Math. Nachr . 193 (1998), 27-36

Compact Sets of Holomorphic Mappings

By CHRISTOPHER BOYD of Galway and SEAN DINEEN of Dublin

(Received January 15,1996)

Abstract. Compactness of locally bounded sets of holomorphic functions with infinite dimen- sional domains is connected, using Heinrich’s density condition, to the Schwartz and semi - Monte1 properties on the domain. The metrizability of bounded subsets for various spaces of holomorphic functions is investigated.

1. Introduction

In a recent paper we noted, and indeed were surprised, by the ubiquitous role of locally bounded sets (of holomorphic mappings) in infinite dimensional holomorphy. This prompted consideration of the following questions:

(a) When are locally bounded sets of holomorphic functions topological, i. e., when do they coincide with the bounded sets of a locally convex topology on the space of holomorphic functions?

(b) What topological or topological vector space properties are shared by locally bounded sets of holomorphic mappings?

Let U be an open subset of a locally convex space E , and let Z ( U ) denote the set of all holomorphic functions on U . The 76 topology on Z ( U ) is generated by all semi -norms p with the property that for every increasing countable open cover (Vn)n of U there exists a positive integer .no and C > 0 such that

for all f E Z ( U ) . Results in [13] include an abstract answer to (a) and show, modulo some interpreta-

tion, that in answering this question we can confine ourselves to the 76 topology. We

1991 Mathematics Subject Classification. Primary: 46G20; Secondary: 46A 11, 46M07. Keywords and phrases. Holomorphic function, locally bounded sets, density condition, metrizable

bounded sets.

28 Math. Nachr. 193 (1998)

were interested in a more concrete characterization or, at least, in results which would immediately yield concrete examples.

As regards (b) we were interested in properties, which were independent of solutions to (a). Nevertheless, both questions impact on one another and the results of this paper and an examination of known examples led us to the following conjecture which involves parts of both (a) and (b).

Conjecture 1.1. If the T,5 -bounded subsets of X ( U ) , U an open subset of a locally convex space E , are relatively compact, then the 7 6 -bounded subsets of X ( U ) are locally bounded.

This of course suggests the following question which is the main focus of our attention in this article.

(c) When are the locally bounded sets of holomorphic functions on a domain in a locally convex space 7 6 -relatively compact?

We denote by ro the topology on X ( U ) of uniform convergence on compact subsets of U . A semi-norm p on X ( U ) is said to be ported by the compact subset K of U if for each open set V , K c V U , there is CV > 0 such that

P(f) L cv l l f l lv for all f E X ( U ) . The rw topology on R ( U ) is the topology generated by all semi- norms ported by compact subsets of U . Let

G(U)={q5 E H(U) ' : q5 restricted to the locally bounded subsets of Z ( U ) is continuous}

By [20], G(U): = ( X ( U ) , 7 6 ) and the equicontinuous subsets of G(U)' can be iden- tified with the locally bounded subsets of X ( U ) . Hence, as we shall see, the study of (c) is related to locally convex properties of the space G(U) .

We use [lo] as a standard reference for infinite dimensional holomorphy and [17, 181 for the theory of locally convex spaces.

2.

In considering question (a) we first note that the rw and 7 6 topologies coincide on locally bounded sets of holomorphic functions and thus we may confine our attention if necessary to the usually more flexible topology rw. Our next step is to show that we may confine our investigation to a certain class of semi - Montel spaces on one side and to Schwartz spaces on the other. We show ,

(*I Schwartz ==$ positive solution to (c) j semi - Montel

By considering the compact open topology on certain F'rCchet - Montel spaces we show that neither of these implications can be reversed, i,e., neither yield necessary and sufficient conditions on a locally convex space to fully answer (c).

Boyd/Dineen, Compact Sets of Holomorphic Mappings 29

We use Ei to denote the inductive dual of E . That is Ei = lim (E')u. where U ( E )

is the collection of all absolutely convex neighbourhoods of 0 in E. We note that this is just the space E' endowed with the T~ (or 76) topology. We let EL = ( E ' , T ~ ) .

To prove the right-hand implication in (*) we require an abstract result related to the invariance of ultrapowers ([15]). We recall that a locally convex space E has the density condition if

For every X:U(E) + lR+\{O} and every V E U ( E ) there is a finite subset Ul of U ( E ) and B bounded in E such that

* U ( E )

Proposition 2.1. Let E be a complete locally convex space, then the following are equivalent:

(i) E is semi- Monte1 and satisfies the density condition, (ii) For every X : U ( E ) + lR+\{O} there i s a compact subset K of E such that for

every V E U ( E ) there is a finite subset U1 of U ( E ) such that

(iii) For every X : U ( E ) + R+\{O} and every V E U ( E ) there i s a finite subset U1 of U ( E ) and K compact in E such that

Proof . We have (i) implies (ii) by [14, Proposition 51 and clearly (ii) implies (iii). If (iii) holds, then E satisfies the density condition. Taking polars we see that for

every V E U ( E ) and every X : U ( E ) + lR+/{O} there is a finite subset UI of U(E) and a compact subset K of E such that

Since sets of the form I? (U,,, X(V)- 'U"), are a fundamental system of neighbour- hoods of 0 in E:, EL and E,! induce the same topology on the equicontinuous subsets of E'. By [18, Theorem 9.3.71 we conclude that EL = E,!. In particular, EL = Ei which

Using [15, Proposition 1.51 we note that the conditions in the above proposition are equivalent to saying that for every countably incomplete N(E)+ -good ultrafilter D, ( E ) D = E. Condition (iii) is got by replacing the bounded set in the definition of the density condition by a compact set.

implies that E is semi-Monte1 and so (iii) implies (i).

Motivated by the above result and by [14, Proposition 91 we have:

30 Math. Nachr. 193 (1998)

Proposition 2.2. Consider the following conditions on a complete locally convex space E:

(i) For every X : U(E) 4 lR+\{O} there is a compact subset K of E such that for every V weakly open neighbourhood of 0 in E there is a finite subset Ui of U ( E ) such that n X ( U ) U ~ K + V ,

UEUI

(ii) Ef = Ei1 (iii) The equicontinuous subsets of E‘ are relatively compact in Ei, (iv) E i s semi- Montel and inductively s emi - reflexive, i. e. , (E,!)’ = El Then (i) implies (ii) and (ii), (iii) and (iv) are equivalent. If E is infrabarrelled then

all the above conditions are equivalent.

P r o o f . Suppose (i) holds. A neighbourhood basis for Ei is given by sets of the form

for any choice of {X(V) > 0 : U E U}. Taking polars in (i) it follows that for any choice of {X(U) > 0 : U E U } there is K compact in E such that for every zero- neighbourhood V in (E,a(E,E’)) there is Ul finite in U such that

Taking the union over all weakly open zero neighbourhoods V we have

This implies EL = E,! and hence (i) implies (ii). If (ii) holds then EL is bornological and therefore, if E is infrabarrelled, (i) holds by

[14, Proposition 91. From [17, Theorem 3.4.1 and Proposition 3.9.81 we have (ii) implies (iii) and by [18,

Theorem 9.3.71 we see that (iii) implies (ii). Condition (ii) holds if and only if Ef = EL and EL = E,!. The first equality says that

E is semi - Montel while the second says that E is reinforced regular. By [3, Theo- rem 3.51 this is equivalent to E being semi - Montel and inductively semi - reflexive.

0

It is clear that the conditions in Proposition 2.2 are implied by those in Proposi- tion 2.1. All the conditions in both propositions are equivalent for F’rkhet spaces and DF spaces and for the space C(”). Montel (LF) spaces which satisfy condition (M) (resp. (Mo)) of Retakh also satisfy the conditions in Proposition 2.1 (resp. Proposi- tion 2.2) by [14, Proposition 81 (resp. [23, Lemma 4.2 I) .

By (161 there exist complete semi - Montel spaces without the density condition. In [19] an example is given of a Montel non-bornological perfect sequence space, A,

Boyd/Dineen, Compact Sets of Holomorphic Mappings 31

with X and Xb complete. Hence there exist, by Propositions 2.1 and 2.2, examples of complete Montel spaces which do not have the density condition.

Since the space G(U) is complete we can apply Proposition 2.2, (iii)*(iv), to obtain the first half of the following Theorem which is a partial answer to question (c).

Theorem 2.3. I f the locally bounded subsets of Z ( U ) , U open in E , are 76 - relatively compact then E is semi-Montel. Conversely, if E is Schwartz, and U is balanced or Lindelof, then the locally bounded subsets of Z ( U ) are 76 -relatively compact.

A

Proof. We first consider the case when U is balanced. For each integer n, @ E - the space of n - fold symmetric tensors completed with respect to the projective or a topology - is Schwartz and hence semi - Montel and satisfies the density condition. By Propositions 2.1 and 2.2 the locally bounded subsets of (P("E),T,) are relatively compact and [12, Lemma 3.41 implies that the locally bounded subsets of X ( U ) are 7w - relatively compact.

Next, we suppose that U is Lindelof. Let B be a locally bounded subset of X ( V ) . Then, for each x in U there is an absolutely convex, closed neighbourhood V, of zero such that the functions in B are uniformly bounded on x + V,. As E is Schwartz, we can find an open neighbourhood W, of zero such that W, is compact in Eve. Since U is Lindelof the open cover (x + W2)zEu of U contains a countable subcover W = (xi + WZi),Env. By Montel's Theorem, we can find a net consisting of elements of B , (fo)aEr, which converges to a holomorphic function f in the compact-open topology. We define f,"' on V,, by f,"I(y) = fa(.] + y). By Liouville's Theorem, we may suppose without loss of generality, that the f,"l are holomorphic functions on the unit ball of Eve , . Since the (f,",) are locally bounded they extend to unique holomorphic functions on Bv=,, the unit ball of &,, . Using Montel's Theorem again we can find a subnet ( f ~ ~ , ~ ) o , E r l which converges to a holomorphic function fl on compact subsets of the unit ball of Eve, . In particular, (f~~,l)alErl will converge uniformly to f l on the image of W,, in EvZl and hence (fol,l)orlErI will converge uniformly to fl07r1 on 51 +W,,, where a1 is the canonical projection from E onto Eva, . As ( fo)oEr converges pointwise to f, it follows that f = f l on1 and so (fal,l)alErl will converge uniformly to f on XI + W,, . By induction we can find a sequence of decreasing subnets of (fa)eEr, (fan,n)anErn, such that (fo,,m)a,Erm converges uniformly to f on each of the open sets XI + W,, , . . . , x, + W,, .

Let F = { (a ,n) : Q E rn, n E IN} be ordered by (a ,n ) 2 (P,m) if n 2 m and Q 2 p. Using the map q5 : F --t r sending (a , n) t o a , we see that (fo)aEi; is a subnet of (fa)aEr. Furthermore the net (f0),,F converges uniformly to f on each W Z i . Therefore (fo)aEF converges to f in Zoo(W) and thus in ( Z ( U ) , q ) = li_mZoo(U).

0

Thus we see that the semi - Montel property is necessary and the Schwartz property Sufficient to imply that locally bounded sets are 76 - relatively compact. We now show that neither condition on its own is necessary and sufficient.

n,s,n

U

This shows that B is 76 -relatively compact.

32 Math. Nachr. 193 (1998)

Proposition 2.4. If E is a he'chet Montel space and U is a balanced open subset of E then the locally bounded subsets of X ( U ) are T, -relatively compact if and only i f 7 , = T, on 3c(U).

Proof . If the closed locally bounded sets are T, -compact then, since 7w 2 T,, we have that T~ and T, agree on the locally bounded subsets of X ( U ) . By modifying the proof of [l] we see that T, = 7, on 3t (U) .

Conversely, if 70 = 7,, it suffices to show that locally bounded sets are 70 -relatively compact. The Cauchy inequalities imply that locally bounded sets of holomorphic mappings are equicontinuous and an application of Ascoli's theorem completes the proof. 0

By ANSEMIL - TASKINEN, [2], we know that there exists a F'rCchet - Montel space with T, # T, and F'rCchet non-Schwartz spaces with T, = T,, [ll], and this shows that, our conditions are not necessary and sufficient.

If the 7-6 -bounded subsets of holomorphic functions are locally bounded we obtain, as a corollary to Theorem 2.3, the following generalisation of [13, Prop. 41 and we also note that the converse to our conjecture is true for Schwartz spaces.

Corollary 2.5. If U i s a balanced or Lindelof open subset of a Schwartz space E and the 76 -bounded subsets o f ' f l ( U ) are locally bounded then ( ? f ( U ) , 7 ~ ) is a Montel space.

Our next proposition verifies the conjecture for a rather different class of spaces. We recall that a locally convex space E is said to be holomorphically Mackey if for every open subset U of E and for every Banach space F we have Z(U, F ) = R(U, Fn), where F, is the Banach space F endowed with the weak a(F, F ' ) topology.

Proposition 2.6. Let U be a n open subset of a holomorphically Mackey space E . I; the 7 6 - bounded sets are relatively compact then the 7-6 -bounded subsets of X ( U ) a n locally bounded and ( % ( U ) , T ~ ) is a Montel space.

Proof . Since the 7 6 -bounded subsets are relatively compact and the 76 topology it always barrelled it follows that ( X ( U ) , 7 6 ) is Montel. By (ii) and (iii) of Proposition 2.: we have G(U)I, = G(U): = ( X ( U ) , T ~ ) . Since U is holomorphically Mackey it follow: from [20, Theorem 4.51 that G(U) is Mackey. By [22, IV.4.181, G(U) is reflexive and in particular, barrelled. Hence the 76 -bounded subsets of Z ( U ) are locally bounded

c Most of the standard classes of locally convex space are holomorphically Macke!

A uncountable, is not, however, but in this case the 7 6 - (IS, 101). The space bounded sets are not necessarily relatively compact nor locally bounded (see [7]).

The above Proposition, [8] and (6, Corollary 3.21 imply the following result.

Proposition 2.7. Let E = Cz=, Em denote a nontrivial direct sum of &&he Schwartz spaces. Then ('U(U),76), U an arbitrary open subset of E , is Montel if an only if each Em is isomorphic to 6" for some positive n (depending on m) or 6".

Boyd/Dineen, Compact Sets of Holomorphic Mappings 33

3.

We now present results suggested more by the method of proof in Propositions 2.1 and 2.2 than by any attempts to establish our conjecture. The conditions in proposition 2.1 are necessary and sufficient for the invariance of an ultrapower. This, as shown in [15], is connected with the density condition which in turn is related, for metrizable locally convex spaces, with the metrizability of the bounded subsets of the dual (41. Now ( R ( U ) , 76) is neither metrizable nor the dual of a metrizable space so the results in [4] cannot be applied directly. Nevertheless, the analogous conditions remain equivalent for holomorphic functions on a balanced domain in a F'rdchet space. A result similar to (b) in the following proposition was proved in (91 for F'rdchet spaces with a different type of decomposition.

Proposition 3.1. Let {En},, be a n S - absolute decomposition for the locally convex

(a) Then the bounded subsets of E are rnetn'zable if and only if the bounded subsets

(b) E wall satisfy the density condition if and only if each En satisfies the density

space E (see [lo, p. 1141).

of each En are metrizable.

condition.

Proof . (a) It suffices to use the fact that, a bounded net (xa)= in E , converges to 0 if and only if xt + 0 in En as (Y + co for all n.

(b) Since the density condition is inherited by complemented subspaces we see that each En has the density condition whenever E has. Suppose conversely that each En has the density condition. Consider X : U(E) + R+ \ {0} and V an absolutely convex neighbourhood of 0 in E. Without loss of generality we may suppose that the gauge, p , of V satisfies

for every C,"=l 5, in E . Since the decomposition {En}n is S-absolute, the semi- norm q on E defined by

\n=1 1 n=l

is continuous. Let W denote the unit ball of q. Choose no so that

for all C:il z, in W . If U is a neighbourhood of 0 in E then U" = U n En is a neighbourhood of 0 in En.

Since each En has the density condition we can find, for each n, a finite subset U, of U ( E ) and a bounded subset Bn of En such that

34 Math. Nachr. 193 (1998)

If we let B = x:Ll B, and U = { W } U U:L, U,, then B is bounded, U is finite and

This proves that E satisfies the density condition. 0

As a consequence we have:

Theorem 3.2. Let E be a metrizable locally convex space, then the following con-

(1) 8 E , (equivalently G ( K ) , G(U)) satisfies the density condition f o r each integer

ditions are equivalent h

n.6.n n (respy'each balanced compact subset K of E , each balanced open subset U of E ) .

( 2 ) The bounded subsets of (P(nE), 7,) (equivalently (X(K),-r,), ( X ( U ) , 7 6 ) ) are metrizable f o r each integer n (resp. each balanced compact subset K of E , each balanced open subset U of E ) .

h

Proof . If 8 E has the density condition it is distinguished and by [4] the bounded

subsets of ( 6 E ) : = ( 8 E ) = ( P ( n E ) , ~ w ) are metrizable. Conversely, if the

bounded subsets of (P("E) , 7,) are metrizable, (5 , Theorem 2, Section 21 implies that

satisfies the dual density condition and [21, Corollary 2) implies that 8 E 0

( n i r E ) satisfies the density condition. It suffices to apply Proposition 3.1.

If U is an open subset of a metrizable locally convex space E and B is a closed bounded subset of X ( U ) then B is TO-compact and, on B , the T~ topology coincides with the topology generated by the point evaluations of a dense sequence of U . Hence, if E is separable then B is metrizable. Conversely, if the bounded subsets of ( ? f ( U ) , r o ) are metrizable then, as relatively compact sets, they are separable. In particular, the bounded sets in EL are separable and hence E is separable. We conclude that 7,-bounded sets in X ( U ) are metrizable if and only if E is separable. We can of course vary the topologies and obtain similar results.

Let K be a compact subset of a locally convex space E. In [ll], a sequence { A n } n

of bounded subsets of E is said to converge to K if for every neighbourhood U of K there is an integer no such that K 5 A , 5 U for all n 2 no. The 7 b topology on ? f ( U ) , U balanced open in E , is defined to be the topology generated by all semi -norms of the form

n,s,x h I

, 8 1 7 n,s,r

h

n , s , r

where is any sequence of bounded subsets of E which converge to a balanced compact subset of U. It is easily seen that { (P("E) , p)}, is an S-absolute decompo- sition for ( N ( U ) , Tb). In many cases, e. g. Banach spaces or F'rCchet - Schwartz spaces, 7 6 coincides with rw and this gives a projective description of rw.

Boyd/Dineen, Compac t Sets of Holomorphic Mappings 35

We recall that Rb(U) denotes the subspace of R ( U ) consisting of holomorphic func- tions which are bounded on the strictly bounded subsets of U . The topology of uniform convergence on strictly bounded subsets of U , /3, endows ‘flb(u) with the structure of a Frkchet space when E is normed. If U is balanced then ( x b ( U ) , p ) has {(P(nE),/3)}n as an S - absolute decomposition. For these two spaces we have the following result.

Proposition 3.3. Let U be a balanced open subset of a metrizable locally convex space E , then the bounded subsets of (%b(U),,d) or ( x ( U ) , ~ b ) are metrizable if and only if E satisfies the density condition.

Proof . Since EL is complemented in (?&,(U) , /3) and (R(U) ,Tb) we see that the condition is necessary. Conversely, for each integer n we have

& , ( “ E ) = & , ( E , L b ( ” - ’ E ) ) .

Thus if E has the density condition then, by induction and [5 , Proposition 6, Section 11, the bounded subsets of each J!+,(”E) are metrizable. Since (P(”E), /3) is complemented in Lb(”E) we see that the bounded subsets of each (P(”E),/3) are metrizable. The

0 result now follows from Proposition 3.1.

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Department of Mathematics University College Galway Galway Belfield Ireland Dublin 4

Ireland e -mail: sdineen@ollamh.ucd.ie

Department of Mathematics University College Dublin

Current address of the first author: Department of Mathematics university College Dublin Belfield Dublin 4 Ireland