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• W2091683089 abstract "THEOREM I. If E is a separable Banach space such that E has a complemented subspace isomorphic to 11(r) with r uncountable then E contains a complemented, o(EI, E) closed subspace isomorphic to M(A), the Radon measures on the Cantor set. THEOREM II. If E is a separable Banach space such that E' has a subspace isomorphic to 11(r) with r uncountable, then E contains a subspace isomorphic to 11. THEOREM III. Let E be a Banach space. The following are equivalent: (i) E 'is isomorphic to 1li(r); (ii) every absolutely summing operator on E is nuclear; (iii) every compact, absolutely summing operator on E is nuclear; (iv) if X is a separable subspace of E, then there exists a subspace Y such that X C Y C E and Y 'is isomorphic to 11. THEOREM IV. If E is a 20, space then (i) E is isomorphic to 1l(r) for some set r or (ii) E contains a complemented subspace isomorphic to M(^. COROLLARY. If E is a separable ?P space, then E is (i) finite dimensional, or (ii) isomorphic to 1, or (iii) isomorphic to M(A). COROLLARY. If Ll(u) is isomorphic to the conjugate of a separable Banach space, then LI(C) is isomorphic to 11 or M(A). Introduction. In the past few years, a great deal of information has been obtained about Banach spaces E such that E , the dual of E (the space of continuous linear functionals on E), is isometric to some L1i() = LI(S, 1, I), the Banach space of (equivalence classes of) measurable, absolutely integrable functions on some abstract measure space (S, 1, y). See [91 for the most recent results in this area. There apparently have been very few results characterizing those Banach spaces E such that E' is isomorphic to some LQ(S, i, fL). Closely related to this problem is the problem raised by Dieudonne [21 of determining those L 1(L) spaces which are isomorphic to dual spaces. The study of this problem began with Gelfand who proved that LJ[O, 1 is not isomorphic to a dual space [31; recently, Pe,'czyn'ski proved that if 1i is u-finite and not purely atomic then L(QL) is not isomorphic to a dual space [161. (See [81 and the references of [161 Received by the editors December 27, 1971. AMS (MOS) subject classifications (1969). Primary 4610, 4635." @default.
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• W2091683089 date "1973-02-01" @default.
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• W2091683089 title "Banach Spaces Whose Duals Contain l 1 (Γ) With Applications to the Study of Dual L 1 (μ) Spaces" @default.
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