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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Ultrapowers and local properties of Banach spaces


Author: Jacques Stern
Journal: Trans. Amer. Math. Soc. 240 (1978), 231-252
MSC: Primary 46B99; Secondary 03C20, 46E30
MathSciNet review: 489594
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Abstract: The present paper is an approach to the local theory of Banach spaces via the ultrapower construction. It includes a detailed study of ultrapowers and their dual spaces as well as a definition of a new notion, the notion of a u-extension of a Banach space. All these tools are used to give a unified definition of many classes of Banach spaces characterized by local properties (such as the $ {\mathcal{L}_p}$-spaces). Many examples are given; also, as an application, it is proved that any $ {\mathcal{L}_p}$-space, $ 1 < p < \infty $, has an ultrapower which is isomorphic to an $ {L_p}$-space.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1978-0489594-0
PII: S 0002-9947(1978)0489594-0
Article copyright: © Copyright 1978 American Mathematical Society