FIVE MAJOR RESULTS IN ANALYSIS AND TOPOLOGY Aaron ...

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

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

5. THE BAIRE CATEGORY THEOREM 32 Corollary. No complete metric space is of the first category. Proof. Let {An} be a sequence of nowhere-dense subsets of a complete metric space X. Then X − An is a countable collection of dense open sets of X. By the preceding theorem, let x ∈ n (X − An). Then x ∈ X − n An, so that X = n An. The implications of the Baire category theorem are found wherever complete metric spaces are studied. Considering that the theorem applies to R, C and arbitrary Banach spaces, this includes most of analysis. One familiar result follows immediately from the corollary above. Although its proof is not as flashy as the one provided by G. Cantor it is, however, significantly shorter. Theorem. R is uncountable. Proof. Let Ax = {c} for each x ∈ R. Each Ax is nowhere dense, since if y ∈ R − Ax, then B|x−y|(y) is a neighborhood of y which is disjoint from Ax. If R were countable, then R would be the countable union of nowhere-dense sets, so that R is not a complete metric space, which is absurd. The following two theorems were mentioned at the beginning of the section on the Hahn- Banach theorem as essential results in functional analysis. They follow immediately from the Baire category theorem. The first, the Banach-Steinhaus theorem, implies that either the set M of all bounded linear functions from a Banach space X into a normed linear space Y is itself bounded in the space of all linear functions from X into Y , or that the set of all functionals fx from X into R given by fx(y) = f(x), where f ranges through M, is unbounded in M for all x in some dense subset of X. In other words, either the set of all bounded linear functions from X to Y is nice everywhere, or it is nasty almost everywhere. Theorem (Banach-Steinhaus Theorem 2 ). Suppose X is a Banach space, Y is a normed linear space, and {∆α} is a collection of bounded linear functions from X into Y , where α ranges over some index set A. Then either there exists an M < ∞ such that ∆α ≤ M for every α ∈ A, or sup α ∆αx = ∞ for all x belonging to some dense Gδ subset of X. 2 Taken from [10].
5. THE BAIRE CATEGORY THEOREM 33 The second result is called the Open Mapping theorem. It uses linearity to show that it is not only the case that surjective bounded linear transformations are continuous, but that they are open maps. Definition. Let X be a normed linear space. Then the unit ball in X, B1(0), is the set of all elements of X whose norm is less than unity. Theorem (The Open Mapping Theorem 3 ). Let U and V be the open unit balls of the Banach spaces X and Y , respectively. To every bounded linear transformation ∆ of X onto Y , there corresponds a δ > 0 so that δV ⊂ ∆(U). That is, every surjective bounded linear transformation of one Banach space onto another is an open map. 3 Taken from [10].
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 20 and 21: that 1 −1 cn(1 − x 2 ) n dx =
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Page 28 and 29: 4. THE HAHN-BANACH THEOREM 24 This
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Page 32 and 33: 4. THE HAHN-BANACH THEOREM 28 We co
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Page 38 and 39: CHAPTER 6 The Stone-˘Cech Compacti
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