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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Ultrapowers and local properties of Banach spaces
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by Jacques Stern PDF
Trans. Amer. Math. Soc. 240 (1978), 231-252 Request permission

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
  • © Copyright 1978 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 240 (1978), 231-252
  • MSC: Primary 46B99; Secondary 03C20, 46E30
  • DOI: https://doi.org/10.1090/S0002-9947-1978-0489594-0
  • MathSciNet review: 489594