Abstract
In this chapter we give various characterizations of C(T, ℝ) and C(T, ℂ). In section 9 we present the Banach lattice and Banach algebra characterizations of these spaces and include a discussion of abstract M spaces and the Banach spaces of continuous functions which vanish at infinity on a locally compact Hausdorff space. Section 10 takes up the study of C(T, ℝ) as a Banach space. This means that geometric conditions are given on a Banach space X and its dual X* in order to characterize when X is linearly isometric to C(T, ℝ) for some compact Hausdorff space T. In section 11 we characterize those Banach spaces which have the property that they are complemented with respect to a contractive projection in each Banach space which contains them.
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