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NUCLEAR AND INTEGRAL POLYNOMIALS ON C

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, I UNCOUNTABLE

CHRISTOPHER BOYD

Abstract

We show that forIan uncountable index set andn3 the spaces of alln-homogeneous poly- nomials, alln-homogeneous integral polynomials and alln-homogeneous nuclear polynomials onC(I)are all different. Using this result we then show that the class of locally Asplund spaces (see [10], [6] for definition) is not preserved under uncountable locally convex direct sums nor is separably determined.

1. Introduction

Given an uncountable index setI we consider the locally convex direct sum C(I). Holomorphic functions on this space were first considered in [14] where it is shown that C(I) is not holomorphically Mackey. In [5] we investigated holomorphic functions on C(I) proving the three natural topologies, τ0, τω

andτδ, coincided onH(C(I))if and only ifIhas cardinality less than the first measurable cardinal. Holomorphic functions on the Cartesian productCIwere considered by Barroso and Nachbin, [3], who proved that the compact open and Nachbin ported topologies differed onP(2CI)forIuncountable. In this paper we conclude our examination by showing that the space ofn-homogeneous polynomials,n-homogeneous integral polynomials andn-homogeneous nuc- lear polynomials onC(I) are all different whenI is an uncountable set and n≥3.

Defant, [10], introduced and studied the concept of a space whose dual has the local Radon Nikodým property as a locally convex generalisation of the concept of Asplund Banach space. This property was renamed locally Asplund in [6] and further studied in [7], [8]. In the final section we use our results concerning polynomials on C(I) to show that local Asplundness is neither preserved under locally convex direct sums nor is separably determined. We refer the reader to [15] for further information on homogeneous polynomials on locally convex spaces.

Received September 1, 2001; in revised form January 2, 2002.

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2. Nuclear and Integral polynomials onC(I)

In his thesis, [1], Alencar says that ann-homogeneous polynomialP on a loc- ally convex spaceEis (Pietsch-)integralif there is an absolutely convex closed neighbourhood of 0,U, and a finite regular measureµon(U, σ(E, E))so

that P (x)=

Uφ(x)ndµ(φ)

for all xE. This definition generalises the concept of integral polyno- mial on a Banach space introduced by Dineen, [13]. The space of all n- homogeneous integral polynomials onE is denoted byPI(nE). Clearly we have thatPI(nE) =

U∈UPI(nEˆU). We makePI(nE)into a locally convex space by giving it the topology, τI, defined by the locally convex inductive limit

indU∈U(PI(nEˆU),.(U,I)) where

PU,I =inf

µU :P (x)=

Uφ(x)ndµ(φ)

is the integral norm onPI(nEˆU).

It can be shown, see [6], that then-fold symmetric-tensor product,

s,n,E, is an inductive predual of(PI(nE), τI), i.e.

s,n,E

i =(PI(nE), τI). Ann-homogeneous polynomialP on a locally convex spaceEis said to benuclearif there is an absolutely convex closed neighbourhood of 0,U, in E,k)kbounded inEˆU andk)k1so that

P (x)=

k=1

λkφk(x)n

for allxinE. The space of alln-homogeneous nuclear polynomials onEis denoted byPN(nE). ForAan absolutely convex subset ofEwe letAbe the semi-norm onPN(nE)defined by

A(P )=inf

i=1

iinA:P =

i=1

λiφin

SincePN(nE)=

U∈UPN(nEˆU), we may define a topologyω onPN(nE) by (PN(nE), ω)=indU∈U(PN(nEˆU), U).

In general we havePN(nE)PI(nE)P(nE)for any locally convex space Eand any integern.

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Let us show that forE=C(I)withIuncountable these inclusions are strict.

Theorem2.1.LetIbe a set andn≥3. Then(PI(nC(I)), τI)=(P(nC(I)), τω) if and only ifI is countable.

Proof. If I is countable then C(I) is a DF N space and hence by [15, Proposition 2.12] we havePN(nC(I))= P(nC(I))for alln. Assume now that (PI(nC(I)), τI)=(P(nC(I)), τω). We have that

s,n,C(I)

i=(PI(nC(I)), τI) and the equicontinuous subsets of

s,n,C(I)

correspond to the locally bounded subsets ofPI(nC(I)). Furthermore,

s,n,πC(I)

i = (P(nC(I)), τω) with the equicontinuous subsets of

s,n,πC(I)

corresponding to the locally bounded subsets ofP(nC(I)). This gives us that

s,n,C(I) =

s,n,πC(I). SinceC(I)is stable, [2, Theorem 4.1] implies that

n,C(I) =

n,πC(I). If n≥3 then [19, Theorem] implies immediately thatC(I) is nuclear and hence Iis countable.

Theorem2.2. LetIbe a set. Then then-homogeneous polynomial onC(I), P ((xi)i∈I)=

i∈Ixin, is integral.

Proof. First we observe that the set DI = {(xi)i∈I : |xi| ≤ 1 all i}is bounded inCI =(C(I))b. AsC(I)is barrelledDIis an equicontinuous subset of (C(I)). Furthermore, it follows from [17, Proposition 3.14.3] thatσ(CI,C(I)) induces the product topology onDI. For eachiI we letµi be the Radon measure onD= {z:|z| ≤1}such that

Dzki(z)=

1 ifk =n; 0 otherwise.

For each iI let πi denote the natural projection from DI onto the ith coordinate. By [4, p. 112] (see also [23]) there is a (unique) Radon measureµ onDI such thatµπi =µi for eachiinI. Since

DIφ(x)ndµ(φ)=

(i1,...,in)∈In

xi1. . . xin

DI φi1. . . φindµ(φ)=

i∈I

xin for allx ∈C(I),P (x)=

i∈Ixinis integral.

Given a setI consider the Cartesian productCI. It is shown in [16, The- orem 2.3.7], [22] and [21] thatCI is separable if and only ifIhas cardinality less than or equal toc. However, [20] (see also [9, Example 2.5.7]) show that C(I)with the topology induced fromCI is separable if and onlyIis countable.

Lemma2.3. Let I be uncountable andn ≥ 2. Then then-homogeneous polynomial onC(I)P ((xi)i∈I)=

i∈Ixni is not nuclear.

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Proof. ForiI letei be the element ofC(I)or CI which is 1 in theith coordinate and 0 in each other coordinate. SinceP (eˇ i)n−1= nei we see that P (ˇ C(I))n−1isC(I)with the topology induced fromCI. Let us now suppose that Pis nuclear. Then we can find a sequencek)k=1inCIso that

k=1φknU <

∞for some neighbourhood of zero,U, inC(I)andP (x)=

k=1φk(x)nfor allx ∈ C(I). For eachx ∈ C(I), P (x)ˇ n−1 ∈ CI, and is given byP (x)ˇ n−1 = n

k=1φk(x)n−1φk. Since this series converges inCI we see thatP (ˇ C(I))n−1 is a separable subspace ofCI. However, as we noted before the statement of the Lemma,C(I)with the topology induced fromCI is not separable and our assumption thatP is nuclear is not true.

Corollary 2.4. Let I be uncountable and n ≥ 3. Then PN(C(I)) = PI(C(I))=P(C(I)).

Finally we observe that since every bounded subset of C(I) is finite di- mensional every homogeneous polynomial onC(I) is weakly continuous on bounded sets.

3. Application to Locally Asplund spaces

In modern Banach space theory the Radon-Nikodým property plays a central role. There are many equivalent ways in which this concept can be introduced.

For example, a Banach space has the Radon-Nikodým property if and only if every closed nonempty convex bounded set is the closed convex hull of its strongly exposed points. Alternatively, a Banach spaceEhas the Radon- Nikodým property if and only if every integral operator with values in E is nuclear. The Radon-Nikodým property is dual to another Banach space property – Asplundness. Again there are many equivalent definitions of an Asplund Banach space. For example, a Banach spaceEis Asplund if and only if every separable subspace ofEhas a separable dual or equivalently if every integral operator onEis nuclear. We refer the reader to [12] for more details.

In [10] Defant defined what is meant for a locally convex space to have dual with the locally Radon-Nikodým property. A Banach space has dual with the local Radon-Nikodým property if and only if it is Asplund. In [7] this property was renamed locally Asplund. Locally Asplund spaces have good stability properties. It is shown in [10] that the class of locally Asplund spaces is closed under the formation of subspaces, quotients, arbitrary projective limits and countable locally convex inductive limits. In [7] it is shown that this class is also closed under Schwartz-products. In this section we show that the locally convex spaceC(I), withIuncountable is not locally Asplund. This will prove that the class of locally Asplund locally convex spaces is not closed under the formation of uncountable direct sums and, contrary to the Banach space case, is not separably determined.

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GivenEandF locally convex spaces we letLE;Fdenote the space of all linear maps fromEintoFtransforming some neighbourhood of zero into an equicontinuous set. ThereforeTLE;Fif and only if there exists an absolutely convex closed neighbourhood of zeroV inF such thatT factors continuously through the Banach spaceFV.

Let ((, ), µ) be a finite measure space andX be a Banach space. An operatorT:L1(µ)Xis said to berepresentable, [12], if there is a Bochner- integrablefL(µ;X)such that

T φ=

φf dµ for allφL1(µ).

Given a locally convex space E an operatorTLL1(µ);E is said to belocally representableif there is a neighbourhood of zero V inEand a representable operatorTˆ ∈ L(L1(µ);EV)such that the following diagram commutes

L1(µ)−−−−−−−−−−→T E

Tˆ

EˆV

Defant, [10], says that a locally convex spaceEhas a dual with the local Radon-Nikodým property if for every finite measure space((, ), µ)all op- erators inLL1(µ);Eare locally representable. As in [7] we rename this property and from this point on say thatEislocally Asplund. It is shown in [10] that a locally convex spaceEis locally Asplund if and only if for every ab- solutely convex neighbourhoodUof 0 inEand every positive Radon measure νon(U, σ (E, E))there isVan absolutely convex neighbourhood of 0 inE, VU, such that the embedding(U, σ(E, E)) -→ Visν-measurable.

Defant [10] proceeds to give many reformalizations of the concept of a locally Asplund spaces. Among these is that a locally convex spaceEis locally Asplund if and only if given any locally convex spaceFevery integral bilinear form on E ×F is nuclear. In [6] the author proved that if E is a locally Asplund locally convex space for any positive integer nthe locally convex space(PI(nE), τI)is isomorphic to the locally convex space(PN(nE), ω).

Proposition3.1.IfIis an uncountable set thenC(I)is not locally Asplund.

Proof. Apply [6, Theorem 3], Theorem 2.2 and Lemma 2.3.

In contrast to the Banach space case we get:

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Corollary3.2.Local Asplundness is not separably determined.

Proof. Every separable subspace ofC(I)is either isomorphic toC(N)orCn which is aDF N space and hence is locally Asplund.

We can also obtain the following result.

Proposition3.3.IfIis an uncountable set then the bilinear formB:C(I)× C(I)→C,B((xi)i∈I, (yi)i∈I)=

i∈Ixiyiis integral but not nuclear.

Given locally convex spacesEandF we say thatzE

πF has aseries representationif there is a sequencen)n1and bounded sequence(xn)n

and(yn)n in EandF respectively so thatz =

n=1λnxnyn. From [11,

§4 Remark] we get:

Proposition3.4.There is anLBanach spaceF andz∈CI

πF which does not admit a series representation.

Acknowledgement.On the 11th of July 2001 I presented the result in this talk at a seminar at the University of Valencia. I would to thank those present who pointed out to me that my proof of Lemma 2.3 was much more subtle that I had originally appreciated.

REFERENCES

1. Alencar, R., Aplicações nucleares e integrais e a propriedade de Radon-Nikodým, Ph.D.

Thesis, Universidade de Saõ Paulo, (1982).

2. Ansemil, J. M. and Floret, K.,The symmetric tensor product of a direct sum of locally convex spaces, Studia Math. 129 (3), (1998), 285–295.

3. Barroso, J. A. and Nachbin, L.,Some topological properties of spaces of holomorphic map- pings in infinitely many variables, Advances in Holomorphy, (J. A. Barroso, ed.), North- Holland Math. Stud. 34 (1979), 67–91.

4. Bourbaki, N.,Intégration, Chapitre V, Hermann, Paris (1956).

5. Boyd, C.,Holomorphic mappings onC(I),Iuncountable, Results Math. 36 (1999), 21–25.

6. Boyd, C.,Duality and reflexivity of spaces of approximable polynomials on locally convex spaces, Monatsh. Math. 130 (2000), 177–188.

7. Boyd, C., Dineen, S. and Rueda, P., Locally Asplund spaces of holomorphic functions, Michigan Math. J. 50 (2002), 493–506.

8. Boyd, C., Dineen, S. and Rueda, P.,Locally Asplund preduals of spaces of holomorphic functions, Acta. Math. Hungar. 96 (1-2) (2002), 105–116.

9. Pérez Carreras, P. and Bonet, J.,Barrelled Locally Convex spaces, North-Holland Mathem- atics Stud. 131 (1987).

10. Defant, A.,The local Radon Nikodým property for duals of locally convex spaces, Bull. Soc.

Roy. Sci. Liège 53 (1984), 233–246.

11. Defant, A.,Topological tensor product and the approximation property of locally convex spaces, Bull. Soc. Roy. Sci. Liège 58 (1989), 29–51.

12. Diestel, J. and Uhl, J. J.,Vector Measures, Amer. Math. Soc. Math. Surveys 15 (1977).

13. Dineen, S.,Holomorphic types on a Banach space, Studia Math. 39 (1971), 241–288.

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14. Dineen, S.,Holomorphic functions on inductive limits ofCN, Proc. Roy. Irish. Acad. Sect. A 86 (2) (1986), 143–146.

15. Dineen, S.,Complex Analysis on Infinite Dimensional Spaces, Monographs Math. (1999).

16. Engelking, R.,Outline of General Topology, North Holland Publishing Company, Amsterdam (1968).

17. Horváth, J.,Topological Vector Spaces and Distributions, Vol. 1, Addison-Wesley, Massachu- setts (1966).

18. Jarchow, H.,Locally Convex Spaces, B. G. Teubner, Stuttgart (1981).

19. John, K.,Tensor products of several spaces and nuclearity, Math. Ann. 269 (1984), 333–356.

20. Lohman, R. H. and Steiles, W. J.,On separability in linear topological spaces, Proc. Amer.

Math. Soc. 42 (1974), 236–237.

21. Marczewski, E,Séparabilité et multipication cartésienne des espaces topologiques, Fund.

Math. 34 (1947), 127–143.

22. Pondiczery, E. S.,Power problems in abstract spaces, Duke Math. J. 11 (1944), 835–837.

23. Prohkorov, U. V.,Convergence of random measures and limit theorems in probability theory, Theory Prob. Appl. I (1956), 157–214.

DEPARTMENT OF MATHEMATICS UNIVERSITY COLLEGE DUBLIN BELFIELD

DUBLIN 4 IRELAND

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