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Transfinite duals of quasireflexive Banach spaces


Author: Steven F. Bellenot
Journal: Trans. Amer. Math. Soc. 273 (1982), 551-577
MSC: Primary 46B10
DOI: https://doi.org/10.1090/S0002-9947-1982-0667160-2
MathSciNet review: 667160
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Abstract: The transfinite duals of a space with a neighborly basis are constructed until they become nonseparable. Let $ s(X)$ be the first ordinal $ \alpha $ so that $ {X^\alpha }$ is nonseparable. It is shown that if $ X$ is nonreflexive, $ s(X) \leqslant {\omega ^2} + 1$ (this is best possible) and that $ \{ s(X):X{\text{separable quasireflexive of order one}}\} = \{ \omega + 1,\omega + 2,2\omega + 1,2\omega + 2,{\omega ^2} + 1\} $. A quasireflexive space $ X$ is constructed so that $ {X^\omega }$ is isomorphic to $ X \oplus {c_0}$ and no basic sequence in $ X$ is equivalent to a neighborly basis. It is shown that the $ {\omega ^2}$th dual of James space and James function space are isomorphic to subspaces of one another. Also, perhaps of interest on its own, a reflexive space with a subsymmetric basis is constructed whose inversion spans a nonreflexive space.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1982-0667160-2
Keywords: Neighborly, equal signs additive, invariant under spreading, quasireflexive, transfinite duals, separable and nonseparable, James space, inversion of a basis
Article copyright: © Copyright 1982 American Mathematical Society

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