FIVE MAJOR RESULTS IN ANALYSIS AND TOPOLOGY Aaron ...

More documents

Recommendations

Info

$Lecture notes for Math 522 Spring 2012 (Rudin chapter 7)$

$Math 320 - Fall 2010 HW #7 Selected Solutions Problem 5.1.32 Let ...$

2. THE ASCOLI-ARZELÀ THEOREM 12 Theorem (Ascoli-Arzelà). 2 Let (X,ρX), (Y,ρY ) be compact metric spaces, and let D ⊂ C(X,Y ). Then a necessary and sufficient condition that D be compact in C(X,Y ) is that D be equicontinuous. Proof. (Sufficiency) Let Y X = {f| f : X → Y }, and give Y X the uniform metric ρ∞. Since convergence with respect to ρ∞ is equivalent to uniform convergence, and the point- wise limit of a uniformly convergent sequence of continuous functions is again continuous, C(X,Y ) is closed in Y X . Hence it is sufficient to show that D is compact in Y X , since the intersection of a compact subspace and a closed subspace is again compact. Given ǫ > 0, choose δ > 0 so that for every function f ∈ D, ρX(x1,x2) < δ implies that -dense in X. Let ρY (f(x1),f(x2)) < ǫ. Since X is compact, let {xi} n i=1 be δ 2 and for 1 < i ≤ n, set H1 = B δ(x1), 2 Hj. i−1 Hi = B δ(xi) − 2 j=1 The Hi form a pairwise disjoint cover of X, such that if x1,x2 ∈ Hi then ρX(x1,x2) < δ 2 . Similarly, the compactness of Y allows us to pick a finite ǫ m -dense set {yj} 2 j=1 . Let G be the set of all functions from X to Y which assume, on each Hi, the value of exactly one yj. It is clear that |G| = m n < ∞. It remains to be shown that G is ǫ-dense in Y X with respect to D. Let f ∈ D. For each 1 ≤ i ≤ n, choose yj so that ρY (f(xi),yj) < ǫ/2, and choose g ∈ G so that g(xi) = yj. Given x ∈ X, choose Hi so that x ∈ Hi. Then ρY (f(x),g(x)) ≤ ρY (f(x),f(xi)) + ρY (f(xi),g(xi)) + ρY (g(xi),g(x)) < ǫ ǫ + + 0 = ǫ. 2 2 (Necessity) Let ǫ > 0. Since D is compact, we can choose a finite ǫ-dense subset G of 3 D. The Heine-Cantor theorem allows us to select, for each g ∈ G, a constant δg > 0 so that if ρX(x1,x2) < δg, then ρY (g(x1),g(x2)) < ǫ 3 . Let δ = ming {δg}. 2 The sufficiency portion of this proof is adapted from [6]
2. THE ASCOLI-ARZELÀ THEOREM 13 Hence, if f ∈ D, then a g ∈ G can be chosen so that ρ∞(f,g) < ǫ. It follows that, for 3 x1,x2 which satisfy ρX(x1,x2) < δ, ρY (f(x1),f(x2)) ≤ ρY (f(x1),g(x1)) + ρY (g(x1),g(x2)) + ρY (g(x2),f(x2)) < ǫ. Its appearance in the basic results of analysis makes the Ascoli-Arzelà theorem particu- larly interesting and powerful. For this reason, the applications of the Ascoli-Arzelà theorem are many and varied, extending to a number of diverse branches of mathematics. We now present several important results which follow from it. These examples, taken from geom- etry, complex analysis, and the theory of differential equations, hint at the scope of the Ascoli-Arzelà theorem. A curve in a metric space X from a point a to a point b is a continuous function f : [0, 1] → X such that f(0) = a and f(1) = b. The length of f is defined by n sup 0=x0
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 18 and 19: 2. THE ASCOLI-ARZELÀ THEOREM 14 ne
Page 20 and 21: that 1 −1 cn(1 − x 2 ) n dx =
Page 22 and 23: 3. THE STONE-WEIERSTRASS THEOREM 18
Page 24 and 25: 3. THE STONE-WEIERSTRASS THEOREM 20
Page 26 and 27: 1 3. THE STONE-WEIERSTRASS THEOREM
Page 28 and 29: 4. THE HAHN-BANACH THEOREM 24 This
Page 30 and 31: 4. THE HAHN-BANACH THEOREM 26 The p
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
Page 40 and 41: 6. THE STONE- ˘ CECH COMPACTIFICAT
Page 42 and 43: 6. THE STONE- ˘ CECH COMPACTIFICAT

FIVE MAJOR RESULTS IN ANALYSIS AND TOPOLOGY Aaron ...

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?