary 4.121 we gave an explicit description of G"(U). By examining the proofs of [15, Lemma 4.6 and Theorems 4.5 and 4.111, we can sharpen the conclusion of [15, Corollary 4.121 as follows. 5.1. THEOREM. Let U be an open subset of a Banach space E . Then G"(U) consists of all linear functionals u E G"(U)' of the form 00
u =
p1 S,,,
3=1
with
( a j )E
1' and
(zj)c
U . Aloreover,
IIuII = i n f
00
C IajI,
j=1
where the infimum is taken over all such representations of u . 5.2. THEOREM. Let E and F be Banach spaces, and let H E L ( E ; F ) be surjective. Let U be an open subset of E , atid let 1f = .(U). Let S : G"(U) 4 G"(V) be the unique continuous linear mapping such that the following diagram is commutative:
u " . v
SlJ
1
Gm(U)
5
1 Sv G"(V)
Then S maps the ball { u E G"(U) : I I I L I I < 1) onto the ball { v E G"(V) : [lull < 1). Proof. Since SV o T E 'H"(U; Gm(U)), [15, Theorem 2.11 guarantees that llSll = 116~o H I I = 1. IIence llSiill < 1 for evcry u E G"(U) with IIuJI < 1. If, conversely, v E G"(V), with ((v1(< 1, thcn by Tlicorem 5.1 we can find (y,) c V and (a,) E l', with C,"=,(a31< 1 , such that
La3by,. 00
v =
3=1
Write yI = r ( z 3 )with , zl E U , for every j E If we define
r'-
N.
Linearization of holomofphic mappings of bounded type
157
then u E G " ( U ) , llull < 1 arid
5.3. THEOREM. Let E and F be Banach spaces, and let A E C ( E ; F ) be surjective. Let U be a balanced, open subset of E , and let U = (U,) be a sequence of balanced, bounded, open subsets of U such that U = uF=lU, and p,Un c Un+,, with pn > 1, for every n E N . Let V = A ( U ) , and let V = (V,), where V, = A(U,) f o r every n E N . Let S : G w ( U ) --+ G " ( V ) be the unique continuous linear mapping such that the following diagram is comnautative:
U
L
G"(U)
--%
611
1.
V
1 6v Gm(V)
Then S is surjective and open. Each bounded subset of G"(V) is the image under S of some bounded subset of G"(U).
-
Proof. For each n E N we have a commutative diagram GCO(Un) Sn 1
Gw(Vn)
-+
G"(U)
1s
G"(V)
Let B be a bounded subset of G m ( Y ) . By Theorem 3.2, B is included in some G"(Vn) and is bounded there. By Theorem 5.2, there is a boundcd sct A in Gw(Un) such that B = & ( A ) . Thus B = S ( A ) ,with A bounded in G"(U). In particular we have shown that S is surjective, and therefore open, by an open mapping theorem due t o Grothendieck (see [9, Introduction, p. 17, Thd.orL:me B ] )or [lo, p. 200, T h d o r h e 21). 6. HOLOMORPHIC FUNCTIONS ON PRODUCT SPACES
6.1. THEOREM. Let E arid F be Banach spaces, and let U and V be open sets in E and F , rcsycctively. Then: (a) X"(U x V ) is isometrically tsoniorphic to X"(U; X W ( V ) ) . (b) G"(U x V ) is isonieti~icullyisomorphic to G " ( U ) & G " ( V ) .
Proof. (a) Clearly t h e mapping
f E X"(U x V )
--+
?€
X " ( U ; %"(\'))
defined by ?(z)(g) = f ( z , y ) for all z E U arid y E V , is an isometric isomorphism. (b) O n one hand the mapping (T, y)
E IJ x V
--t
6, @ 6, E G"(U)&G"(V)
158
J. Mujica
is holomorphic and ~ ~ 6 , @ 6=, Il6,ll ~ ~ 116,ll = 1 for all z E U and y E V . By the universal property of G"(U x V ) (see [15, Theorem 2.1]), there is a continuous linear mapping
S : G"(U x V ) -+ G"(U)&G"(V), with llSll = 1, and such that hand the mapping
SS[,,,)= 6,
@
(u,U ) E G"(U) x G"(V)
6, for all -+
z E
U and y E V . On the other
u x u E G"(U x V )
defined by (ux v)(f) = u ( v o f) for every u E G"(U), u E G"(V) and f E 'H"(U x V ) , is bilinear and IIu x 011 5 llull 1 1 ~ 1 1for all u E G"(U) and v E G"(V). By the universal property of tensor products, there is a continuous linear mapping
T : C"(U)&G"(V)
+ Gm(U x
V),
with llTll 5 1 , and such that T ( u @ v )= u X U for all u E G"(U) and u E G"(V). Hence T o S(6(,, ,I) = b(,, ), and S o T(6, @ 6,) = 6, 86, for all z E U and y E V . It follows that S is an isometric isomorphism, with inverse T . 6.2. THEOREM. Let E and F be Banach spaces, and let U and V be balanced, open sets in E and F , respectively. Let U = (Un) be a sequence of balanced, bounded, open subsets of U such that U = Uz=lUn and p,Un c Un+l, with pn > 1, f o r every n E fir. Likewise, let V = (Vn) be a sequence of balanced, bounded, open subsets of V such that V = Ur=p=,Vn and anVn c Vn+,, with an > 1, f o r every n E N . If U x V denotes the sequence (U, x Vn), then: (a) 'FI"(U x V ) is topologically isomorphic to 'H"(U; 'H"(V)). (b) G"(U x V ) is topologically isomorphic t o G"(U)&G"(V).
Proof. (a) Clearly the mapping
f E R"(U x V ) -+ JE R"(U; 'H"(V)) defined by f ( z ) ( y )= f ( z , y ) for all z E U and y E V , is a topological isomorphism. (b) A glance at the diagram
Un x Vn
1
U x V
6un@6vn +
'*
G"(Un)@,G"(Vn)
1
G"(U)G3,G"(V)
shows that the mapping (z, y) E
ux v
-+
6, @ 6, E G ~ ( U ) @ , G " ( V )
belongs to X"(U x V ; G"(U)&G"(V)). By Theorem 2.1 there is a continuous linear mapping S : Gm(Ux V ) + G"(U)&G"(V)
Linearization of holomorphic mappings of bounded type
159
such that S6(,,), = 6,@ 6, for all x E U and y E V . On the other hand, since G"(U) = ind G"(Un) and G"(V) = ind Gm(Vn),the bilinear mapping ( u , v) E G"(U) x G"(V) -+ u x
2,
E
C"(U x V )
defined by (ux v)( f ) = u(v o f)for every u E C"(U), u E G"(V) and f E 'H"(U x V ) , is separatcly continuous. By a theorem of Grothendieck on (DF)-spaces (see [8, p. 66, Corollaire] or [lo, p. 226, Corollaire l]), this bilinear mapping is actually continuous. From now on the proof proceeds exactly as the proof of Theorem 6.1.
7. HOLOMORPHIC MAPPINGS OF BOUNDED TYPE
+
Let U be an open subset of a Banach space E . If U E and x E U , then dU(x) denotes tlie distance from 5 to E\U. Since ldu(z) - d u ( y ) l 5 llz - yII for all x, y E U , we see that dU is a continuous function on U . If U = E , then for convenience we define d u ( x ) = co for every 2 E U . A set A c U is said to be U-bounded if A is bounded and infIEA &(I) > 0. If F is a locally convex space, then 'Hb(U; F ) denotes the locally convex space
'Hb(U; F ) = { f E 'H(U; F ) : f ( A ) is bounded in F for each U-bounded set A } , equipped with the topology of uniform convergence on all U-bounded sets. If F = C, then we write 'Hb(U) instead of 'Hb(U; 6').The mernbers of 'Hb(U; F ) are called holomorphic mappings of bounded type. It is clear that 7&(U; F ) = 'H"(2A; F ) if U = (Un) is any fundamental sequence of open U-bounded sets. In the next proposition we give a fundamental sequence U = (Un) of open U-bounded sets which satisfics the hypotheses of all the results in the preceding sections.
7.1. PROPOSITION. Let U be an open subset of a Banach space El and let (Un) be defined b y 11, = {x E U : llzll < PZ and d ~ ( x>) 2-"}. Then: (a) (Un) is a fuiidaniental scqueiice oJ open, U-bounded sets. (b) If U is balanced, then each U,, is balanced as well and, furthermore, there exists with pn > 1, such that pnUn c U,,, for every 12 E N. The proof of this proposition is straightforward, and is left to the reader. For each
nE
N we may take pn = 1 + 1/n2".
Theorem 2.1 yields Galindo et al. [7].
ill
particular the following result, which improves a result of
160
J. Mujica
7.2. PROPOSITION. Let U be an open subset of a Banach space E . Then there are a complete, barrelled (DF)-space Gb(U) and a mapping 6u E Ftb(U; Gb(U)) with the following universal property: For each complete locally convex space F and each mapping f € ?fb(u; F ) , there is a continuous h e a r mapping T, E L(Gb(U); F ) such that T, o 6u = f . This property characterizes Gb(U) uniquely up to a topological isomorphism.
Theorem 3.2 yields in particular the following result, which improves a result of Galindo et a1 [7]. 7.3. PROPOSITION. Let U be a balanced, open subset of a Banach space E , and let (Un)be a fundamental sequence of open U-bounded sets such that each U,, is balanced and pnUn c Un+l, with pn > 1, f o r every 11 E IN. Then Gb(U) = ind Gm(Un) and this inductive limit of boundedly retinctive. It is clear too that Proposition 3.4 yielcls in particular a result of Ansemil and Ponte [2] and Isidro [12]. Theorem 4.2 yields in particular the following result.
7.4. PROPOSITION. Let U be a balanced, open subset of a Banach space E . Thcn Gb(U) has the approximation property if and only if E has the approximation property. Theorem 5.3 yields in particular the following result. 7.5. PROPOSITION. Let E and F be Banach spaces, and let R E L ( E ; F ) be surjective. Let V be a balanced, open subset o f F , let U = R-'(V), and let S : Gb(U) -+ Gb(V) be the unique continuous linear mapping such that the following diagram is commutative: U
A
V
Gb(U)
3
Gb(V)
bu
1
1 bv
Then S is surjective and open. Each bounded subset of Gb(V) is the image under S of some bounded subset ofGb(U).
Proof. Let (Un) be a fundamcntal sequence of open U-bounded sets such that each U,, is balanced and p,Un c Unfl, with p,, > 1, for every 11 E N . Let V, = .(Un) for every n E A'. Since U = .-'(V), it follows that each V-bounded set is the image under a of some U-bounded set. Whence it follows that (Vn) is a fundamental sequence of open V-bounded sets. Thus Theorem 5.3 applics and yields the desired conclusion.
Remark. There are Banach spaces E and F , there is a surjective mapping a convex, balanced, open set U in E such that, if we set v = r ( U ) , then the canonical mapping s E L(Gb(U); Gb(V))is not surjective. Indeed, if S were surjective, then S would be open, by an open mapping theorem due 7.6. R
E
L ( E ; F ) , and there is
Linearization of holomorphic mappings of bounded type
161
to Grothendieck (see [9, Introduction, p. 17, ThCorkme B] or [lo, p. 200, Thdorkme 21). Then, by a theorem of Grothendieck on (DF)-spaces (see [8, p. 76, Proposition 51 or [lo, p. 228, Proposition 4]), each bounded subset of Gb(V)would be included in the closure of the image under S of some bounded subset of Gb(U). Then the mapping g E 'Hb(v)+ g o n E 'Hb(u)would be an embedding, that is, a topological isomorphism between 'Hb(v) and a subspace of 'Hb(u).But this is not always true, as a counterexample of Ansemil et a]. [I] shows. Finally Theorem 6.2 yields in particular tlie following result.
7.7. P R O P O S I T I O N . Let E and F bc Banach spaces, and let U and V be balanced, open sets in E and F , respectively. Then: (a) 'Ftb(U x V ) is topologzcally isomorphic to ' F t b ( U ; 'Hb(V)). (b) Gb(U x V ) is topologically isomorphic to Gb(U)& Gb(V).
REFERENCES
[l] J . M. Ansemil, R. M. Aron a n d S. P o n t e , Embeddings of spaces of holomorphic functions of bounded type. Preprint, 1990.
[2] J. M. Anseniil a n d S. P o n t e , Ail example of a quasi-normable Fre'chet function space which is not a Scliwcirfz space. In: Functional Analysis, Holomorphy and Approximation Theory, edited by S. hfacliado, pp. 1-8. Lecture Notes in Mathematics, vol. 843. Springer, Berlin, 1SS1. [3]K. D. B i e r s t e d t , A n introduction to locally convex inductive limits. In: Functional Analysis and its Applications, editcd by 1-1. IIogbe-Nlend, pp. 35-133. World Scientific, Singapore, 19SS.
[4] K. D. Bierstedt a n d J. B o n e t , Bidnulity in Fre'chet and (LB)-spaces. In: Progress in Functional Analysis, edited by J. Bonet et al. North-Holland Mathematics Studies. North-Holland, Amsterdam, to appear. [ 5 ] K . D. B i e r s t e d t a n d R. Meise, Bemerkungen uber die Approximationseigenschaft
lokalkonvexer Fur~ktionenriurne.hlatli. Ann. 209 (1974), 99-107.
[6] S. D i n e e n , Coniplcx Analysis in Locally Convex Spaces. North-Holland Mathematics Studies, vol. 57. North-IIolland, Amsterdam, 1981.
[7] P. Galindo, D. G a r c i a a n d M. M a e s t r e , IIoloniorphic mappings of bounded type. J. hlath. Anal. Appl., to appear.
J. Mujica
162
[8] A. Grotliendieck, Sur les espaces ( F ) et ( D F ) . Summa Brasil. Math. 3 (1954), 57-123. [9] A. Grotliendieck, Produits tensoriels topologiques et espaces nucle'aires. Amer. Math. SOC. 16 (1955).
[lo] A. G r o t h e n d i e c k , Espaces Paulo, 1964.
Mem.
Vectoriels Topologiques, 33 ediq8o. Universidade de SiLo
[ll] J. H o r v a t h , Topological Vector Spaces and Distributions, vol. I. Addison-Wesley, Reading, Massachusetts, 1966. [12] J. M. Isidro, O n the distinguished character of the function spaces of holomorphic mappings of bounded type. J. Funct. Anal. 3s (19SO), 139-145. [13] P. M a z e t , Analytic Sets in Locally Convex Spaces. North-Holland Mathematics Studies, vol. S9. North-IIolland, Amsterdam, 1954. [14] J . Mujica, Complex Analysis i.1~Banach Spaces. Studies, vol. 120. North-Holland, Amsterdam, 19%.
North-Holland Mathematics
[15] J . M u j i c a , Linearization of bounded holomorphic mappings on Banach spaces. Trans. Amer. Math. SOC.324 (1991), 867-887. [16] J. M u j i c a a n d L. Nachbin, Linearization of holomorphic mappings on locally convex spaces. J. hilath. Pures Appl., to appear.
[17]K . F. Ng, On a theorem of Dixmier. Math. Scand. 29 (1971), 279-280. [18] R. A. Ryan, Applications of topological tensor products to iiifiiiite dimensional holomorphy. Ph. D. thesis, Trinity College Dublin, 19SO. [19] M. S c h o t t e n l o h e r , E-products and continuation of analytic mappings. In : Analyse Fonctionnelle et Applications, edit6 par L. Nachbin, pp. 261-270. Hermann, Paris, 1975.