Ultrapowers and local properties of Banach spaces

Stern, Jacques

doi:10.1090/S0002-9947-1978-0489594-0

Ultrapowers and local properties of Banach spaces
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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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