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Ultraproducts of Type \(\mathcal{L}_{\infty }\)

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Separably Injective Banach Spaces

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2132))

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Abstract

The Banach space ultraproduct construction is perhaps the main bridge between model theory and the theory of Banach spaces and its ramifications. Ultraproducts of Banach spaces, even at a very elementary level, proved very useful in local theory, the study of Banach lattices, and also in several nonlinear problems, such as the uniform and Lipschitz classification of Banach spaces. We refer the reader to Heinrich’s survey paper [126] and Sims’ notes [234] for two complementary accounts. Traditionally, the main investigations about Banach space ultraproducts have focused on the isometric theory, reaching a quite coherent set of results very early, as can be seen in [132]. We will review some results on the isometric theory of ultraproducts in Sect. 4.7.4, but most of the Chapter is placed in the isomorphic context.

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Authors and Affiliations

  1. Dpto. de Matemáticas, Universidad de Murcia, Murcia, Spain

    Antonio Avilés

  2. Dpto. de Matemáticas, Universidad de Extremadura, Badajoz, Spain

    Félix Cabello Sánchez & Jesús M. F. Castillo

  3. Dpto. de Matemáticas, Universidad de Cantabria, Santander, Spain

    Manuel González

  4. Dpto. de Matemáticas, Universidad de Extremadura, Cáceres, Spain

    Yolanda Moreno

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  1. Antonio Avilés
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  2. Félix Cabello Sánchez
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  3. Jesús M. F. Castillo
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  4. Manuel González
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  5. Yolanda Moreno
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Avilés, A., Sánchez, F.C., Castillo, J.M.F., González, M., Moreno, Y. (2016). Ultraproducts of Type \(\mathcal{L}_{\infty }\) . In: Separably Injective Banach Spaces. Lecture Notes in Mathematics, vol 2132. Springer, Cham. https://doi.org/10.1007/978-3-319-14741-3_4

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