If a Banach space X contains a complemented subspace isomorphic to Co (respectively, I 1 ), then X contains complemented almost isometric copies of Co (respectively, (}). If a Banach space X is such that X* contains a subspace isomorphic to L^O, 1] (respectively, t°°), then X" contains almost isometric copies of L l [0,1] (respectively, £°°).In [3], James proved that if a Banach space contains a subspace which is isomorphic to Co (respectively, 1}), then it contains almost isometric copies of Co (respectively, I 1 ). In this short note we shall prove complemented versions of these results and show that a dual Banach space containing a subspace isomorphic to L 1^, 1] (respectively, £°°) must contain almost isometric copies of L^O, 1] (respectively, t°°). In particular, the L^O, 1] result is in sharp contrast to a result of Lindenstrauss and Pelczyriski PROOF: Let Y be a complemented subspace of X which is isomorphic to c 0 . Let P be a bounded linear projection from X onto Y.

Connections between the shape of the unit ball of a Banach space and analytic properties of the Banach space have been studied for many years. In this article, some geometric properties of spaces related to n-homogeneous polynomials are considered. In particular, the rotundity and smoothness of spaces of continuous n-homogeneous polynomials and its preduals are studied. Furthermore, an inequality relating the product of the norms of linear functionals on a Banach space with the norm of the continuous n-homogeneous polynomial determined by the product of the linear functionals is derived. This inequality is used to study the strongly exposed points of the predual of the space of continuous 2-homogeneous polynomials.

Abstract. We give a basic sequence characterization of relative weak compactness in c 0 and we construct new examples of closed, bounded, convex subsets of c 0 failing the fixed point property for nonexpansive self-maps. Combining these results, we derive the following characterization of weak compactness for closed, bounded, convex subsets C of c 0 : such a C is weakly compact if and only if all of its closed, convex, nonempty subsets have the fixed point property for nonexpansive mappings.

No abstract