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POLYNOMIAL DUNFORD-PETTIS PROPERTIES

arXiv:math/9211210v1 [math.FA] 17 Nov 1992POLYNOMIAL SCHUR ANDPOLYNOMIAL DUNFORD-PETTIS PROPERTIESJeff Farmer and William B. JohnsonAbstract. A Banach space is polynomially Schurif sequential convergence against analytic polynomials implies norm convergence.Carne, Cole and Gamelin show that a space has this property and the Dunford-Pettisproperty if and only if it is Schur.

Herein is defined a reasonable generalization ofthe Dunford–Pettis property using polynomials of a fixed homogeneity. It is shown,for example, thata Banach space will has the PN Dunford–Pettis property if and only if everyweakly compact N−homogeneous polynomial (in the sense of Ryan) on the space iscompletely continuous.

A certain geometric condition, involving estimates on spread-ing models andimplied by nontrivial type,is shown to be sufficient to imply that a space is polynomially Schur.1. IntroductionThe relationship between holomorphic functions defined on an infinite dimen-sional Banach space and (geometric or topological) properties of the space has beenof recent interest (see, for example, [AAD], [ACG], [CCG], [CGJ], [F], [R 1]).

Asin the one–dimensional case, holomorphic functions are defined in terms of Taylorseries, which in the infinite–dimensional case have terms consisting of homogeneousanalytic polynomials. Just as in the case of linear functionals (1–homogeneous poly-nomials), one can consider properties of the topologies induced by the polynomialson the space.

In this paper we consider the properties which are analagous to theSchur property and the Dunford–Pettis property; i.e., those obtained by replacingweak sequential convergence with sequential convergence against an arbitrary N–homogeneous analytic polynomial. We relate these properties to one another andto the geometric property of type and the existence of certainspreading models.X will be a complex infinite–dimensional Banach space.

An N−homogeneousanalytic polynomial on X is the restriction to the diagonal of an N−linear formon the N−fold Cartesian product of X with itself, or equivalently, a linear func-tional on the N−fold projective tensor product of X with itself. Indeed, given anN−homogeneous analytic function P on X, one obtains an N–linear form on X by1991 Mathematics Subject Classification.

Primary 46B05, 46B20 Secondary 46G20.This paper forms a portion of the first author’s doctoral dissertation written under the super-vision of the second author.The first author was supported in part by NSF Grant #DMS-9021369.The second author was supported in part by NSF Grant #DMS-9003550.This paper is in final form and no version of it will be submitted for publication elsewhere.Typeset by AMS-TEX1

2JEFF FARMER AND WILLIAM B. JOHNSONtaking the Nth derivative and dividing by N! ; the form is related to the polynomialby the polarization formula:AP (x1, .

. .

, xn) = Avg{ǫi = ±1}(Πǫi)P nXi=1ǫixi.The form AP is clearly symmetric (invariant under permutations of the coor-dinates).Likewise any bounded symmetric N−linear form will give rise to anN−homogeneous analytic polynomial. Such a form can be linearized by taking theprojective tensor product of X with itself N times and extendingthe form to a linear functional on this tensor product.

The subspace of symmetriclinear functionals is the dual of the symmetric N−fold projective tensor product,which is a complemented subspace of the N−fold projective tensor product. Theprojection is given by extending the following map linearly:(x1 ⊗x2 ⊗· · · ⊗xn) →1n!Xπ∈Sn(xπ1 ⊗xπ2 ⊗· · · ⊗xπn).We denote the symmetric projective tensor product by b⊗Ns X .

The N–linear formAP associated with P can now be considered a linear functional on b⊗Ns X . Thesupremum norm of the polynomial is related to that of the linear functional asfollows:||P|| ≤||AP || ≤N NN!

||P||If we call the space of polynomials PN the above simply says that PN is isomorphicto (b⊗Ns X)∗. Since for our purposes the index N will be fixed, we will suppressreference to this isomorphism and use the same label for a polynomial and itsassociated symmetric linear functional.

More details about the above relationshipsmay be obtained from [M] or [R 1]. We will study the topologies generated by thesepolynomials, especially with respect to sequential convergence.We define the PN−weak topology on X to be the topology generated by the allof the homogeneous analytic polynomials of degree less than or equal to N; that is,a net {xα} converges to x in the PN−weak topology if, for every M ≤N, for everyM−homogeneous analytic polynomial P, P(xα) →P(x).

Note that for N = 1 thisis the usual weak topology, and that for M > N the PM−weak topology is finerthan the PN−weak one. We call the weak polynomial topology the topology whichis generated by the union of PN for all N ∈Z+.

In analogy to the Schur property,we say a space is PN−Schur if whenever P(xn) →0 for all P ∈PN then xn isnorm null. If P(xn) →0 for all P ∈PN for all N implies that xn is norm null,then we say X is P−Schur.

It is evident (multiply linear functionals) that everyPN−Schur space is P−Schur and that every Schur space is PN−Schur for every N(and P−Schur).These topologies were introduced in [R 1], and the weak polynomial topology alsoappeared in [CCG] which considered relations between the Dunford-Pettis property,the Schur property and the P−Schur property (in the terminology of [CCG], “Xis P−Schur ”= “X is a Λ−space”).Let θ : X →b⊗Ns X by θ(x) = x ⊗· · · ⊗x (N times) and define θ(X) =∆N(X).This set is a non-convex, norm-closed subset of b⊗Ns X with the property thatλx ∈∆N(X) whenever x ∈∆N(X).

POLYNOMIAL SCHUR AND POLYNOMIAL DUNFORD–PETTIS PROPERTIES3Now θ is a continuous N−homogeneous function, which is the (nonlinear) pread-joint of the isomorphism between PN and (b⊗Ns X)∗. Notice that θ is an N−to−onemap; we haveθ−1(x ⊗· · · ⊗x) = {e2πniN x|n ∈Z}(Use separating functionals to the Nth power to show equality.) We reserve thesymbol θN for this function.If P(xα) →P(x) for all P ∈PN(X) then the net need not converge againstpolynomials in PM for all M < N, but since P(x) = P(y) for all P ∈PN impliesthat xy is a complex Nth root of unity, if also xα →x weakly, then xαPN−weakly converges to x.

Thus in practice it is easy to pass from convergenceagainst all N−homogeneous polynomials to PN−weak convergence.Although for any one polynomial P, P(x −xα) →0 and P(xα) →P(x) are notin general equivalent, the following known fact is useful.Lemma 1.1. xα →x in the PN−weak topology, if and only if x −xα →0 in thePN−weak topology.Proof (sketch). Let P be an N-homogeneous polynomial and let xα →x in thePN−weak topology.

Then, letting AP be the N–linear form associated with P wehaveP(x −xα) =NXi=1(−1)iNiAP (x, x, . .

. , x, xα, .

. .

, xα)where in each term x appears i times and xα appears N −i times. Since convergencein the PN−weak topology implies convergence against any polynomial of lesserhomogeneity, we consider each term as an (N −i)−homogeneous polynomial (xbeing fixed), to see that the sum indeed converges to zero.

The converse is obtained,using the same expansion, by induction on N.2. Polynomial Dunford–Pettis SpacesOne result of [CCG] is that a Banach space is Schur if and only if it is polyno-mially Schur and has the Dunford–Pettis property.

We can obtain an analagousresult for polynomials of fixed homogeneity by defining an appropriately analagousDunford-Pettis property. Our first task is to adapt Lemma 7.3 of [CCG] for ourpurposes.Propopsition 2.1.

The following areequivalent for any Banach space X, and any fixed positive integer N.(i) Any polynomial on X is PN−weakly sequentially continuous. (ii) If {xn}∞n=1 is a PN−weakly null sequence in X (i.e.if {xk ⊗· · · ⊗xk}(N times) is weakly null), then {xk ⊗· · · ⊗xk} (m times) is weakly null inb⊗mX for m > N.(iii) For m > N the function θ = θ(N, m) which takes θ : ∆N(X) →∆m(X) byx ⊗· · · ⊗x (N times) 7→x ⊗· · · ⊗x (m times)is weak to weak sequentially continuous.

4JEFF FARMER AND WILLIAM B. JOHNSONProof. The proof of these equivalences is an exercise, and can be adapted easilyfrom the proof of Lemma 7.3 in [CCG].For n = 1, the equivalent properties of 2.1 were shown to be implied by theDunford–Pettis property.

We will now define a polynomial Dunford–Pettis propertywhich will imply the conditions of 2.1 for each positive integer.We say that a space X has the PN Dunford–Pettis property provided that itsatisfies any of the equivalent conditions of Proposition 2.2.Proposition 2.2. Let X be a Banach space.

For fixed N, the following are equiv-alent :(i) Whenever {Pn}∞n=1 is a weakly null sequence of N−homogeneous polynomi-als (or equivalently, symmetric bounded N−linear forms on X) and {xn}∞n=1converges PN−weakly to x in X, then Pn(xn) →0. (ii) Every weakly compact operator on b⊗Ns X is completely continuous when re-stricted to ∆N(X) .

(iii) If K is a weakly compact set in any Banach space Y, and J is PN−weaklycompact in X, then θN(J) ⊗K is a weakly compact set in (b⊗Ns X)b⊗Ywhere by θN(J) ⊗K we mean simply the set θN(J) × K, that is, the set of allθN(j) ⊗k with j ∈J and k ∈K.In the case N = 1 these conditions reduce to known equivalent statements of theclassical Dunford–Pettis property; this proposition justifiesthe definition of the PN Dunford–Pettis property. Before giving the proof, wemake the following remark.R.

Ryan in [R 2] considers N−homogeneous polynomials from X →Y ; as in thescalar case, we can equivalently consider linear operators from b⊗Ns X to Y ; sucha polynomial is weakly compact if it maps bounded sets to weakly compact ones(i.e. if the associated linear operator is weakly compact).

Ryan investigates someconditions which are equivalent to weak compactness of such polynomials. Usingthis definition it is easy to see that (ii) above is equivalent to.

(ii)′ Every weakly compact N–homogeneous polynomial from X to any Banach spaceY is completely continuous (on X).Proof. (i)⇒(iii) We want to show that θN(J) × K is weakly compact in b⊗Ns X ⊗Y .

Take asequence θN(xn) ⊗kn in θN(J) ⊗K; by hypothesis assume we have passedto a subsequence such that θN(xn) and kn are weakly convergent to θN(x)and k in b⊗Ns X and Y , respectively. That is, θN(xn)−θN(x) and kn −k areweakly null.

Ifφ ∈(b⊗Ns X ⊗Y )∗≡BY, (b⊗Ns X)∗then φ(kn −k) is weakly null by continuity and (i) applies. We then haveDφ(kn −k), θN(xn) −θN(x)E=φ(kn), θN(xn)−φ(k), θN(xn)−φ(kn), θN(x)+φ(k), θN(x)

POLYNOMIAL SCHUR AND POLYNOMIAL DUNFORD–PETTIS PROPERTIES5Taking limits as n →∞we see0 = limn→∞φ(kn), θN(xn)−2φ(k), θN(x)+φ(k), θN(x)= limn→∞φ(kn), θN(xn)−φ(k), θN(x)This says exactly that θN(xn) ⊗kn is weakly convergent to θN(x) ⊗k. (iii)⇒(ii) Let T : b⊗Ns X →Y be weakly compact and θN(xn) →θN(x) weakly inthe symmetric tensor product.

Choose φn to be norming functionals forT (θN(xn) −θN(x)) in the sphere of Y ∗. Since T ∗is also weakly compact,assume by passing to a subsequence that T ∗(φn) converges weakly, say toψ.

By passing to a subsequence we can also assume that {T ∗(φn) −ψ}∞n=1is basic (or norm null, in which case the argument is simpler). Apply (iii)to the setsK =T ∗(φn) −ψ∞n=1 ∪{0}andJ = {xn}∞n=1 ∪{x}to see that θN(J)⊗K is weakly compact in (b⊗Ns X)b⊗(b⊗Ns X)∗.

The sequenceθN(xn) ⊗{T ∗(φn) −ψ} thus has a convergent subsequence (we pass tothat). First we claim that this subsequence must go weakly to zero; indeed,it goes to zero in a weaker Hausdorfftopology, namely that generated byconsidering the weak topology on the second co-ordinate.

Since it is clearthat θN(x) ⊗{T ∗(φn) −ψ} is weakly null, we can conclude thatw −lim[θN(xn) −θN(x)] ⊗{T ∗(φn) −ψ}= 0Now consider the functional associated with the identity operator; call it Γ.We have0 = limn→∞Γ{T ∗(φn) −ψ} ⊗[θN(xn) −θN(x)]= limn→∞{T ∗(φn) −ψ}, θN(xn) −θN(x)= limn→∞T ∗(φn), θN(xn) −θN(x)−ψ, θN(xn) −θN(x)= limn→∞φn, T (θN(xn)) −T (θN(x))= limn→∞∥T (θN(xn)) −T (θN(x))∥This gives (ii). (ii)⇒(i) Let {Pn}∞n=1 be the weakly null sequence and {xn}∞n=1 go PN−weakly to xand define a map T from ∆N(X) to c0 byT (θN(z)) =Pn(z)n∀z ∈XThe map T extends linearly (via the polarization formula) to all of b⊗Ns X.Since T ∗(en) = Pn goes weakly to 0 we see that the map T is weakly com-pact.

Applying (ii) we get T (θN(xn)) going in the norm on c0 to T (θN(x)).But since the norm on c0 is the sup norm, this gives (i) and completes theproof.We note that the condition (ii) is sharply stated with the following example.

6JEFF FARMER AND WILLIAM B. JOHNSONExample. We will see momentarily that l2 is P2–Schur and therefore is P2–Dunford Pettis.

Consider the operatorId⊗Q1 : l2 b⊗l2 →c0where Q1 is the projection onto the first basis vector. Consider the symmetrizedversion, that is, restrict the operator to the symmetric tensor product, which is acomplemented subspace.

This operator is weakly compact and therefore completelycontinuous on θ2(l2) by Proposition 2.2 but is clearly not completely continuous onthe entire symmetric tensor product; consider the image of en ⊗e1 + e1 ⊗en, forexample, which is weakly null but whose image is the unit vector basis of c0.Proposition 2.3. If a Banach space is PN Dunford–Pettis then it satisfies theequivalent conditions in Proposition 2.1.Proof.

We prove 2.2(i) implies 2.1(ii).Let {xn}∞n=1 be a PN−weakly null sequence in X, i.e. θN(xn)∞n=1 is weakly nullin b⊗NX and θM(xn)∞n=1 is also weakly null in b⊗MX whenever 1 ≤M < N. Letm = N +1, φ ∈(b⊗N+1X)∗and consider φ as a linear operator from X to (b⊗NX)∗.Since {xn}∞n=1 is weakly null in X, so is its image in (b⊗NX)∗under φ. Thusφ, θN+1(xn)=φ(xn), θN(xn)→0by the first formulation of the PN Dunford–Pettis property (notice that for thisapplication it matters not whether φ is symmetric).

This proves the proposition form = N + 1 and by induction (in an obvious way) for m = qN + 1 for q = 1, 2, . .

. .But we can also write an analogous proof for m = N + k for 2 ≤k < N and extendit by induction as well.Proposition 2.4.

Let X be a Banach space. For fixed N, the following are equiv-alent :(i) X is PN−Schur.

(ii) X has the PN Dunford–Pettis property and is P−Schur. (iii) X satisfies (i)–(iii) of proposition 1.1 and is P−Schur.Proof.

(i)⇒(ii) requires only Lemma 1.1 and (ii)⇒(iii) is Proposition 2.3, so (iii)⇒(i)remains. Let θN(xn) be weakly null.

Then θM(xn) is weakly null for all M by 2.1.But since X is PN−Schur, xn must go to 0 in norm.It is of interest to note that if we are not in the context of the P−Schur property,the conditions of 2.1 are weaker than the PN Dunford–Pettis property; T ∗, theoriginal Tsirelson space (or, in fact any space having the approximation propertywith PN(X) reflexive for all N; see [F] for further discussion of such spaces) willsatisfy 2.1 for all N but fail to be PN Dunford–Pettis.Examples. It is clear from the classical work of Pitt [P] that lp spaces for (1 ≤p <∞) are PN−Schur for N ≥p, and it is proved in [CCG] that Lp spaces (2 ≤p < ∞)are P−Schur (in fact PN−Schur for N ≥p); we can thus conclude that they are PNDunford–Pettis.

The space c0 is Dunford–Pettis and therefore PN Dunford–Pettisfor every N. This implies (for example) that l3 ⊕c0 is P3–Dunford–Pettis but notPN−Schur for any N. In the next section we discuss further exactly which spacesmay be PN−Schur.

POLYNOMIAL SCHUR AND POLYNOMIAL DUNFORD–PETTIS PROPERTIES73. Spaces with Type are Polynomially SchurIn this section we will give some sufficient criteria for spaces to be Polynomi-ally Schur.

We will show, for example, that any space having non-trivial type isP−Schur and indeed is PN−Schur for some N.(Jaramillo and Prieto [JP] have independently shown that every superreflexivespace is polynomially Schur). In particular, Lp spacesare Polynomially Schur for all 1 < p < ∞.It is convenient to use the concept of a spreading model, the construction ofwhich is due to Brunel and Sucheston [BS 1].Finite versions of Ramsey’s Theorem allow that given any property of n-tuplesof elements from a sequence, one can pass to a subsequence with the property thatall n-tuples formed from the subsequence share the property or else all fail it.

Byusing the size of the norm of a sum of n elements as the property one can, byrepeatedly applyling the theorem, approximately stabilize the norm (to within anydesired ǫn) of any finite combination as long as many of the beginning terms arethrown away. More preciesely we have the following fact (see [B] or [BS 1]):Proposition 3.1.

Let (fn) be a bounded sequence with no norm-Cauchy subse-quence in a Banach space X. Then there exists a subsequence (en) of (xn) and anorm L on the vector space S of finite sequences of scalars such that∀ǫ > 0∀a ∈S∃k ∈N s.t.

∀k < k1 < k2 < · · · < kMwe haveXaieki −L(a) < ǫThe completion of [ei] (call it F) under the norm L is called a spreading modelfor the sequence (en). The reason for the terminology is that the sequence (en)is invariant under spreading with respect to the norm F, that is to say, for everyfinite sequence of scalars (ai) and every subsequence σ of the natural numbersMXi=1aieiF=MXi=1aieσ(i)FThus any norm estimate satisfied by sequences in the spreading model will beapproximately satisfied for sequences of finite length to any desired degree providedwe go out far enough in the sequence (xn).

If the original sequence was weaklynull then the resulting sequence will be unconditional; that is to say, we have thefollowing (Lemma 2 of [B], or see [BS 2]).Proposition 3.2. If (xn) is weakly null, then the sequence (en) is unconditionalin F with unconditional constant at most 2.Now we are ready to state the criterion.Theorem 3.3.

Suppose a Banach space X has the property that for every nor-malized weakly null sequence {yn} in X there exists a subsequence and a sequence{fn} in X∗biorthogonal to it which has an (unconditional) spreading model with

8JEFF FARMER AND WILLIAM B. JOHNSONan upper p-estimate for some p > 1. Then X is P−Schur.

If the same p works forevery such sequence and N > p′, then the space X is PN−Schur.We know of no space which is P−Schur but which fails the above property. Thespace (l3⊕l4⊕l5⊕· · · )2 is easily seen to be P−Schur although it fails cotype (andhence type and superreflexivity), yet is reflexive; it does satisfy the hypothesis of3.3.

A Schur space satisfies the hypothesis vacuously.Proof. We will pass to subsequences and relabel without mercy.

Start with anybounded sequence in X which is not norm null; we need to find a polynomial whichis bounded away from zero on a subsequence. By Rosenthal’s theorem [D, ChapterXI] there is either a weakly Cauchy subsequence or a subsequence equivalent to theunit vector basis of l1.

Since the unit vector basis of l1 is not weakly null, we aredone in this case. For the same reason (linear functionals are polynomials) we arefinished if our weakly Cauchy sequence is not weakly null.

So we have reduced tothe case of a bounded weakly null sequence which is not norm null and can applythehypothesis to that sequence(it is purely formal that the “normalized”condition can be replaced by “bounded”).Let {yn, fn} be the biorthogonal system obtained, and assume with no loss ofgenerality that the fn are normalized. By thedefinition of a spreading model we know that for any c > 0 we can find a constantC so that for every M we haveMXi=c log Mfi ≤CMXi=c log M∥fi∥p1p.This is because once we have a spreading model we may always improve the stabilityestimates by passing to a subsequence.Choosing c small enough so that m =c log M ≤M1p for all M ∈N, and letting (ai) be scalars of modulus less than orequal to 1, we obtainMXi=1aifi ≤m−1Xi=1||fi|| +MXi=maifi ≤m + C MXi=m|ai|p1p≤C1M1p .Now by general Banach lattice techniques (see Lemma 3.4 below), we know thatfor any r < p (we choose r so that p′ < r′ < N) we can get a different C so thatfor any sequence (ai) we will haveMXi=1aifi ≤C MXi=1|ai|r1r.Now, given y ∈B choose (ai) ∈lr of lr norm one to norm the sequence (fi(y))Mi=1in lr′.

We then have that MXi=1fi(y)N1N≤ MXi=1|ai|r1r MXi=1|fi(y)|r′1r′. =MXi=1aifi(y)

POLYNOMIAL SCHUR AND POLYNOMIAL DUNFORD–PETTIS PROPERTIES9≤MXi=1aifi ≤C MXi=1|ai|r1r= Cwhich says thatP(y) =∞Xi=1fi(y)N ≤CNhence||P|| ≤CN.But we know that P(yn) = fn(yn)N is bounded away from zero because the se-quence (yn, fn) was biorthogonal. Therefore no sequence bounded away from 0 inX can be polynomially null, and so X is P−Schur.

If there is a uniform value forp then X is PN−Schur for N > p′.It remains for us to prove the following standard fact.Lemma 3.4. Suppose that {xi} is a normalized sequence in a Banach space satis-fyingXi∈Baixi ≤C|B|1p maxi∈B |ai|for all scalars (ai), and all finite subsets B of the natural numbers.

Then for any1 < r < p there exists a constant D so that for all (ai) and all MMXi=1aixi ≤D MXi=1|ai|r1r.Proof. We use the convention that 1p + 1q = 1 and 1r + 1s = 1.

Given the scalars,assume by homogeneity that PMi=1 |ai|r = 1. DefineBn = {i 2−n−1 < |ai| ≤2−n}and writeMXi=1aixi ≤XnXi∈Bnaixi ≤CXn2−n|Bn|1p r.Now just compute:Xn2−n|Bn|1p =Xn2−nrp 2n( rp −1)|Bn|1p ≤Xn2−nr|Bn| 1p Xn2nq( rp −1) 1q= 2rpXn2−(n+1)r|Bn| 1p D(p, r) ≤2rp MXi=1|ai|r1pD(p, r)= 2rp D(p, r) MXi=1|ai|r1r.The condition in Theorem 3.3 is rather hard to check.

However, a much simplercriterion is sufficient.

10JEFF FARMER AND WILLIAM B. JOHNSONTheorem 3.5. Suppose the dual space of X, X∗, has type p > 1.

Then for everynormalized weakly null sequence {yn} in X there exists a subsequence and a sequence{xn} in X∗biorthogonal to it which has an upper p-estimate. In particular, Xsatisfies the hypothesis of 3.3, and thus is PN−Schur for all N > p.In view of the fact that every space with non-trivial type also has a dual withsome non-trivial type, we can state the following corollary.Corollary 3.6.

Suppose X has type. Then for some N, X is PN−Schur.Proof (of 3.5).

Recall that the fact that X∗has type p means that there is aconstant Tp so thatZ 10MXi=1ri(t)xidt ≤Tp MXi=1∥xi∥p1p∀M ∈Nfor any finite number of elements x1, . .

. xn (where ri are the Rademacher functions).Now suppose that (yn) is a normalized weakly null sequence, which we can assumeis basic by passing to a subsequence.

We can find a bounded sequence (xn) in X∗of functionals biorthogonal to (yn). Since l1 i s not embeddable in X∗we know byRosenthal’s theorem that we can find a weakly Cauchy subsequence of (xn).

Passto the odd terms of (yn), relabel and replace (xn) with (x2n+1 −x2n). Then wehave a biorthogonal system (xn, yn) with (xn) →0 weakly.

By proposition 3.1 andthe remark following it we know that (xn) has an unconditional spreading model F.Now F is finitely representable in X∗(this means that given any finite dimensionalsubspace of F we can find a 1+ǫ–isomorphic copy of that subspace in X∗, see again[B] or [BS 1]). Since the definition of type is local, F will also have type p withconstant ≤Tp.

Since the basis en of F is unconditional with constant at most 2,it has an upper p–estimate with constant less than or equal to 2Tp. Thus we havesatisfied the hypothesis of Theorem 3.3.References[AAD] R. Alencar, R. Aron and S. Dineen, A Reflexive space of holomorphic functions in infinitelymany variables, Proc.

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[CGJ] B. Cole, T. Gamelin, and W. B. Johnson, Analytic disks in fibers over the unit ball of aBanach space (to appear).[D]J. Diestel, Sequences and series in Banach spaces, Springer–Verlag, New York, 1984.[F]J.

Farmer, Polynomial reflexivity in Banach spaces (to appear).[JP]J. Jaramillo and Prieto, Weak-polynomial convergence on a Banach space (to appear).[M]J.

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POLYNOMIAL SCHUR AND POLYNOMIAL DUNFORD–PETTIS PROPERTIES11[R 1]R. Ryan, Applications of topological tensor products to infinite dimensional holomorphy,Thesis, Trinity College, Dublin, 1980. [R 2]R. Ryan, Weakly compact holomorphic mappings on Banach spaces, Pac.

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