Real and Complex Analysis (Rudin)

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CONTENTS xi Appendix: Hausdorff's Maximality Theorem 395 Notes <strong>and</strong> Comments 397 Bibliography 405 List of Special Symbols 407 Index 409
Page 2: REAL AND COMPLEX ANALYSIS
Page 5 and 6: REAL AND COMPLEX ANALYSIS INTERNATI
Page 8 and 9: CONTENTS Preface xiii Prologue: The
Page 10 and 11: CONTENTS ix Chapter 10 Elementary P
Page 14 and 15: PREFACE This book contains a first-
Page 16 and 17: PROLOGUE THE EXPONENTIAL FUNCTION T
Page 18 and 19: PROLOGUE: THE EXPONENTIAL FUNCTION
Page 20 and 21: CHAPTER ONE ABSTRACT INTEGRATION To
Page 22 and 23: ABSTRACT INTEGRA nON 7 If no two me
Page 24 and 25: ABSTRACT INTEGRA nON 9 It would per
Page 26 and 27: ABSTRACT INTEGRATION 11 1.8 Theorem
Page 28 and 29: ABSTRACT INTEGRATION 13 1.12 Theore
Page 30 and 31: ABSTRACT INTEGRATION 15 Corollaries
Page 32 and 33: ABSTRACT INTEGRA nON 17 As the proo
Page 34 and 35: ABSTRACT INTEGRATION 19 The cancell
Page 36 and 37: ABSTRACT INTEGRATION 21 Next, let s
Page 38 and 39: Corollary If aij :2= ° for i and j
Page 40 and 41: ABSTRACT INTEGRATION 25 each of the
Page 42 and 43: ABSTRACT INTEGRA nON 27 PROOF Since
Page 44 and 45: ABSTRACT INTEGRA TlON 29 1.37 The f
Page 46 and 47: ABSTRACT INTEGRATION 31 PROOF Let a
Page 48 and 49: CHAPTER TWO POSITIVE BOREL MEASURES
Page 50 and 51: POSITIVE BOREL MEASURES 35 [The con
Page 52 and 53: POSITIVE BOREL MEASURES 37 2.6 Theo
Page 54 and 55: POSITIVE BOREL MEASURES 39 will mea
Page 56 and 57: POSITIVE BOREL MEASURES 41 (b) I-'(
Page 58 and 59: POSITIVE BOREL MEASURES 43 Put V =
Page 60 and 61: POSITIVE BOREL MEASURES 45 PROOF If
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POSITIVE BOREL MEASURES 47 Since hi
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POSITIVE BOREL MEASURES 49 PROOF Pu
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POSITIVE BOREL MEASURES 51 (c) m is
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POSITIVE BOREL MEASURES 53 2.21 Rem
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POSITIVE BOREL MEASURES 55 If T is
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POSITIVE BOREL MEASURES 57 Since th
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POSITIVE BOREL MEASURES S9 13 Is it
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CHAPTER THREE LP-SPACES Convex Func
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If-SPACES 63 (4) becomes exp U (Xl
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I!'-SPACES 65 Hence the left side o
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H-SPACES 67 Suppose j, g, and h are
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I!-SPACES 69 The simple functions p
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l!'-SPACES 71 Given / E Co(X) and
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If-SPACES 73 (b) Prove that equalit
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I!'-SPACES 75 (b) If 1 :5 P < 00 an
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ELEMENTARY HILBERT SPACE THEORY 77
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ELEMENTARY IDLBERT SPACE THEORY 8S
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CHAPTER FIVE EXAMPLES OF BANACH SPA
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EXAMPLES OF BANACH SPACE lECHNIQUES
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EXAMPLES OF BANACH SPACE TECHNIQUES
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EXAMPLFS OF BANACH SPACE TECHNIQUES
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COMPLEX MEASURES 117 This notation
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COMPLEX MEASURES 119 We have thus s
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COMPLEX MEASURES 121 (f) Since A2 .
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COMPLEX MEASURES 123 Hence g(x) E [
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COMPLEX MEASURES 125 By analogy wit
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COMPLEX MEASURES 127 then
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COMPLEX MEASURES 129 The first part
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COMPLEX MEASURES 131 Once we have t
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COMPLEX MEASURES 133 4 Suppose 1 :5
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CHAPTER SEVEN DIFFERENTIATION In el
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DIFFERENTIATION 137 7.3 Lemma If W
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DIFFERENTIATION 139 PROOF Define fo
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DIFFERENTIA nON 141 PROOF Let x be
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DIFFERENTIATION 143 7.14 Theorem Su
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DIFFERENTIATION 145 2- n - 1
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DIFFERENTIATION 147 Assume next tha
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DIFFERENTIATION 149 The next theore
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DIFFERENTIATION 151 stronger hypoth
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DIFFERENTIATION, 153 Since r" = m(B
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DIFFERENTIATION 155 Theorem 7.8 tel
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DIFFERENTIATION 157 7 Construct a c
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DIFFERENTIATION 159 21 Iffis a real
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INTEGRATION ON PRODUCT SPACES 161 I
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INTEGRATION ON PRODUCT SPACES 163 P
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INTEGRATION ON PRODUCT SPACES 165 (
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INTEGRATION ON PRODUCT SPACES 167 T
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INTEGRATION ON PRODUCT SPACES 169 T
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INTEGRATION ON PRODUCT SPACES 171 w
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INTEGRATION ON PRODUCT SPACES 173 F
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INTEGRATION ON PRODUCT SPACES 175 S
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INTEGRATION ON PRODUCT SPACES 177 S
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FOURIER TRANSFORMS 179 The formal p
Page 196 and 197:
FOURIER TRANSFORMS 181 Let us see w
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FOURIER TRANSFORMS 183 The integran
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FOURIER lRANSFORMS 185 and Theorem
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FOURIER lRANSFORMS 187 Theorem 9.2(
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FOURIER TRANSFORMS 189 orthogonal p
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FOURIER TRANSFORMS 191 follows: We
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FOURIER TRANSFORMS 193 Then f'! (d
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FOURIER TRANSFORMS 195 What does (*
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ELEMENTARY PROPERTIES OF HOLOMORPHI
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Page 220 and 221:
ELEMENTARY PROPERTIES OF HOLOMORPlD
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ELEMENTARY PROPERTIES OF HOLOMORPlD
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CHAPTER ELEVEN HARMONIC FUNCTIONS T
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HARMONIC FUNCTIONS 233 The Poisson
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HARMONIC FUNCTIONS 235 Note: This t
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HARMONIC FUNCTIONS 237 The Mean Val
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HARMONIC FUNCTIONS 239 Hence
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HARMONIC FUNCTIONS 241 The regions
Page 258 and 259:
HARMONIC FUNCTIONS 243 Hence, setti
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HARMONIC FUNCTIONS 245 (a) If P. is
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HARMONIC FUNCTIONS 247 PROOF To say
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HARMONIC FUNCTIONS 249 As before, L
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HARMONIC FUNCTIONS 251 If {u.} is a
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CHAPTER TWELVE THE MAXIMUM MODULUS
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THE MAXIMUM MODULUS PRINCIPLE 255 T
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THE MAXIMUM MODULUS PRINCIPLE 257 F
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THE MAXIMUM MODULUS PRINCIPLE 259 F
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THE MAXIMUM MODULUS PRINCIPLE 261 1
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THE MAXIMUM MODULUS PRINCIPLE 263 N
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THE MAXIMUM MODULUS PRINCIPLE 265 S
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APPROXIMATIONS BY RATIONAL FUNCTION
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APPROXIMA nONS BY RA nONAL FUNCTION
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APPROXIMA TIO~S BY RATIONAL FUNCTIO
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CONFORMAL MAPPING 279 Here /'(zo) =
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CONFORMAL MAPPING 281 Let us discus
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CONFORMAL MAPPING 283 Under these c
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CONFORMAL MAPPING 285 14.9 Remarks
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CONFORMAL MAPPING 287 Divide Eqs. (
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CONFORMAL MAPPING 289 PROOF Iff= I/
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CONFORMAL MAPPING 291 Theorem 14.18
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CONFORMAL MAPPING 293 Put y(t) = .J
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CONFORMAL MAPPING 295 18 Suppose 0
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CONFORMAL MAPPING 297 (d) Let ex be
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ZEROS OF HOLOMORPHIC FUNCTIONS 299
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ZEROS OF HOLOMORPlDC FUNCTIONS 305
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ZEROS OF HOWMORPHIC FUNCTIONS 307 J
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ZEROS OF HOLOMORPIDC FUNCTIONS 309
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ZEROS OF HOWMORPHIC FUNCTIONS 311 i
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CHAPTER SIXTEEN ANALYTIC CONTINUATI
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ANALYTIC CONTINUATION 321 integers.
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ANALYTIC CONTINUATION 323 the radiu
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ANALYTIC CONTINUATION 325 There are
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ANALYTIC CONTINUATION 327 16.15 Tbe
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ANALYTIC CONTINUATION 329 We claim
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ANALYTIC CONTINUATION 331 PROOF Let
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ANALYTIC CONTINUATION 333 P(f1' 01)
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CHAPTER SEVENTEEN This chapter is d
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HP-SPACES 337 some r > 0 we have D(
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HP-SPACES 339 PROOF Note first that
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For each z E U, 11 - e-ilz I and P(
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log I 9 I are 0 a.e. (Theorem 11.22
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Since Ilog+ u - log+ V I ~ I u - v
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X n , the one-dimensional spaces sp
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So there exists a qJ E Y such that
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HP-SPACES 351 Let us recall that ev
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5 Suppose fe HP, qJ e H(U), and qJ(
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ffI' -SPACES 355 exists (and is fin
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ELEMENTARY TIIEORY OF BANACH ALGEBR
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ELEMENTARY THEORY OF BANACH ALGEBRA
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CHAPTER NINETEEN HOLOMORPHIC FOURIE
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HOLOMORPHIC FOURIER TRANSFORMS 373
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UNIFORM APPROXIMATION BY POLYNOMIAL
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UNIFORM APPROXIMATION BY pOLYNOMIAL
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APPENDIX HAUSDORFF'S MAXIMALITY THE
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NOTES AND COMMENTS Chapter 1 The fi
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NOTES AND COMMENTS 399 Chapter 6 Se
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NOTES AND COMMENTS 401 orthonormal
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NOTES AND COMMENTS 403 CbapterlO Se
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406 BIBLIOGRAPHY 24. F. Riesz and B
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408 LIST OF SPECIAL SYMBOLS AND ABB
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410 INDEX Cartesian product, 7,160
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412 INDEX Halmos, P. R., 398,403 Ha
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414 INDEX Outer regular set, 47 Ove
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416 INDEX Uniform continuity, 51 Un
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Real and Complex Analysis (Rudin)

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