18.100C Lecture 18 Summary Deﬁnitions of B(X) (the space of bounded functions), C(X) (the space of continuous functions) as metric spaces. Also, for X = [a, b], deﬁnition of B 1 (X) (the space of functions with bounded derivative) as a metric space. Theorem 18.1. B(X) is a complete metric space. Theorem 18.2. C(X) is a closed subspace of B(X), hence itself complete. Theorem 18.3. Take a bounded subset of B1 (X), consider it as a subset of C(X), and take its closure with respect to the metric of C(X). Then that closure is a compact subset of C(X). Uniform approximation by step functions and by piecewise linear functions. Theorem 18.4. Let X be a compact metric space. Suppose that A ⊂ C(X) is a subset with the following properties: (i) if f, g ∈ A, then max(f, g) ∈ A and min(f, g) ∈ A; (ii) for any two points x = y and any real numbers a, b, there is an f ∈ A such that f (x) = a, f (y) = b. Then A is dense in C(X). Theorem 18.5 (Stone-Weierstrass). Let X be a compact metric space. Suppose that A ⊂ C(X) is a subset with the following properties: (i) all constant func tions are in A; (ii) if f, g ∈ A, then f + g ∈ A; (iii) if f, g ∈ A, then f · g ∈ A; (iv) for any two points x = y, there is an f ∈ A such that f (x) = f (y). Then A is dense in C(X). Application: polynomials, trigonometric polynomials. 1

18.100C Real Analysis: Lecture 18 Summary

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