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.
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.
We show that for the Hardy class of functions H 1 with domain the ball or polydisc in C N , a certain type of uniform convexity property (the uniform Kadec-Klee-Huff property) holds with respect to the topology of pointwise convergence on the interior; which coincides with both the topology of uniform convergence on compacta and the weak * topology on bounded subsets of H 1 .Also, we show that if a Banach space X has a uniform Kadec-Klee-Huff property, then the Lebesgue-Bochner space L p (µ, X) 1 ≤ p < ∞ must have a related uniform Kadec-Klee-Huff property. Consequently, by known results, the above spaces have normal structure properties and fixed point properties for non-expansive mappings.
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