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Some Properties in Spaces of Multilinear Functionals and Spaces of Polynomials

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Some Properties in Spaces of Multilinear Functionals and Spaces of Polynomials 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 <h<k<∞ ∗ such that h ≤k xn k≤ k, n =1, 2,... If the sequence (xn) is basic, then (xn)isthe 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.
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