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Banach Spaces and Banach Lattices

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Banach Spaces of Continuous Functions as Dual Spaces

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Abstract

We shall now give some background in the theory of normed and Banach spaces, including the key definitions of dual and bidual spaces and of an isomorphism and an isometric isomorphism between two normed spaces. In particular, we shall show how certain bidual spaces can be embedded in other Banach spaces. In \(\S\) 2.3, we shall also recall some basic results and theorems concerning Banach lattices. We shall define complemented subspaces of a Banach space in \(\S\) 2.4, and also we shall discuss, in \(\S\) 2.5, the projective and injective objects in the category of Banach spaces and bounded operators. We shall conclude the chapter by discussing dentability and the Krein–Milman property for Banach spaces in \(\S\) 2.6.

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Notes

  1. 1.

    An immediate contradiction can be obtained at this point by an appeal to a lemma of Rosenthal (see [131, Proposition 7.21]); we provide a somewhat simpler, self-contained argument here.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Lancaster, Lancaster, UK

    H. G. Dales

  2. Center of Excellence in Computation Algebra, and Topology (CECAT), Chapman University, Orange, CA, USA

    F. K. Dashiell Jr.

  3. Department of Mathematical Sciences, University of Alberta, Edmonton, AB, Canada

    A. T.-M. Lau

  4. Department of Pure Mathematics, University of Leeds, Leeds, UK

    D. Strauss

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Dales, H.G., Dashiell, F.K., Lau, A.TM., Strauss, D. (2016). Banach Spaces and Banach Lattices. In: Banach Spaces of Continuous Functions as Dual Spaces. CMS Books in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-32349-7_2

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