SOME PROPERTIES IN SPACES OF MULTILINEAR FUNCTIONALS AND SPACES OF POLYNOMIALS
By Manuel Valdivia Departmento de Analisis´ Matematico,´ Universidad de Valencia
(Communicated by S. Dineen, m.r.i.a.)
[Received after revision 30 January 1998. Read 16 March 1998. Published 30 September 1998.]
Abstract In this paper we study some properties related to the biduals of certain Banach spaces whose elements are multilinear functionals, or polynomials, defined in Banach spaces. In particular, we obtain a simple description of the n-fold Schwartz -product. We give examples of Banach spaces that are not quasi-reflexive and so, for each positive integer n, P(nX)∗∗ identifies with P(nX∗∗) in a way clearly specified. In the last fifteen years the theory of polynomials and multilinear forms has been extensively developed. The relationship between reflexivity and weak continuity was found by Ryan [26], the connection between reflexivity and weak sequential continuity was obtained by Alencar et al. [2], the use of upper and lower estimates and spreading models in connection with reflexivity was obtained by Farmer [12] and developed by Farmer and Johnson [13], Gonzalo [16] and Gonzalo and Jaramillo [17], [18]. Furthermore, Ryan [26] introduced the square order system for tensors and polynomials and Alencar [1] (see also [11]) discovered the connection between reflexivity of spaces of polynomials and the existence of a basis. Alencar et al. [2] first discussed polynomials on Tsirelson’s space and its dual, while Aron and Dineen [5] introduced Q-reflexive spaces and the theory of polynomials on the James–Tsirelson space. In this paper we discuss refinemets of some of the above results. In particular we show that the approximation property is not required for some of these results and we also give an example of a Q-reflexive space that is not quasi-reflexive. We also give new proofs and presentations of some results from the papers quoted above. Throughout this paper we use infinite dimensional linear spaces defined over the field of real or complex numbers. N is the set of positive integers. For a set B, |B | denotes its cardinal number. If X is a Banach space, then X∗ and X∗∗ represent its conjugate and double conjugate, respectively. B(X) is the closed unit ball of X and k·k its norm. If A is a closed and bounded absolutely convex subset of X, XA will denote the linear hull of A normed by its Minkowski functional; XA is then a Banach space. If x ∈ X and ∗ u ∈ X , hx, ui means u(x). If (xn) is a sequence in X,[xn] will be its closed linear span. We say that (xn) is a seminormalised sequence if there exist 0 1980 Mathematics Subject Classification: Primary 46 A 25, Secondary 46 A 32 Mathematical Proceedings of the Royal Irish Academy, 98A (1), 87–106 (1998) c Royal Irish Academy 88 Mathematical Proceedings of the Royal Irish Academy ∗ sequence in [xn] formed by the linear functionals associated with the Schauder basis (xn)of[xn]. We say that X is quasi-reflexive if it has finite codimension in its bidual; in particular, every reflexive Banach space is quasi-reflexive. A Banach space X is said to be Asplund when every separable closed subspace of X has separable dual, or equivalently, X∗ has the Radon–Nikodym property. A sequence (xn) in a Banach space X has a lower p-estimate (1 ≤ p<∞) if there is a constant c>0 such that ! m m 1/p X X a x ≥ c | a |p n n n n=1 n=1 for every finite sequence of scalars a1,a2,...,am.If(xn) is a basic sequence, it follows that if X∞ x = anxn ∈ [xn], n=1 then (an)isinlp and there is a one-to-one continuous linear map T from [xn] into lp such that Txn = en,n=1, 2,..., where (en) is the unit vector basis of lp. We say that the sequence (xn) has an upper p-estimate if there is a constant c>0 such that ! m m 1/p X X a x ≤ c | a |p n n n n=1 n=1 P∞ for every finite sequence a1,a2,...,am. Similarly, if (an) belongs to lp, then n=1 anxn converges in X and hence there is a continuous linear map T from lp into X such that Ten = xn,n=1, 2,... When the sequence (xn) is basic the map T is then one-to-one. If X1,X2,...,Xn are Banach spaces, we take the product B(X1) × B(X2) ×···× B(Xn) as the closed unit ball of X1 × X2 ×···×Xn. Now, if Y is another Banach space, we put L(X1,X2,...,Xn; Y ) for the Banach space of all continuous n-linear mappings from X1 × X2 × ··· × Xn into Y with the usual norm. When Y is the Banach space of scalars, we write L(X1,X2,...,Xn). K(X1,X2,...,Xn; Y )is the subspace of L(X1,X2,...,Xn; Y ) formed by the compact n-linear mappings. ∗ ∗ ∗ ∗ ∗ ∗ Lw∗ (X1 ,X2 ,...,Xn ; Y ) will be the subspace of L(X1 ,X2 ,...,Xn ; Y ) formed by ∗ ∗ ∗ ∗ those elements whose restrictions to B(X1 ) × B(X2 ) × ··· × B(Xn ) are weak - ∗ ∗ ∗ norm continuous; it is norm-closed in L(X1 ,X2 ,...,Xn ; Y ) and it is immedi- ∗ ∗ ∗ ate from the definition that each f ∈Lw∗ (X1 ,X2 ,...,Xn ; Y ) is a compact n- ∗ ∗ ∗ linear mapping, i.e. f ∈K(X1 ,X2 ,...,Xn ; Y ). Again, if Y is the space of scalars, ∗ ∗ ∗ ∗ ∗ ∗ we write Lw∗ (X1 ,X2 ,...,Xn ). By L(w∗)(X1 ,X2 ,...,Xn ) we denote the closure of ∗ ∗ ∗ ∗ ∗ ∗ Lw∗ (X1 ,X2 ,...,Xn )inL(X1 ,X2 ,...,Xn ) with respect to the topology of the uni- Valdivia—Spaces of polynomials 89 ∗ ∗ ∗ form convergence over the compact subsets of X1 × X2 × ··· × Xn . We sup- ∗ ∗ ∗ pose that L(w∗)(X1 ,X2 ,...,Xn ) is endowed with the norm induced by that of ∗ ∗ ∗ L(X1 ,X2 ,...,Xn ). We represent by Lw(X1,X2,...,Xn; Y ) the subspace of L(X1,X2,...,Xn; Y ) whose elements, when restricted to B(X1) × B(X2) ×···×B(Xn), are weak-norm continuous. Again, if Y is the space of scalars, we write Lw(X1,X2,...,Xn). Whenever n n n ∗ n ∗ X = X1 = X2 = ··· = Xn, we shall write L( X; Y ), L( X), Lw∗ ( X ), L(w∗)( X ) n and Lw( X), respectively, for the corresponding spaces defined above. ∗ ∗ We recall Schwartz’s definition of the -product of Banach spaces: Xγ is X - equipped with the topology γ of uniform convergence on all compact sets of the Banach space X; note that γ = σ(X∗,X)onB(X∗). Now the -product is defined by ∗ ∗ ∗ ∗ XY := L(Xγ ; Y ) ⊂L(X ; Y )=L(X ,Y ) equipped with the operator norm; (see [21, pp 343–50]) for more details, in particular for the associativity (X1X2)X3 = X1(X2X3) which allows us to define the n-fold -product X1X2...Xn of the Banach spaces Xj , j =1, 2,...,n. Theorem 1. Let X1,X2,...,Xn,Y be Banach spaces. Then ∗ ∗ ∗ ∗ ∗ ∗ Lw∗ (X1 ,X2 ,...,Xn ; Y )=L(X1γ,X2γ,...,Xnγ; Y ) ∗ = Lw∗ ((X1X2...Xn) ; Y )=X1X2...XnY , in particular ∗ ∗ ∗ Lw∗ (X1 ,X2 ,...,Xn )=X1X2...Xn. Observe that the latter equality is an isometry since both spaces carry the norm ∗ ∗ ∗ from L(X1 ,X2 ,...,Xn ). ∗ ∗ Proof. By the theorem of Banach–Dieudonne´ [22, p. 272], a subset D ⊂ X1γ ×X2γ × ∗ ∗ ∗ ∗ ···×Xnγ is closed if and only if for all λ>0 the set D∩λ[B(X1 )×B(X2 )×···×B(Xn )] is weak∗-closed. This implies ∗ ∗ ∗ ∗ ∗ ∗ Lw∗ (X1 ,X2 ,...,Xn ; Y ) ⊂L(X1γ,X2γ,...,Xnγ; Y ), the converse inequality being obvious. In particular, if X is a Banach space, ∗ ∗ Lw∗ (X ; Y )=L(Xγ ; Y )=XY . We have ∗ ∗ e ∗ e e ∗ (X1X2...Xn)γ = X1γ⊗πX2γ⊗π ···⊗πXnγ (see [21, p. 346]), hence we obtain ∗ ∗ ∗ ∗ ∗ ∗ ∗ e ∗ e e ∗ Lw∗ (X1 ,X2 ,...,Xn ; Y )=L(X1γ,X2γ,...,Xnγ; Y )=L(X1γ⊗πX2γ⊗π ···⊗πXnγ; Y ) ∗ ∗ = L((X1X2...Xn)γ ; Y )=Lw∗ ((X1X2...Xn) ; Y )=X1X2...XnY . 90 Mathematical Proceedings of the Royal Irish Academy Using the associativity (X1X2...Xr)(Xr+1Xr+2...Xn), we have Corollary 1. ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Lw∗ (X1 ,X2 ,...,Xn )=Lw∗ (X1 ,X2 ,...,Xr ; Lw∗ (Xr+1,Xr+2,...,Xn )), in particular ∗ ∗ ∗ ∗ ∗ ∗ Lw∗ (X1 ,X2 ,...,Xn )=Lw∗ (X1 ,X2 ,...,Xn−1; Xn). ∗ ∗ Again, these equalites are isometries. Note that every element in Lw∗ (X1 ,X2 ,..., ∗ Xn ) defines a compact operator ∗ e ∗ e e ∗ X1 ⊗πX2 ⊗π ···⊗πXn−1 −→ Xn. Given a Banach space X and a positive integer n, if we consider the restrictions to the diagonal of the elements of L(nX), we obtain a linear space that we represent as P(nX) so that if P ∈P(nX) there is an element f of L(nX) such that P (x):=f(x,x,...,x),x∈ X. We then say that P is a continuous n-homogeneous polynomial defined in X.We define kP k:= sup{|P (x)|: kxk≤ 1} and assume that P(nX) is provided with this norm. As for tensor products, we shall use the notation of [23], extended here in the case of tensor products of more than two factors. n Given the Banach space X and the positive integer n, Pw( X) is the subspace n n ∗ of P( X), whose elements are weak-continuous when restricted to B(X). Pw∗ ( X ) is the subspace of P(nX∗) formed by those elements whose restrictions to B(X∗) ∗ n ∗ n ∗ n ∗ are weak -continuous. P(w∗)( X ) is the closure of Pw∗ ( X )inP( X ) for the topology of the uniform convergence over the compact subsets of X∗. We assume n ∗ n ∗ that P(w∗)( X ) is provided with the norm induced by that of P( X ). Later on we shall make use of the following results that we proved in [28]: ∗ ∗ ∗ (a) If X1,X2,...,Xn are Asplund spaces, then Lw∗ (X1 ,X2 ,...,Xn ) is also an Asplund space. n ∗ (b) Let X be a Banach space and let n be a positive integer. If Pw∗ ( X ) does not contain copies of l1, or, in particular, if X is an Asplund space, then n ∗ ∗∗ n ∗ Pw∗ ( X ) = P(w∗)( X ). n ∗ (c) Let X be a Banach space and n a positive integer. If Pw∗ ( X ) has no copy ∗ of l1, or, in particular, if X is Asplund, then, if X has the approximation property, we have that n ∗ ∗∗ n ∗ Pw∗ ( X ) = P( X ). Valdivia—Spaces of polynomials 91 The following result [19] will also be needed: ∗ (d) Let X be a Banach space with no copies of l1.IfA is a weak -compact subset of X∗, then its norm closed absolutely convex hull is weak∗-compact. ∗ ∗ ∗ Theorem 2. If X1,X2,...,Xn are Banach spaces such that Lw∗ (X1 ,X2 ,...,Xn ) has no copy of l1, then ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ Lw∗ (X1 ,X2 ,...,Xn ) = L(w∗)(X1 ,X2 ,...,Xn ). Proof. If we call ∗ ∗ ∗ Y := Lw∗ (X1 ,X2 ,...,Xn ), ∗ e ∗ e e ∗ ∗ let ψ be the continuous linear map from X := X1 ⊗πX2 ⊗π ···⊗πXn into Y such ∗ that, if f is in Y and uj is in Xj , j =1, 2,...,n, then hf,ψ(u1 ⊗ u2 ⊗···⊗un)i = f(u1,u2,...,un). We set M := {u1 ⊗ u2 ⊗···⊗un ∈ X: kuj k≤ 1,j =1, 2,...,n}. It is immediate that ψ(M) is weak∗-compact in Y ∗ and that B(Y ) is the polar set of ψ(M)inY . By result d, the closed absolutely convex hull of ψ(M)inY ∗ coincides with B(Y ∗). Besides, the closed absolutely convex hull of M in X equals B(X). Thus, ψ(B(X)) is dense in B(Y ∗) and, after the closed graph theorem, we have that ψ(B(X)) contains the interior of B(Y ∗)inY ∗, [20, p. 296]. Consequently, ψ is an onto map. If ψ∗: Y ∗∗ −→ X∗ is the conjugate map of ψ, then ψ∗ is one-to-one and ψ∗(Y ∗∗)isσ(X∗,X)-closed [22, p. 18]. On the other hand, ψ∗(B(Y ∗∗)) is B(X∗) ∩ ψ∗(Y ∗∗). Hence if we identify X∗ ∗ ∗ ∗ with L(X1 ,X2 ,...,Xn ) in the usual manner, it follows that, bearing in mind that in B(Y ∗∗) the topology σ(Y ∗∗,Y∗) is the same as that of the uniform convergence ∗ ∗ ∗ ∗ ∗ ∗∗ over the compact subsets of Y , L(w∗)(X1 ,X2 ,...,Xn ) coincides with ψ (Y ) and the result follows. Corollary 2. If X1,X2,...,Xn are Asplund spaces, then ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ Lw∗ (X1 ,X2 ,...,Xn ) = L(w∗)(X1 ,X2 ,...,Xn ). ∗ ∗ ∗ Proof. After result (a), Lw∗ (X1 ,X2 ,...,Xn ) is an Asplund space and hence it contains no copy of l1. Our former theorem now applies. Lemma 1. Let X1,X2,...,Xn,Y be Banach spaces. Let r be a non-negative integer less than n.Iff ∈Lw(X1,X2,...,Xn; Y ) and Yr f(x1,x2,...,xr, ·,...,·), (x1,x2,...,xr) ∈ B(Xj ), (1) j=1 92 Mathematical Proceedings of the Royal Irish Academy Qn is the family of (n − r)-linear mappings defined in j=r+1 Xj taking values in Y , with (x1,x2,...,xr) as parameter, we then have that the family formed by the restrictions of Qn the elements of (1) to j=r+1 B(Xj ) is weakly equicontinuous at the origin. Proof. We proceed by induction over n.Forn = 1, it happens that r = 0 and the family in (1) has only one element; the property then clearly holds. We assume now that the property is true for n =1, 2,...,m− 1. We fix n = m and proceed inductively over r.Ifr = 0, again (1) has only one function and the property follows. Let s be a positive integer less than m and assume that the property holds for r = s − 1, but it does not hold for r = s. Then there is δ>0 such that, for each weak neighborhood U of the origin in B(Xs+1) × B(Xs+2) ×···×B(Xm), we have that sup{kf(x1,x2,...,xm)k: xj ∈ B(Xj ),j =1, 2,...,s,(xs+1,xs+2,...,xm) ∈ U} >δ. (2) Now, since the property is true for r = s − 1, there is a weak neighborhood Uj of the origin in B(Xj ), j = s, s +1,...,m, so that, for any value of the parameter (x1,x2,...,xs−1), 1 kf(x1,x2,...,xs,xs+1,...,xm)k< 4 δ, xj ∈ Uj ,j= s, s +1,...,m. (3) We put U1j := Uj , j = s +1,s+2,...,m, and find, bearing (2) in mind, x1j ∈ B(Xj ),j=1, 2,...,s, x1j ∈ U1j ,j= s +1,s+2,...,m, such that kf(x11,x12,...,x1m)k>δ. Proceeding again by complete induction, let us suppose that, for a positive integer k, we have obtained a weak neighborhood Ukj of the origin in B(Xj ), j = s +1,s+ 2,...,m, and xkj ∈ B(Xj ),j=1, 2,...,s, xkj ∈ Ukj,j= s +1,s+2,...,m, such that kf(xk1,xk2,...,xkm)k >δ. (4) We have now that f(·,...,·,xks, ·,...,·)isaY -valued (m − 1)-linear mapping whose restriction to B(X1) ×···×B(Xs−1) × B(Xs+1) ×···×B(Xm) is weakly continuous Qm and hence the restrictions to j=s+1 B(Xj ) of the (m − s)-linear mappings Ys−1 f(x1,x2,...,xs−1,xks, ·,...,·), (x1,x2,...,xs−1) ∈ B(Xj ) j=1 form a family which is weakly equicontinuous at the origin. Consequently, there is a weak neighborhood U(k+1)j of the origin in B(Xj ), U(k+1)j ⊂ Ukj, j = s+1,s+2,...,m, so that, for any value of the parameter (x1,x2,...,xs−1) 1 kf(x1,...,xs−1,xks,xs+1,...,xm)k< 4 δ, xj ∈ U(k+1)j ,j= s +1,s+2,...,m. (5) Valdivia—Spaces of polynomials 93 We apply (2) now and obtain x(k+1)j ∈ B(Xj ),j=1, 2,...,s, x(k+1)j ∈ U(k+1)j ,j= s +1,s+2,...,m, such that kf(x(k+1)1,x(k+1)2,...,x(k+1)m)k>δ, ∞ which coincides with inequality (4) replacing k by k + 1. The sequence (xks)k=1 has ∗ ∗∗ ∗∗ a weak -cluster point x0 in B(Xs ). We find a neighborhood V of the origin of Xs ∗ for the weak topology, absolutely convex, such that V ∩ B(Xs) ⊂ Us. We find two positive integers p 1 I(x1,x2,...,xs−1):=f(x1,x2,...,xs−1, 2 (xqs − xps),xq(s+1),...,xqm), J(x1,x2,...,xs−1):=f(x1,x2,...,xs−1,xps,xq(s+1),...,xqm), H := f(xq1,xq2,...,xqm). It follows that 1 2 (xqs − xps) ∈ Us,xqj ∈ Uqj ⊂ U(p+1)j ⊂ Uj ,j= s +1,s+2,...,m, and therefore, applying (3), (5) and (4), respectively, we obtain that 1 1 kI(x1,x2,...,xs−1)k< 4 δ, kJ(x1,x2,...,xs−1)k< 4 δ,kH k>δ. Then 1 1 1 1 1 1 4 δ>kI(xq1,xq2,...,xq(s−1))k=k 2 H − 2 J(xq1,xq2,...,xq(s−1) k> 2 δ − 8 δ>4 δ, which is a contradiction. Theorem 3. Let X1,X2,...,Xn,Y be Banach spaces. Let r be a non-negative integer less than n.Iff ∈Lw(X1,X2,...,Xn; Y ) and Yr f(x1,x2,...,xr, ·,...,·), (x1,x2,...,xr) ∈ B(Xj ), (6) j=1 Qn is the family of (n − r)-linear mappings defined in j=r+1 Xj with values in Y , (x1,x2,...,xr) as parameter, then we have that the family formed by the restrictions Qn Qn of the elements of (6) to j=r+1 B(Xj ) is weakly uniformly equicontinuous in j=r+1 B(Xj ). Proof. For each integer j, r (x1,...,xj−1,xj+1,...,xn) ∈ B(X1) ×···×B(Xj−1) × B(Xj+1) ×···×B(Xn) 94 Mathematical Proceedings of the Royal Irish Academy formed by the restrictions of the linear functionals f(x1,...,xj−1, · ,xj+1,...,xn) to B(Xj ) is weakly equicontinuous at the origin. Thus, given >0, there is an absolutely convex weak neighborhood of the origin Vj in Xj such that 1 kf(x ,...,x ,v ,x ,...,x )k< , v ∈ V ∩ B(X ), 1 j−1 j j+1 n 2n j j j xi ∈ B(Xi),i=1, 2,...,j− 1,j+1,...,n. Then, given the points, Yn (yr+1,yr+2,...,yn), (zr+1,zr+2,...,zn) ∈ B(Xj ),yj −zj ∈ Vj ,j= r+1,r+2,...,n, j=r+1 1 it follows that, for every xi ∈ B(Xi), i =1, 2,...,r, noticing that 2 (yj −zj ) ∈ Vj ∩B(Xj ), j = r +1,r+2,...,n, kf(x1,x2,...,xr,yr+1,yr+2,...,yn) − f(x1,x2,...,xr,zr+1,zr+2,...,zn)k= 1 k2f(x1,x2,...,xr,yr+1,yr+2,...,yn)+2f(x1,x2,...,xr,zr+1, 2 (yr+2 − zr+2),...,zn) 1 + ···+2f(x1,x2,...,xr,yr+1,yr+2,..., 2 (yn − zn)k<. Corollary 3. Let X1,X2,...,Xn,Y be Banach spaces. If f ∈Lw(X1,X2,...,Xn; Y ), then f is weakly uniformly continuous in B(X1) × B(X2) ×···×B(Xn). Corollary 4 [6]. Let P be an n-homogeneous polynomial defined in a Banach space X with values in a Banach space Y .IfP is weakly continuous on B(X) then P is weakly uniformly continuous on B(X). We recall that Corollary 3 can also be obtained by conveniently modifying the method described in [6] to n-linear mappings. Note 1. Notice that the method used in the proofs of Lemma 1 and Theorem 3 can be applied as to achieve the following result: Let E1,E2,...,En,F be locally convex spaces. Let Aj be an absolutely convex pre- Qr compact subset of Ej , j =1, 2,...,n. Let f be an n-linear mapping of j=1 Ej into F Qr whose restriction to j=1 Aj be continuous. If r is a non-negative integer less than n and Yr f(x1,x2,...,xr, ·,...,·), (x1,x2,...,xr) ∈ Aj (7) j=1 Qn is the family of (n − r)-linear mappings defined in j=r+1 Ej with values in F, with parameter (x1,x2,...,xr), then we have that the family formed by the restrictions of Qn Qn the elements of (7) to j=r+1 Aj is uniformly equicontinuous in j=r+1 Aj . Valdivia—Spaces of polynomials 95 Let X1,X2,...,Xn be Banach spaces. If f belongs to Lw(X1,X2,...,Xn), it may ˜ ∗∗ ∗∗ ∗∗ be extended to an element f of Lw∗ (X1 ,X2 ,...,Xn ), according to Corollary 3. The mapping that takes f into f˜ is a linear isometry. By means of this mapping, ∗∗ ∗∗ ∗∗ we identify Lw(X1,X2,...,Xn) with Lw∗ (X1 ,X2 ,...,Xn ). Similarly, for a Banach n n ∗∗ space X and a positive integer n, we may identify Pw( X) with Pw∗ ( X ), according to Corollary 4. Then, using Theorem 2 and Corollary 2, or else result (b), respectively, we obtain the following results. Proposition 1. If X1,X2,...,Xn are Banach spaces such that Lw(X1,X2,...,Xn) con- ∗ ∗ ∗ tains no copy of l1, or, in particular, when X1 ,X2 ,...,Xn are Asplund, then ∗∗ ∗∗ ∗∗ ∗∗ Lw(X1,X2,...,Xn) = L(w∗)(X1 ,X2 ,...,Xn ). n Proposition 2. If X is a Banach space and n is a positive integer such that Pw( X) ∗ has no copy of l1, or, in particular, when X is Asplund, then n ∗∗ n ∗∗ Pw( X) = P(w∗)( X ). The reflexivity of spaces of polynomials as well as for spaces of multilinear functionals has been studied by a number of authors, e.g. [1]–[4], [12], [26]. We obtain here a few new results concerning reflexivity. Proposition 3. Let X1,X2,...,Xn be Banach spaces. The following conditions are equiv- alent: ∗ ∗ ∗ (1) Lw∗ (X1 ,X2 ,...,Xn ) is reflexive. ∗ ∗ ∗ (2) Xj is reflexive, j =1, 2,...,n, and each element of L(w∗)(X1 ,X2 ,...,Xn ) is ∗ ∗ ∗ sequentially weakly continuous in X1 × X2 ×···×Xn . ∗ ∗ ∗ ∗ ∗ ∗ Proof. We write X := Lw∗ (X1 ,X2 ,...,Xn ), Y := L(w∗)(X1 ,X2 ,...,Xn ). 1 ⇒ 2. X1⊗e X2⊗e ···⊗e Xn identifies canonically with a closed subspace of X, hence we deduce, keeping in mind that in this paper we only deal with non-trivial linear spaces, that Xj is reflexive, j =1, 2,...,n. On the other hand, X coincides with its bidual Y and the conclusion follows. 2 ⇒ 1. Let ψ denote the canonical mapping ∗ ∗ ∗ ∗ from X1 × X2 ×···×Xn into X such that ∗ hf,ψ(u1,u2,...,un)i = f(u1,u2,...,un),f∈ Y, uj ∈ Xj ,j=1, 2,...,n. We set ∗ ∗ ∗ M := ψ(B(X1 ) × B(X2 ) ×···×B(Xn )). ∗ Let (zr) be a sequence in M. We choose ujr in B(Xj ), j =1, 2,...,n, such that ψ(u1r,u2r,...,unr)=zr,r=1, 2,... We find a subsequence (u )of(u ) that converges in X∗ to u for the weak jrm jr j j topology. Then if z := ψ(u1,u2,...,un) and f ∈ Y, 96 Mathematical Proceedings of the Royal Irish Academy it follows that limhf,zr i = lim f(u1r ,u2r ,...,unr )=f(u1,u2,...,un)=hf,zi, m m m m m m and thus M is σ(X∗,X)-compact. Now, since B(X∗) is the closed convex hull of M in X∗, we apply Krein’s theorem [21, p. 235] to obtain that B(X∗) is weakly compact and hence X is reflexive. Proposition 4. Let X be a Banach space and n a positive integer. The following con- ditions are equivalent: n ∗ (1) Lw∗ ( X ) is reflexive. r ∗ (2) X is reflexive and, for each integer r, 1 ≤ r ≤ n, every element of L(w∗)( X ) is sequentially weakly continuous in the origin. r ∗ n ∗ Proof. 1 ⇒ 2. If 1 ≤ r ≤ n, Lw∗ ( X ) is isomorphic to a subspace of Lw∗ ( X ) and is thereby reflexive. This conclusion follows from Proposition 3. 2 ⇒ 1. To apply n ∗ Proposition 3, take f ∈L(w∗)( X ). Using with I := {1, 2,...,n} the formula X f(x + x)=f(x )+ f (P x ) n n PJ (x) I\J n ∅6=J⊂I from [3] (see proof of theorem 3.1.; here f := f (P (·)) is just the (n −|J |)- J PJ (x) I\J linear mapping f with x(j)fixedinthej-th component for j ∈ J) and observing n−|J| ∗ that fJ ∈L(w∗)( X ), one immediately obtains the sequential weak continuity of f. Proposition 5. Let X be a Banach space and n a positive integer. The following con- ditions are then equivalent: n ∗ (1) Pw∗ ( X ) is reflexive. n ∗ (2) X is reflexive and each element of P(w∗)( X ) is weakly sequentially continuous in X∗. s n ∗ n ∗ Proof. 1 ⇒ 2. Let Lw∗ ( X ) be the subspace of Lw∗ ( X ) formed by the symmetric n ∗ elements. If P belongs to Pw∗ ( X ), we write T (P ) for the n-linear symmetric functional obtained from P by means of the polarisation formula. Then n ∗ s n ∗ T : Pw∗ ( X ) −→ L w∗ ( X ) ∗ s n ∗ is an isomorphism. Now, take u0 in X , u0 6= 0. For every g in Lw∗ ( X ), let Sg be the element of X such that, for each u in X∗, (Sg)(u)=g(u, u0,u0,...,u0). Then n ∗ S ◦ T : Pw∗ ( X ) −→ X is a continuous onto linear map and hence X is reflexive. On the other hand, we Valdivia—Spaces of polynomials 97 n ∗ n ∗ apply result (b) to obtain that Pw∗ ( X ) coincides with P(w∗)( X ) and the result ∗ n ∗ ∗ follows. 2 ⇒ 1. Let ψ be the injection from X into Pw∗ ( X ) such that n ∗ ∗ hP,ψ(u)i = P (u),P∈Pw∗ ( X ),u∈ X . Proceeding now as in the proof of Proposition 3, we obtain that ψ(B(X∗)) n ∗ n ∗ is a σ(Pw∗ ( X ), P(w∗)( X ))-compact and, by Krein’s theorem, it follows that n ∗ ∗ n ∗ B(Pw∗ ( X ) ) is weakly compact and thus Pw∗ ( X ) is reflexive. Proposition 6. Let X be a Banach space and let n be a positive integer. The following conditions are equivalent: n ∗ (1) Pw∗ ( X ) is reflexive. r ∗ (2) X is reflexive and, for each integer r, 1 ≤ r ≤ n, every element of P(w∗)( X ) is sequentially weakly continuous in the origin of X∗. Proof. 1 ⇒ 2. We consider T and u0 with the same meaning as in the proof of s n ∗ Proposition 5. We fix 1 ≤ r ≤ n. For each g in Lw∗ ( X ), let Vg be the element of r ∗ ∗ Pw∗ ( X ) such that, for each u in X , (Vg)(u)=g(u,...,u,u0,...,u0). Then, n ∗ r ∗ V ◦ T : Pw∗ ( X ) −→ P w∗ ( X ) r ∗ is linear, continuous and onto and therefore Pw∗ ( X ) is reflexive. We apply Propo- r ∗ sition 5 and thus have that X is reflexive and that every element of P(w∗)( X )is sequentially weakly continuous in the origin of X∗.2⇒ 1. The proof is analogous r ∗ to that of Proposition 4, keeping in mind the isomorphism between Pw∗ ( X ) and s r ∗ Lw∗ ( X ) and taking f to be symmetric. Proposition 7. Let X1,X2,...,Xn be Banach spaces. The following conditions are equiv- alent: (1) For each closed subspace Yj of Xj , j =1, 2,...,n, L(Y1,Y2,...,Yn) is reflexive. (2) Xj is reflexive and, for each closed subspace Yj of Xj , j =1, 2,...,n, the elements of L(Y1,Y2,...,Yn) are sequentially weakly continuous. ∗ Proof. 1 ⇒ 2. Since Xj is isomorphic to a subspace of L(X1,X2,...,Xn), it is reflexive. Take now a closed subspace Yj of Xj . Let (xjm) be a sequence in Yj weakly convergent to xj that does not converge in norm. We may find a subsequence (yjm)of(xjm) such that (yjm − xj ) is a basic sequence, j =1, 2,...,n. Let Zj be the linear span of {xj ,yj1,yj2,...,yjm,...}. It follows that Zj has a Schauder basis (zjm). ∗ Now, let (ujm) be a bounded sequence of Yj such that the restrictions of vjm to Zj , ∗ m =1, 2,..., are the functionals associated with the basis (zjm). There is an order < in the set M := {z1m ⊗ z2m ⊗···⊗znm: m =1, 2,...} ∗ such that (M,< ) is a Schauder basis of Z := Z1⊗e πZ2⊗e π ···⊗e πZn, [14]. Then the 98 Mathematical Proceedings of the Royal Irish Academy linear hull of {u ⊗u ⊗···⊗u : m ,m ,...,m ∈ N} is dense in L(Z ,Z ,...,Z ). 1m1 2m2 nmn 1 2 n 1 2 n Given the positive integers r1,r2,...,rn, we have that limhy1m ⊗ y2m ⊗···⊗ynm,u1r ⊗ u2r ⊗···⊗unr i m 1 2 n = lim u1r (y1m)u2r (y2m) ···unr (ynm) m 1 2 n = hx ⊗ x ⊗···⊗x ,u ⊗ u ⊗···⊗u i 1 2 n 1r1 2r2 nrn and hence, for each element g of L(Y1,Y2,...,Yn) we have that lim g(y1m,y2m,...,ynm)=g(x1,x2,...,xn), m which concludes the proof. 2 ⇒ 1. We take a closed subspace Yj of Xj , j = ∗ ∗ ∗ ∗ 1, 2,...,n.WesetUj := Yj , j =1, 2,...,n. Then Lw∗ (U1 ,U2 ,...,UN ) coincides with L(Y1,Y2,...,Yn) and, applying Proposition 3, this space happens to be reflexive. Making use of Propositions 5 and 6, respectively, we obtain the following two propositions. Proposition 8. Let X be a Banach space and let n be a positive integer. The following conditions are equivalent: (1) For every closed subspace Y of X, P(nY ) is reflexive. (2) X is reflexive and, for every closed subspace Y of X, the elements of P(nY ) are sequentially weakly continuous. Proposition 9. Let X be a Banach space and let n be a positive integer. The following conditions are equivalent: (1) For every closed subspace Y of X, P(nY ) is reflexive. (2) X is reflexive and, for every closed subspace Y of X and every integer r, 1 ≤ r ≤ n, the elements of P(rY ) are sequentially weakly continuous in the origin. In the setting of reflexive Banach spaces with the approximation property, Ryan [26] characterised the reflexivity of P(nX) by the weak continuity of each element of P(nX) on the unit ball and Alencar et al. [2, proposition 4] showed that it is enough to check this at the origin. In Propositions 8 and 9 we have removed the approximation property hypothesis. In order to prove the coming proposition, we shall need the following results that we obtained in [29]: (e) In a Banach space X, let (xm) be a Schauder basis which has a lower p- estimate. If n is an integer with n ≥ p and we put, for each P of P(nX), T (P ):=(P (xm)), then the mapping n T : P( X) −→ l∞ is linear, continuous and onto. (f) In a Banach space X, let (xm) be a sequence that converges weakly to the Valdivia—Spaces of polynomials 99 origin. If n is an even positive integer and P is a continuous n-homogeneous polynomial defined in X such that (P (xm)) does not converge to zero, then there is a subsequence (ym)of(xm) with a lower n-estimate. (g) In a Banach space X, let (xm) be a sequence weakly convergent to the origin. Let P be a continuous n-homogeneous polynomial on X for some positive integer n.If(P (xm)) does not converge to zero, then there is a basic subsequence (zm)of(xm) and a constant a>0 such that X ∗ n−1 k zm k≤ a |B | n m∈B for each non-empty finite subset B of N. We shall also make use of the following result to be found in [13]: (h) Let (xm) be a semi-normalised sequence in the Banach space X such that there are two constants a>0 and b>0 with X b k xm k≤ a |B | m∈B 1 for each non-empty finite subset B of N.Ifp is a number such that 1 Proposition 10. Let X be a Banach space and let n be an even positive integer. The following conditions are equivalent: (1) For each closed subspace Y of X, P(rY ),r≤ n is reflexive. (2) X is reflexive and every semi-normalised sequence that converges weakly to the origin in X has no lower n-estimate. Proof. 1 ⇒ 2. From what has been said before, X is reflexive. Let us assume that there is a semi-normalised sequence (ym)inX weakly convergent to the origin and with a lower n-estimate. We may assume that it is a basic sequence. Let Y be n the closed linear span of (ym). We apply result (e) and obtain P in P( Y ) such that P (ym)=1,m =1, 2,... Then, keeping Proposition 9 in mind, we achieve a contradiction. 2 ⇒ 1. Let us assume that (1) is not satisfied. Again, Proposition 9 gives us an integer r,1≤ r ≤ n, a closed subspace Y of X and an element P of P(rY ) which is not sequentially weakly continuous in the origin. If r = n, it suffices to apply result (f) to reach the conclusion. If r ∗ T : l n −→ [y ] n−1 m ∗ so that, if (e ) is the unit vector basis of l n , then Te = y , m =1, 2,.... Now, if m n−1 m sm ∗ T :[ym] −→ ln is the conjugate map of T , then the restriction of T ∗ to [y ] is a continuous one- sm to-one linear map that takes (y ) into the unit basis of l . It follows then that (y ) sm n sm has a lower n-estimate. Corollary 5. If X is a Banach space, the following conditions are equivalent: (1) For each closed subspace Y of X and each positive integer n, P(nY ) is reflexive. (2) X is reflexive and every semi-normalised sequence in X that weakly converges to the origin has no lower n-estimate for any positive integer n. The implication 2 ⇒ 1 of Corollary 5 was obtained by Farmer [12, theo- rem 1.3 (i)], for closed subspaces Y with the approximation property. For the proof of the next proposition we need the following result, [29] (see also [18, proposition 1.9]): (i) In a Banach space X, let (xm) be a semi-normalised unconditional basic sequence weakly convergent to the origin. If there is, for a positive integer n,a continuous n-homogeneous polynomial P defined in X such that (P (xm)) does not converge to zero, then there is a subsequence (ym)of(xm) with a lower n-estimate. Proposition 11. Let X be a Banach space such that every semi-normalised sequence in X that weakly converges to the origin has an unconditional basic subsequence. Let n be a positive integer. The following conditions are equivalent: (1) For each closed subspace Y of X, P(nY ) is reflexive. (2) X is reflexive and every semi-normalised sequence in X that converges weakly to the origin has no lower n-estimate. Proof. It suffices to follow an analogous argument to the one used in the proof of Propostion 10, making use of result (i) instead of result (f). We do not know whether Proposition 10 holds for odd n. Nevertheless, using result (g), one can show the following. Proposition 12. Let X be a Banach space and let n be a positive integer. If there is a certain p>nsuch that every semi-normalised sequence in X that converges weakly to Valdivia—Spaces of polynomials 101 the origin has no lower p-estimate, then P(nY ) is reflexive for every closed subspace Y of X. Proposition 12 removes the approximation property hypothesis in [16, corol- lary II.4.5] and [12, theorem 1.3] (see the remark above Proposition 10). ∗ ∗ ∗ Next we study some properties of the space Lw∗ (X1 ,X2 ,...,Xn ) using the approximation property in Banach spaces. The following theorem extends a well- known characterisation of the approximation property via compact linear operators or the -product and a property of spaces of polynomials which can be found in [6]. Theorem 4. Let X1,X2,...,Xn−1 be Banach spaces. The following conditions are equiv- alent: (1) X1,X2,...,Xn−1 have the approximation property. ∗ ∗ ∗ (2) For each Banach space Xn, X1 ⊗X2 ⊗···⊗Xn is dense in Lw∗ (X1 ,X2 ,...,Xn ). (3) For each reflexive Banach space Xn, X1 ⊗ X2 ⊗ ··· ⊗ Xn is dense in ∗ ∗ ∗ Lw∗ (X1 ,X2 ,...,Xn ). ∗ ∗ (4) For each Banach space Y , (X1⊗X2⊗···⊗Xn−1)⊗Y is dense in Lw∗ (X1 ,X2 ,..., ∗ Xn−1; Y ). Proof. 1 ⇒ 2. Since X ⊗ Y is dense in XY , if the Banach space X or the Banach space Y has the approximation property, it follows that Xn−1⊗Xn is dense in Xn−1Xn and so Xn−2 ⊗(Xn−1Xn) is dense in Xn−2(Xn−1Xn) and therefore Xn−2 ⊗Xn−1 ⊗Xn is dense in Xn−2Xn−1Xn and hence we obtain that X1 ⊗ X2 ⊗···⊗Xn is dense in ∗ ∗ ∗ X1X2...Xn = Lw∗ (X1 ,X2 ,...Xn ). 2 ⇒ 3. Obvious. 3 ⇒ 4. Using [9] (and [10, p. 33]), it is easy to see that for a compact set A ⊂ Y there is an absolutely convex compact set B ⊂ Y such that YB is reflexive and A is also compact in YB. Since all ∗ ∗ ∗ f ∈Lw∗ (X1 ,X2 ,...,Xn−1; Y ) are compact, this implies that one may assume Y to be reflexive in 4. Now ∗ ∗ ∗ ∗ ∗ ∗ ∗ Lw∗ (X1 ,X2 ,...,Xn−1; Y )=Lw∗ (X1 ,X2 ,...,Xn−1,Y ) by Corollary 1, which gives 3 ⇒ 4 and 4 ⇒ 2. 2 ⇒ 1. To show that X := X1...Xn−1 has the approximation property (and hence X1,...,Xn−1) it is enough to show (see [10, p. 60]) that, for each reflexive Banach space Xn, the space X ⊗ Xn is dense in ∗ ∗ ∗ ∗ K(Xn ; X)=XnX = Lw∗ (X1 ,X2 ,...,Xn ). This is true by hypothesis. Note 2. Schwartz [27] defines the -product of n locally convex spaces Ei to be the set of n-linear mappings 0 0 0 φ:(E1)γ × (E2)γ ×···×(En)γ −→ K 0 that are hypocontinuous with respect to the equicontinuous sets (i.e. if Ai ⊂ Ei are equicontinuous for i 6= i then there is a neighborhood V of the origin in (E )0 0 i0 i0 γ with ! Y sup |φ A × V |≤1 i i0 i6=i0 102 Mathematical Proceedings of the Royal Irish Academy and proves that E1 ⊗E2 ⊗···⊗En is dense if E1,E2,...,En−1 have the approximation property. Corollary 6. Let X1,X2,...,Xn be Banach spaces. If X1,X2,...,Xn−1 have the approx- imation property, then ∗ ∗ ∗ e e e Lw∗ (X1 ,X2 ,...,Xn )=X1⊗X2⊗ ···⊗Xn. As in [14], we endow N2 with the rectangular order <∗. Proceeding by induction, assume that for an integer n − 1 ≥ 2 we have the elements of Nn−1 ordered in the n−1 sequence (ar). We then order, in the rectangle manner, the pairs (ar,s), ar ∈ N , s ∈ N, and thus define an order <∗ in Nn. This is the square ordering described inductively by Ryan [26]. This ordering has also been used in [11]. Corollary 7. If (xjm) is a Schauder basis in the Banach space Xj , j =1, 2,...,n, then {x ⊗ x ⊗···⊗x :(m ,m ,...,m ) ∈ Nn,<∗} 1m1 2m2 nmn 1 2 n ∗ ∗ ∗ is a Schauder basis in Lw∗ (X1 ,X2 ,...,Xn ). Proof. It follows immediately, applying [14] and Corollary 6. Corollary 8. If (xjm) is a shrinking basis in the Banach space Xj , j =1, 2,...,n, then {x∗ ⊗ x∗ ⊗···⊗x∗ :(m ,m ,...,m ) ∈ Nn,<∗} 1m1 2m2 nmn 1 2 n is a Schauder basis in Lw(X1,X2,...,Xn). ∗ ∗ ∗ Proposition 13. Let X1,X2,...,Xn be Banach spaces such that Lw∗ (X1 ,X2 ,...,Xn ) ∗ ∗ ∗ contains no copy of l1.IfX1 ,X2 ,...,Xn−1 have the approximation property, then ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ Lw∗ (X1 ,X2 ,...,Xn ) = L(X1 ,X2 ,...,Xn ). ∗∗ ∗ ∗ Proof. Given uj ∈ Xj , j =1, 2,...,n, the element u1 ⊗u2 ⊗···⊗un of L(X1 ,X2 ,..., ∗ ∗ ∗ ∗ Xn ) clearly belongs to L(w∗)(X1 ,X2 ,...,Xn ) and hence it suffices, after Theorem ∗∗ ∗∗ ∗∗ ∗ ∗ ∗ 2, to see that X1 ⊗ X2 ⊗···⊗Xn is dense in X := L(X1 ,X2 ,...,Xn ) for the ∗ ∗ ∗ topology of uniform convergence over each compact subset of X1 × X2 ×···×Xn . The property is obviously true for n = 1. Following an induction argument, we ∗ ∗ ∗ ∗ ∗ ∗ assume that X2 ⊗ X3 ⊗···⊗Xn is dense in Y := L(X2 ,X3 ,...,Xn ) for the topology ∗ ∗ ∗ of the uniform convergence over the compact subsets of X2 × X3 ×···×Xn .We ∗ take a compact subset Kj contained in B(Xj ), j =1, 2,...,n. Let f be an arbitrary ∗ element of X. Let >0. Let T be the map from X1 into Y such that ∗ Tu = f(u, ·, ·,...,·),u∈ X1 . ∗ Then T is continuous and linear. Since X1 has the approximation property, there is Valdivia—Spaces of polynomials 103 ∗ a finite-rank operator S: X1 −→ Y such that 1 kTu1 − Su1 k< 2 , u1 ∈ K1. We may write Xr ∗∗ S = v1s ⊗ fs,v1s ∈ X1 ,fs ∈ Y, s=1, 2,...,r. s=1 P r s ∗∗ We take ∞ >k> s=1 k v1s k and find vih ∈ Xi , h =1, 2,...,h(s), i =2, 3,...,n,so that h(s) X f (u ,u ,...,u ) − vs (u )vs (u ) ...vs (u ) < , s 2 3 n 2h 2 3h 3 nh n 2rk h=1 ui ∈ Ki, i =2, 3,...,n, s =1, 2,...,r. Then, for ui ∈ Ki, i =1, 2,...,n, r h(s) X X f(u ,u ,...,u ) − v (u )vs (u ) ...vs (u ) 1 2 n 1s 1 2h 2 nh n s=1 h=1 r X ≤ f(u ,u ,...,u ) − v (u )f (u ,u ,...,u ) 1 2 n 1s 1 s 2 3 n s=1 r h(s) X X + v (u )(f (u ,u ,...,u ) − vs (u )vs (u ) ...vs (u ) 1s 1 s 2 3 n 2h 2 3h 3 nh n s=1 h=1 Xr ≤kTu − Su k +k <. 1 1 2rk s=1 Corollary 9. Let X1,X2,...,Xn be Banach spaces such that Lw(X1,X2,...,Xn) has no ∗∗ ∗∗ ∗∗ copies of l1.IfX1 ,X2 ,...,Xn−1 have the approximation property, then ∗∗ ∗∗ ∗∗ ∗∗ Lw(X1,X2,...,Xn) = L(X1 ,X2 ,...,Xn ). ∗ Corollary 10. Let Xj be a Banach space such that Xj is Asplund, j =1, 2,...,n.If ∗∗ ∗∗ ∗∗ X1 ,X2 ...,Xn−1 have the approximation property, then ∗∗ ∗∗ ∗∗ ∗∗ Lw(X1,X2,...,Xn) = L(X1 ,X2 ,...,Xn ). n Proposition 14. Let X be a Banach space and let n be a positive integer. If Pw( X) ∗ ∗∗ contains no copy of l1 or, in particular, if X is an Asplund space, then, if X has the approximation property, we have that n ∗∗ n ∗∗ Pw( X) = P( X ). Proof. An immediate consequence of result (c). 104 Mathematical Proceedings of the Royal Irish Academy n n In the case where P( X)=Pw( X), Proposition 14 follows from the proof of [5, theorem 5(b)]. When, for a particular isomorphism, the equality P(nX)∗∗ = P(nX∗∗) holds for every integer n, it is said, after [5], that X is Q-reflexive. The only known examples of Q-reflexive Banach spaces are either reflexive, as the dual of Tsirelson’s space, [24, p. 95], or quasi-reflexive [5]. In the following, we give examples of Q-reflexive Banach spaces that are neither reflexive nor quasi-reflexive. We start out with the following lemma. Lemma 2. Lw(X1,X2,...,Xn) ⊂Lw(X1⊗e πX2,X3,...,Xn). Proof. Each f ∈Lw(X1,X2,...,Xn) defines a bilinear mapping T ∈L(X1,X2; Lw(X3,...,Xn)). Let us prove that T is compact. For this take e ∗∗ ∗∗ ∗∗ f ∈Lw∗ (X1 ,X2 ,...,Xn ), ∗∗ ∗∗ ∗∗ the canonical extension of f to X1 × X2 ×···×Xn . By Corollary 1 we have that ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ Lw∗ (X1 ,X2 ,...,Xn )=Lw∗ (X1 ,X2 ; Lw∗ (X3 ,...,Xn )). e ∗∗ ∗∗ ∗∗ ∗∗ Let T be the element of Lw∗ (X1 ,X2 ; Lw∗ (X3 ,...,Xn )) associated with f. Then e e T is compact and since T is the restriction of T to X1 × X2, we obtain that T is compact also. Let us denote by f1 ∈L(X1⊗e πX2,X3,...,Xn) the mapping related to f and by T1 ∈L(X1⊗e πX2; Lw(X3,...,Xn)) the mapping related to T . It is clear that T1 is compact and that f1(z,w)=T1(z)[w], z ∈ X1⊗e πX2, w ∈ X3 ×···×Xn.For S ∈L(X3,...,Xn) and w =(w3,...,wn) ∈ X3 ×···×Xn we shall use the notations ⊗ ∗ ⊗ S for the elements of (X3⊗e π ···⊗e πXn) associated with S and w := w3 ⊗···⊗wn. Now take zα −→ z0 in B(X1⊗e πX2) weakly, wα −→ w0 in B(X3) ×···×B(Xn) weakly and >0. Since T1 is compact, there are S1,S2, ···,Sm ∈Lw(X3,...,Xn) such that for every z ∈ B(X1⊗e πX2) there is an integer r(z), 1 ≤ r(z) ≤ m, with ⊗ ⊗ kT1(z) − Sr(z) k=kT1(z) − Sr(z) k≤ . ⊗ ⊗ ⊗ It is clear that Sr(z)wα = Sr(z)wα −→ Sr(z)w0 = Sr(z)w0 . Therefore ⊗ ⊗ ⊗ |f1(zα,wα) − f1(z0,w0)|≤|T1(zα)[z0] − T1(z0)[w0]| + |hT1(zα) ,wα − w0 i| . ∗ The first term converges to zero since T1(·)[w0] ∈ (X1⊗e πX2) and the second term can be estimated by max (2 kT (z )⊗ − S ⊗ k + |hS ⊗ ,w⊗ − w⊗i|) α 1 α r(zα) r(zα) α 0 which is smaller than 3 for α sufficiently large. Thus f1 ∈Lw(X1⊗e πX2,X3,...,Xn). Valdivia—Spaces of polynomials 105 ∗ ∗ Let T be the dual of the Tsirelson space T as it appeared in [24, p. 95] and TJ the Tsirelson∗–James’ space from [7] and [25]. Theorem 5. Let Y be a subspace of T ∗ with the approximation property. If e ∗ X := Y⊗πTJ , then the following properties are satisfied: (1) X has a topological complement in X∗∗ which is isomorphic to Y . (2) X is Q-reflexive. ∗ Proof. By [4], the spaces Y and TJ have the property Pα for all α>0 which implies n ∗ n (see [3]) that all 2n-linear continuous mappings, Y × (TJ ) −→ K, are weakly n ∗ n ∗ ∗ sequentially continuous. Since l1 6⊂ Y × (TJ ) , they are in Lw(Y,TJ ,...,Y,TJ ). In particular, by Lemma 2, n n n n L( X)=Lw( X) and P( X)=Pw( X) and hence Proposition 14 implies n ∗∗ n ∗∗ n ∗∗ P( X) = Pw( X) = P( X ), i.e. X is Q-reflexive. Moreover, using Corollary 4, ∗ ∗ ∗∗ ∗∗∗ ∗ e ∗∗ X = Lw(Y,TJ )=Lw∗ (Y ,TJ )=Y ⊗TJ ∗∗∗ ∗ and since TJ = TJ ⊕ K, ∗∗ ∗∗ e ∗∗∗ e ∗ e ∗ X = Y ⊗πTj = Y ⊗π(TJ ⊕ K)=(Y ⊗πTJ ) ⊕ Y = X ⊕ Y. If we consider a reflexive Banach space Y , it can be shown analogously to what e ∗ we have done in Theorem 5, that X := Y ⊗πTJ has a topological complement in ∗∗ X isomorphic to Y . Moreover, if Y = lp (1 Acknowledgements The author wishes to express his gratitude to Professor Klaus Floret, who kindly read the manuscript, simplified the proof of Theorem 4, which was originally much longer and did not make use of the -product, and eliminated the approximation property from Lemma 2 of the former version. This paper was supported in part by Direccion´ General de Ensenanza˜ Superior PB96-0758. Added in proof. After submitting our paper we got a preprint by Gonzalez´ [15] where he obtained examples of Q-reflexive spaces that are neither reflexive nor quasi-reflexive. 106 Mathematical Proceedings of the Royal Irish Academy References [1] R. Alencar, On reflexivity and basis for P(mE), Proceedings of the Royal Irish Academy, 85A, (1985), 135–8. [2] R. Alencar, R. Aron and S. 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