Real and Complex Analysis (Rudin)

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EXAMPLES OF BANACH SPACE TECHNIQUES 111 We now fix z e U and write z = re i9, ° ~ r < 1, lJ real. If un(w) = w n , then un e A for n = 0, 1, 2, ... ; hence (2) shows that (n = 0, 1, 2, ... ). (3) Since U-n = Un on T, (3) leads to This suggests that we look at the real function since (n = 0, ± 1, ±2, ... ). (4)
112 REAL AND COMPLEX ANALYSIS 5.25 Theorem Suppose A is a vector space of continuous complex functions on the closed unit disc O.lf A contains all polynomials, and if sup I f(z)1 = sup I f(z) I (1) zeU ze T for every f E A (where T is the unit circle, the boundary of U), then the Poisson integral representation 1 I" 1 - r2 . f(z) = -2 1 2 (() ) 2 f(e'~ dt 1t _" - r COS - t + r is valid for every f E A and every z E U. (2) Exercises I Let X consist of two points a and b, put JI( { a}) = JI( {b}) =!, and let lJ'(JI) be the resulting real lJ'-space. Identify each real functionf on X with the point (f(a),f(b» in the plane, and sketch the unit balls of lJ'(JI), for ° and only if 1 ~ p ~ 00. For which p is this unit ball a square? A circle? If JI({a}) # JI(b), how does the situation differ from the preceding one? 2 Prove that the unit ball (open or closed) is convex in every normed linear space. 3 If 1 < p < 00, prove that the unit ball of lJ'(Jl) is strictly convex; this means that if f#o, h =!(f + 0), then IIhll, < 1. (Geometrically, the surface of the ball contains no straight lines.) Show that this fails in every IJ(Jl), in every L""(JI), and in every C(X). (Ignore trivialities, such as spaces consisting of only one point.) 4 Let C be the space of all continuous functions on [0, 1], with the supremum norm. Let M consist of allf E C for which r l/2 f(t) dt _ r l f(t) dt = 1. Jo JI/2 Prove that M is a closed convex subset of C which contains no element of minimal norm. S Let M be the set of all f E IJ([O, 1]), relative to Lebesgue measure, such that ff(t) dt = 1. Show that M is a closed convex subset of IJ([O, 1]) which contains infinitely many elements of minimal norm. (Compare this and Exercise 4 with Theorem 4.10.) 6 Let f be a bounded linear functional on a subspace M of a Hilbert space H. Prove that f has a unique norm-preserving extension to a bounded linear functional on H, and that this extension vanishes on M.l. 7 Construct a bounded linear functional on some subspace of some IJ(JI) which has two (hence infinitely many) distinct norm-preserving linear extensions to IJ(Jl). 8 Let X be a normed linear space, and let X" be its dual space, as defined in Sec. 5.21, with the norm II!II = sup {If(x) I: IIxll ~ 1}. (a) Prove that X* is a Banach space. (b) Prove that the mappingf-+f(x) is, for each x E X, a bounded linear functional on X*, of norm IIxli. (This gives a natural imbedding of X in its "second dual" X**, the dual space of X*.)
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REAL AND COMPLEX ANALYSIS
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REAL AND COMPLEX ANALYSIS INTERNATI
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CONTENTS Preface xiii Prologue: The
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CONTENTS ix Chapter 10 Elementary P
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CONTENTS xi Appendix: Hausdorff's M
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xiv PREFACE Experience with the fir
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BIBLIOGRAPHY 1. L. V. Ahlfors: "Com
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LIST OF SPECIAL SYMBOLS AND ABBREVI
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INDEX Absolute continuity, 120 of f
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INDEX 411 Double integral, 165 Dual
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INDEX 413 Lower limit, 14 Lower sem
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INDEX 415 Separable space, 92, 247
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Real and Complex Analysis (Rudin)

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