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# FA.Chap5.1-4 - Chapter 5 The Banach space C X In this...

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Unformatted text preview: Chapter 5 The Banach space C ( X ) In this chapter we single out one of the most extensively studied Banach spaces, C ( X ) – the Banach space of all continuous complex-valued functions on a com- pact topological space X . In the case X = [ a, b ], we will give several important results, including theorems of Korovkin and Bernstein. These will lead us to the Weierstrass approximation theorem, which implies the separability of C [ a, b ]. When X is a general compact topological space, we will study sub-collections of C ( X ) that are closed under “multiplication” in C ( X ); these collections are the “sub-algebras” in C ( X ). This will lead to the Stone-Weierstrass theorem, a far-reaching generalization of the Weierstrass approximation theorem. We will begin by defining equicontinuity , a concept needed for the Arzela- Ascoli theorem , which gives conditions for when a subset of C ( X ) is compact. 5.1 The Arzela-Ascoli theorem The result behind all of calculus is the statement that a subset of R n is compact if and only if it is closed and bounded; this underlies such critical results as the mean value theorem, the uniform continuity of functions on compact sets (which makes Riemann sums converge and thus allows the definition of the integral), the fundamental theorem of algebra, and the spectral theorem for symmetric matrices. In the function space setting of functional analysis, the analogous statement is the Arzela-Ascoli theorem, which says that if X is compact, a subset of C ( X ) is compact if and only if it is closed, bounded, and equicontinuous . Equicontinuity Let X be a topological space and ( Z, ρ ) a metric space. Then f : X → Z is continuous if for each x ∈ X and for all ² > , there is some neighborhood U of 365 366 Chapter 5. The Banach space C ( X ) x such that f ( U ) ⊂ B ( f ( x ) , ² ) . (5.1.1) Let S ⊂ C ( X, Z ), where as usual, C ( X, Z ) denotes the collection of all contin- uous functions X → Z . Then (5.1.1) holds for all f ∈ S , but in general, U depends on f . The case where we can choose a single....
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FA.Chap5.1-4 - Chapter 5 The Banach space C X In this...

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