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Proceedings of the American Mathematical Society

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Infinite dimensional $ L$-spaces do not have preduals of all orders

Author: Thomas E. Armstrong
Journal: Proc. Amer. Math. Soc. 74 (1979), 285-290
MSC: Primary 46B10
MathSciNet review: 524301
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Abstract: It is shown that if E is an infinite dimensional Banach space with first dual E', second dual E", and nth dual $ {E^{[n]}}$ and if $ {E^{[n]}}$ is either an L- or M-space all duals are either L- or M-spaces except possibly E which could be a Lindenstrauss space. If E is an L- or M-space there is an integer $ n(E)$ so that if $ m > n(E)$ there is no Banach space F with $ E = {F^{[m]}}$. The linear isomorphic analogues to these isometric results are also established. In particular if E is an $ {\mathcal{L}_1}$ or $ {\mathcal{L}_\infty }$ space there is an integer $ \bar n(E)$ so that E is not linearly isomorphic to $ {F^{[m]}}$ for any Banach space F when $ m > \bar n(E)$.

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Keywords: L-space, M-space, Lindenstrauss space, dual, predual, simplex, injective Banach space, topological dimension
Article copyright: © Copyright 1979 American Mathematical Society