A fully convex Banach space which does not have the Banach-Saks property

Abstract

Let k 2 2 be an integer. A Banach space X is said to be fully k-convex (kR) if for every sequence {xn> in X, lim,,,,..,n,, o. (l/k) II&l, + ... + x,,J = 1 implies that {xn} is a Cauchy sequence in X. It is known [S] that every uniformly convex space is 2R and that every kR space is reflexive. It is also known [3, 51 that there are 2R spaces which are not superreflexive. Milman [ 10, p. 973 has raised the question whether every reflexive space is isomorphic to a 2R space. A Banach space X is said to have the Banach-Saks property (BS) if for every bounded sequence {x,} in X, there exists a subsequence {y,} of (xn} such that the Cesaro means, { (l/n)( y, + . . . + y,}, of { yn} is convergent in X. It is known that [7] all superreflexive spaces have the (BS) and that [ 1 l] every Banach space with (BS) is reflexive. However, there exist (e.g., [1]) reflexive Banach spaces which do not have the (BS). It is also known [12] that there are Banach spaces with the (BS) which are not superreflexive. Since the (BS) is invariant under isomorphism, it is natural to ask whether every kR space has the (BS). An affirmative answer would yield a negative solution to Milman’s question. The purpose of the paper is to renorm Baernstein’s space [ 1 ] so that it is 2R. Therefore, kR spaces do not necessarily have the (BS). We shall need the following characterization of kR spaces. For the proof when k = 2, see [S, 6, S].