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- W2079800536 abstract "PROOF. (i) Suppose X is quasi-reflexive of order <n. By [1, p. 908] there exist a Banach space (Y, r) (quasi-reflexive of the same order as X) and a linear homeomorphism q from (X, p) onto (Y*, r*). Let J be the canonical map from Y into Y**. Let S= * [JY]. The codimension of S in X* is equal to the codimension of JY in Y**, which is ? n. Suppose q is a norm on X* equivalent to p*. Define q2 on Y** by q2(F) =q(4*F), where k* is the transpose of 7. Then q2 and r** are equivalent norms on Y** since q and p* are equivalent norms on X* and since 4 * is a linear homeomorphism from (Y**, r**) onto (X*, p*). Define q3 on Y by q3(y)=q2(Jy). Note that r(y) =r**(Jy). Now q3 and r are equivalent norms on Y since q2 and r** are equivalent norms on Y** and since J is a linear isomorphism from Y into Y**. Thus q* and r* are equivalent norms on Y*. Define t on X by t(x) =q* (<Ax). Then t and p are equivalent norms on X since q is a linear homeomorphism from (X, p) onto (Y*, r*). Finally, t*| S q I S since for all y E Y we have" @default.
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- W2079800536 date "1969-01-01" @default.
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- W2079800536 title "Quasi-reflexivity and dual norms" @default.
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- W2079800536 doi "https://doi.org/10.1090/s0002-9939-1969-0253020-7" @default.
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