Weak-Polynomial Convergence on a Banach Space

Abstract

Abstract.We show that any super-reflexive Banach space is a A-space (i.e., the weak-polynomial convergence for sequences implies the norm convergence). We introduce the notion of /c-space (i.e., a Banach space where the weakpolynomial convergence for sequences is different from the weak convergence) and we prove that if a dual Banach space Z is a k-space with the approximation property, then the uniform algebra A(B) on the unit ball of Z generated by the weak-star continuous polynomials is not tight.We shall be concerned in this note with some questions, posed by Carne, Cole, and Gamelin in [3], involving the weak-polynomial convergence and its relation to the tightness of certain algebras of analytic functions on a Banach space.Let A' be a (real or complex) Banach space. For m = 1,2, ... let 3°(mX) denote the Banach space of all continuous m-homogeneous polynomials on X, endowed with its usual norm; that is, each P e 3B(mX) is a map of the form P(x) = A(x, ... ,m) x), where A is a continuous, m-linear (scalar-valued) form on Xx---m]xX.Also, let S6(X) denote the space of all continuous polynomials on X; that is, each P e 3°(X) is a finite sum P -Pq + Px-\-\-Pn , where P0 is constant and each Pm e 3B(mX) for m= 1,2, ... , n.In [3] a sequence (Xj) c X is said to be weak-polynomial convergent to x e X if P(xj) -> P(x) for all P e 36(X); and the space X is defined to be a A-space, if whenever (xj) is a sequence in X that is weak-polynomial convergent to 0, then ||jc,-|| -»■ 0.It is shown in [3] that lp is a A-space for 1 < p < co ; it is also shown that Lp(p) is a A-space for 2 < p < oo and Lx[0, 1] is not a A-space, and the question is posed as to whether Lp(p) is a A-space for 1 < p < 2. Our next result will provide an affirmative answer to this question.First we recall that super-reflexive Banach spaces can be defined as those spaces that admit an equivalent uniformly convex norm. In particular, spaces Lp(p) are super-reflexive for 1 < p < oo and any measure p (see, e.g., [8, Chapter 3]).We shall use the following fact: If X is super-reflexive, then there exists

Polynomials on Schreier's Space

“…E is said to be a Λ-space if P-null sequences and norm null sequences coincide in E. Spaces with the Schur property are trivially Λ-spaces. All superreflexive spaces are Λ-spaces [16]. It is proved in [13,Corollary 3.6] that every Banach space with nontrivial type is a Λ-space.…”

Three-Space Problems for Polynomial Properties in Banach Spaces

“…In [7] Carne, Cole and Gamelin defined a L-space as a Banach space such that every sequence that is polynomially null is also norm null, and conjectured that, for 1<p<+O, L p -spaces are L-spaces. In [24], Jaramillo and Prieto proved that if X * has, for some p<+O, property W p (i.e. it is reflexive and every weakly null sequence admits a weakly p-summable subsequence ; see [10]) then X is a L-space.…”

Three-Space Problems for Polynomial Properties in Banach Spaces

“…In [24], Jaramillo and Prieto proved that if X * has, for some p<+O, property W p (i.e. it is reflexive and every weakly null sequence admits a weakly p-summable subsequence ; see [10]) then X is a L-space.…”