FIVE MAJOR RESULTS IN ANALYSIS AND TOPOLOGY Aaron ...

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

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4. THE HAHN-BANACH THEOREM 28 We conclude with a few interesting results which follow from the Hahn-Banach theorem. The first two, the theorems of Runge and Müntz-Szasz, are generalizations of the Weier- strass approximation theorem. The first, by Runge, involves the approximation of analytic functions in compact subsets of the complex plane by rational functions whose singularities are contained in some open subset of the plane. Theorem (Runge 3 ). Let G be an open set in the plane, S 2 be the completed plane, A be a set which has one point in each maximal connected subset of S 2 − G, and assume that f is a analytic function on G. Then there exists a sequence {Rn} of rational functions, with singularities only in A, such that Rn converges uniformly to f on compact subsets of G. If G is taken to be C, then Runge’s theorem states that, for any entire (analytic every- where) function f, one can find a sequence of complex polynomials Pn that converges to f on any compact subsets of C, a clear and useful extension of Weierstrass’ theorem. A novel generalization of the approximation theorem by Herman Müntz and Otto Szasz involves the type of polynomials which can be used in the approximation of real-valued continuous functions. Theorem (Müntz-Szasz 4 ). Suppose 0 < λ1 < λ2 < λ3 < ... and let X be the closure in C ([0, 1], R) of the set of all finite linear combinations of the functions 1,t λ1 ,t λ2 ,.... Then X = C ([0, 1], R) if, and only if, ∞ i=1 1 λi = ∞. More specifically, if we want to use linear combinations of the terms t λ1 ,t λ2 ,... to approximate continuous real functions on the closed interval [0, 1], how must the λi be distributed in R+? When the λi are taken to be the positive integers, we get the Weierstrass approximation theorem (for real-valued functions). The next theorem describes a sufficient condition for a space to be isomorphic to, or have the same structure as, C 3 Taken from [10] 4 Taken from [10].
4. THE HAHN-BANACH THEOREM 29 Definition. A Banach Algebra X over a field A is a Banach space with an additional associative operation, multiplication, such that for any x,y ∈ X, x · y ≤ x · y. Theorem (Gelfand-Mazur 5 ). Every complex Banach algebra with unity in which each nonzero element has an inverse is isomorphic to C This is particularly interesting, since it shows that C is, in some sense, the only complete normed complex linear space that is also an algebraic field. 5 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 =
Page 22 and 23: 3. THE STONE-WEIERSTRASS THEOREM 18
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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 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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