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Introduction

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Banach Spaces of Continuous Functions as Dual Spaces

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Abstract

In this chapter, we shall begin by introducing some basic notations. This will be followed by a discussion in \(\S\) 1.4 of some topological concepts, including those of locally compact spaces and Stonean spaces. We shall later frequently refer to the Stone–Čech compactification β X of a completely regular space X, and this is introduced in \(\S\) 1.5. In \(\S\) 1.6, we shall prove Gleason’s theorem characterizing projective topological spaces as the Stonean spaces, and, in \(\S\) 1.7, we shall also recall some basic theory of Boolean algebras, generalizing this slightly to cover Boolean rings; we shall discuss the Stone space of a Boolean ring and give various important examples of Boolean rings.

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Notes

  1. 1.

    An example of C. H. Dowker shows that the converse fails; see [99, 6.2.20] and [112, 16M(Δ 1)]. In fact there is a (non-normal) zero-dimensional, locally compact space X such that βX is not zero-dimensional [236]; this contradicts a sentence in the book [155, p. 169] of Kelley.

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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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  1. H. G. Dales
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  3. A. T.-M. Lau
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  4. D. Strauss
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Dales, H.G., Dashiell, F.K., Lau, A.TM., Strauss, D. (2016). Introduction. 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_1

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