FIVE MAJOR RESULTS IN ANALYSIS AND TOPOLOGY Aaron ...

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that 1 −1 cn(1 − x 2 ) n dx = 1. Since 3. THE STONE-WEIERSTRASS THEOREM 16 1 −1 (1 − x2 ) n dx = 2 1 0 (1 − x2 ) n dx ≥ 2 1/ √ n (1 − x 0 2 ) ndx ∗ ≥ 2 1/ √ n (1 − nx 0 2 )dx = 4 3 √ n > 1 √ n , then cn < √ n. To justify step ∗, note that (1 − x 2 ) n − (1 − nx 2 ) = 0 if x = 0, and d 2 n 2 2 n−1 (1 − x ) − (1 − nx ) = n(1 − x ) (−2x) + 2nx dx = 2nx (1 − (1 − x2 ) n−1 ) > 0 (x ∈ (0, 1)). Hence, for any 0 < δ ≤ 1, Qn(x) ≤ √ n(1 − δ 2 ) n (δ ≤ |x| ≤ 1), so that Qn → 0 uniformly for all x such that δ ≤ |x| ≤ 1. Now set Pn(x) = 1 −1 f(x + t)Qn(t)dt (0 ≤ x ≤ 1). Because [0, 1] ⊂ [x − 1,x + 1] for x ∈ [0, 1], our assumptions about f guarantee that Pn(x) = x+1 x−1 f(t)Qn(t − x)dt = 0 x−1 f(t)Qn(t − x)dt + 1 0 f(t)Qn(t − x)dt + x+1 1 f(t)Qn(t − x)dt = 0 + 1 0 f(t)Qn(t − x)dt + 0, which is clearly a polynomial in x. Thus, {Pn} is a sequence of polynomials (which are real if f is real). Since f is a continuous on a compact set, for ǫ > 0 choose δ > 0 such that |y − x| < δ guarantees that |f(y) − f(x)| < ǫ . Let M = sup |f(x)|. Since Qn(x) ≥ 0, then 2 x for 0 ≤ x ≤ 1, 1 |Pn(x) − f(x)| = [f(x + t) − f(x)] Qn(t)dt −1 1 ≤ |f(x + t) − f(x)|Qn(t)dt −1 −δ ≤ 2M Qn(t)dt + ǫ δ Qn(t)dt + 2M 2 −1 ≤ 4M √ n(1 − δ 2 ) n + ǫ 2 < ǫ −δ 1 δ Qn(t)dt for sufficiently large n. This completes the proof.
3. THE STONE-WEIERSTRASS THEOREM 17 The Weierstrass approximation theorem allows us to choose a sequence of polynomials which converge uniformly to a given continuous function f. Since the pointwise limit of a uniformly convergent sequence of functions has many of the same properties as the terms in the sequence, this theorem tells us a great deal about C([a,b], C). There is a significant generalization of Weierstrass’ approximation theorem due to Mar- shall Stone, which relaxes two of the conditions of Weierstrass’ theorem. First, set of functions is not required to be polynomials, but rather a type of subclass A of C(K, C), called an algebra, which possesses a few important properties. Also, K is not required to be the closed interval [a,b], but rather is allowed to be any compact topological space. This result is known as the Stone-Weierstrass theorem, and its proof is accomplished in two main steps. First, we prove a special case of the theorem for the space C(K, R) which relies on the classical Weierstrass approximation theorem. We then use this result to prove the theorem in its full strength. Before we begin, a few preliminary results are in order. Definition. Let K be compact, A be a scalar field, and let A be a linear subspace over A of C(K,A) such that for f,g ∈ A, f · g ∈ A, where (f · g)(x) = f(x) · g(x). Then A is called an algebra of continuous functions. A is said to be real or complex if A is R or C. An algebra of continuous functions is merely a ring of continuous functions, inheriting addition and multiplication from A, and which is closed under scalar multiplication. Definition. Let A be a subset of the metric space (C(K,A),ρ∞), where A is R or C. Then the closure of A in C(K,A) is called the uniform closure of A, and A is said to be uniformly closed in C(K,A). The uniform closure of an algebra A can be thought of as the set of all functions in C(K,A) which are the limit of a fundamental sequence in A. Theorem. Let A be an algebra of bounded functions. Then the uniform closure B of A is a uniformly closed algebra. Proof. For f ∈ B, g ∈ B, let {fn}, {gn} be uniformly convergent sequences in A such that fn → f, gn → g. Since the fi and gi are bounded, fn + gn → f + g, fngn → fg,
Page 1: FIVE MAJOR RESULTS IN ANALYSIS AND
Page 4 and 5: Preface During the analysis of scie
Page 6 and 7: N = {1, 2, 3,...} R = {x| x is a re
Page 8 and 9: (1) x + y ∈ G, (2) (x + y) + z =
Page 10 and 11: TOPOLOGY 6 implies that f is contin
Page 12 and 13: A ⊂ β∈J ∗ TOPOLOGY 8 Uβ, t
Page 14 and 15: TOPOLOGY 10 ensures that f(x0) ∈
Page 16 and 17: 2. THE ASCOLI-ARZELÀ THEOREM 12 Th
Page 18 and 19: 2. THE ASCOLI-ARZELÀ THEOREM 14 ne
Page 22 and 23: 3. THE STONE-WEIERSTRASS THEOREM 18
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Page 26 and 27: 1 3. THE STONE-WEIERSTRASS THEOREM
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Page 32 and 33: 4. THE HAHN-BANACH THEOREM 28 We co
Page 34 and 35: CHAPTER 5 The Baire Category Theore
Page 36 and 37: 5. THE BAIRE CATEGORY THEOREM 32 Co
Page 38 and 39: CHAPTER 6 The Stone-˘Cech Compacti
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