Some Properties in Spaces of Multilinear Functionals and Spaces of Polynomials

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 ‖ · ‖ 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∗, 〈x, u〉 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 ≤‖ xn ‖≤ k, n = 1, 2, . . . If the sequence (xn) is basic, then (x∗n) is the 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 ∑ n=1 anxn ∥∥∥∥ ≥ c ( m ∑