A complementary universal conjugate Banach space and its relation to the approximation problem
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LetC 1=(ΣG n ) l 1, where (G n ) is a sequence which is dense (in the Banach-Mazur sense) in the class of all finite dimensional Banach spaces. IfX is a separable Banach space, thenX * is isometric to a subspace ofC 1 * =(ΣG n * ) m which is the range of a contractive projection onC 1 * . Separable Banach spaces whose conjugates are isomorphic toC 1 * are classified as those spaces which contain complemented copies of C1. Applications are that every Banach space has the [metric] approximation property ([m.] a.p., in short) iff (ΣG n * ) m does, and if there is a space failing the m.a.p., thenC 1 can be equivalently normed to fail the m.a.p.
KeywordsBanach Space Equivalent Norm Studia Math Separable Banach Space Unconditional Basis
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