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A complementary universal conjugate Banach space and its relation to the approximation problem

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Abstract

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.

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Authors and Affiliations

  1. Ohio State University, 43210, Columbus, Ohio

    William B. Johnson

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  1. William B. Johnson
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Additional information

The author was supported in part by NSF GP 28719.

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Johnson, W.B. A complementary universal conjugate Banach space and its relation to the approximation problem. Israel J. Math. 13, 301–310 (1972). https://doi.org/10.1007/BF02762804

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