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

Let C 1=(G 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. If X is a separable Banach space, then X * is isometric to a subspace of C 1 * =(G n * ) m which is the range of a contractive projection on C 1 * . Separable Banach spaces whose conjugates are isomorphic to C 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 (G n * ) m does, and if there is a space failing the m.a.p., then C 1 can be equivalently normed to fail the m.a.p. 1972 Hebrew University.

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