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Real and complex analysis

Real and complex analysis

Real and complex

  • Page 2: REAL AND COMPLEX ANALYSIS
  • Page 5 and 6: REAL AND COMPLEX ANALYSISINTERNATIO
  • Page 8 and 9: CONTENTSPrefacexiiiPrologue: The Ex
  • Page 10: CONTENTS ixChapter 10Elementary Pro
  • Page 16 and 17: PROLOGUETHE EXPONENTIAL FUNCTIONThi
  • Page 18 and 19: PROLOGUE: THE EXPONENTIAL FUNCTION
  • Page 20 and 21: CHAPTERONEABSTRACT INTEGRATIONTowar
  • Page 22 and 23: ABSTRACT INTEGRA nON 7If no two mem
  • Page 24 and 25: ABSTRACT INTEGRA nON 9It would perh
  • Page 26 and 27: ABSTRACT INTEGRATION 111.8 Theorem
  • Page 28 and 29: ABSTRACT INTEGRATION 131.12 Theorem
  • Page 30 and 31: ABSTRACT INTEGRATION 15Corollaries(
  • Page 32 and 33: ABSTRACT INTEGRA nON 17As the proof
  • Page 34 and 35: ABSTRACT INTEGRATION 19The cancella
  • Page 36 and 37: ABSTRACT INTEGRATION 21Next, let s
  • Page 38 and 39: Corollary If aij :2= ° for i and j
  • Page 40 and 41: ABSTRACT INTEGRATION 25each of thes
  • Page 42 and 43: ABSTRACT INTEGRA nON 27PROOF Since
  • Page 44 and 45: ABSTRACT INTEGRA TlON 291.37 The fa
  • Page 46 and 47: ABSTRACT INTEGRATION 31PROOF Let a
  • Page 48 and 49: CHAPTERTWOPOSITIVE BOREL MEASURESVe
  • Page 50 and 51: POSITIVE BOREL MEASURES 35[The conv
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    POSITIVE BOREL MEASURES 372.6 Theor

  • Page 54 and 55:

    POSITIVE BOREL MEASURES 39will mean

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    POSITIVE BOREL MEASURES 41(b) I-'(K

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    POSITIVE BOREL MEASURES 43Put V = U

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    POSITIVE BOREL MEASURES 45PROOF If

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    POSITIVE BOREL MEASURES 47Since hi

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    POSITIVE BOREL MEASURES 49PROOF Put

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    POSITIVE BOREL MEASURES 51(c) m is

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    POSITIVE BOREL MEASURES 532.21 Rema

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    POSITIVE BOREL MEASURES 55If T is o

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    POSITIVE BOREL MEASURES 57Sincethe

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    POSITIVE BOREL MEASURES S913 Is it

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    CHAPTERTHREELP-SPACESConvex Functio

  • Page 78 and 79:

    If-SPACES 63(4) becomesexp U (Xl +

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    I!'-SPACES 65Hence the left side of

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    H-SPACES 67Suppose j, g, and h are

  • Page 84 and 85:

    I!-SPACES 69The simple functions pl

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    l!'-SPACES 71Given / E Co(X) and

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    If-SPACES 73(b) Prove that equality

  • Page 90 and 91:

    I!'-SPACES 75(b) If 1 :5 P < 00 and

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    ELEMENTARY HILBERT SPACE THEORY 774

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    ELEMENTARY HILBERT SPACE THEORY 79I

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    ELEMENTARY HILBERT SPACE THEORY 81P

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    ELEMENTARY HILBERT SPACE THEORY 834

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    ELEMENTARY IDLBERT SPACE THEORY 8S4

  • Page 102 and 103:

    ELEMENTARY HILBERT SPACE THEORY 87(

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    ELEMENTARY HILBERT SPACE THEORY 89W

  • Page 106 and 107:

    ELEMENTARY HILBERT SPACE THEORY 91T

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    ELEMENTARY HILBERT SPACE THEORY 93M

  • Page 110 and 111:

    CHAPTERFIVEEXAMPLES OF BANACH SPACE

  • Page 112 and 113:

    EXAMPLES OF BANACH SPACE lECHNIQUES

  • Page 114 and 115:

    EXAMPLES OF BANACH SPACE TECHNIQUES

  • Page 116 and 117:

    EXAMPLES OF BANACH SPACE TECHNIQUES

  • Page 118 and 119:

    EXAMPLES OF BANACH SPACE TECHNIQUES

  • Page 120 and 121:

    EXAMPLES OF BANACH SPACE TECHNIQUES

  • Page 122 and 123:

    EXAMPLES OF BANACH SPACE TECHNIQUES

  • Page 124 and 125:

    EXAMPLES OF BANACH SPACE TECHNIQUES

  • Page 126 and 127:

    EXAMPLES OF BANACH SPACE TECHNIQUES

  • Page 128 and 129:

    EXAMPLFS OF BANACH SPACE TECHNIQUES

  • Page 130 and 131:

    EXAMPLES OF BANACH SPACE TECHNIQUES

  • Page 132 and 133:

    COMPLEX MEASURES 117This notation i

  • Page 134 and 135:

    COMPLEX MEASURES 119We have thus sp

  • Page 136 and 137:

    COMPLEX MEASURES 121(f) Since A2 .1

  • Page 138 and 139:

    COMPLEX MEASURES 123Hence g(x) E [0

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    COMPLEX MEASURES 125By analogy with

  • Page 142 and 143:

    COMPLEX MEASURES 127then

  • Page 144 and 145:

    COMPLEX MEASURES 129The first part

  • Page 146 and 147:

    COMPLEX MEASURES 131Once we have th

  • Page 148 and 149:

    COMPLEX MEASURES 1334 Suppose 1 :5

  • Page 150 and 151:

    CHAPTERSEVENDIFFERENTIATIONIn eleme

  • Page 152 and 153:

    DIFFERENTIATION 1377.3 Lemma If W i

  • Page 154 and 155:

    DIFFERENTIATION 139PROOF Definefor

  • Page 156 and 157:

    DIFFERENTIA nON 141PROOF Let x be a

  • Page 158 and 159:

    DIFFERENTIATION 1437.14 Theorem Sup

  • Page 160 and 161:

    DIFFERENTIATION 1452- n - 1

  • Page 162 and 163:

    DIFFERENTIATION 147Assume next that

  • Page 164 and 165:

    DIFFERENTIATION 149The next theorem

  • Page 166 and 167:

    DIFFERENTIATION 151stronger hypothe

  • Page 168 and 169:

    DIFFERENTIATION, 153Since r" = m(B(

  • Page 170 and 171:

    DIFFERENTIATION 155Theorem 7.8 tell

  • Page 172 and 173:

    DIFFERENTIATION 1577 Construct a co

  • Page 174 and 175:

    DIFFERENTIATION 15921 Iffis a real

  • Page 176 and 177:

    INTEGRATION ON PRODUCT SPACES 161If

  • Page 178 and 179:

    INTEGRATION ON PRODUCT SPACES 163Pr

  • Page 180 and 181:

    INTEGRATION ON PRODUCT SPACES 165(c

  • Page 182 and 183:

    INTEGRATION ON PRODUCT SPACES 167To

  • Page 184 and 185:

    INTEGRATION ON PRODUCT SPACES 169Th

  • Page 186 and 187:

    INTEGRATION ON PRODUCT SPACES 171wh

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    INTEGRATION ON PRODUCT SPACES 173Fo

  • Page 190 and 191:

    INTEGRATION ON PRODUCT SPACES 175S

  • Page 192 and 193:

    INTEGRATION ON PRODUCT SPACES 177Su

  • Page 194 and 195:

    FOURIER TRANSFORMS 179The formal pr

  • Page 196 and 197:

    FOURIER TRANSFORMS 181Let us see wh

  • Page 198 and 199:

    FOURIER TRANSFORMS 183The integrand

  • Page 200 and 201:

    FOURIER lRANSFORMS 185and Theorem 3

  • Page 202 and 203:

    FOURIER lRANSFORMS 187Theorem 9.2(d

  • Page 204 and 205:

    FOURIER TRANSFORMS 189orthogonal pr

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    FOURIER TRANSFORMS 191follows: We k

  • Page 208 and 209:

    FOURIER TRANSFORMS 193Thenf'! (d (X

  • Page 210 and 211:

    FOURIER TRANSFORMS 195What does (*)

  • Page 212 and 213:

    ELEMENTARY PROPERTIES OF HOLOMORPHI

  • Page 214 and 215:

    ELEMENTARY PROPERTIES OF HOLOMORPHI

  • Page 216 and 217:

    ELEMENTARY PROPERTIES OF HOLOMORPHI

  • Page 218 and 219:

    ELEMENTARY PROPERTIES OF HOLOMORPHI

  • Page 220 and 221:

    ELEMENTARY PROPERTIES OF HOLOMORPlD

  • Page 222 and 223:

    ELEMENTARY PROPERTIES OF HOLOMORPHI

  • Page 224 and 225:

    ELEMENTARY PROPERTIES OF HOLOMORPHI

  • Page 226 and 227:

    ELEMENTARY PROPERTIES OF HOLOMORPHI

  • Page 228 and 229:

    ELEMENTARY PROPERTIES OF HOLOMORPHI

  • Page 230 and 231:

    ELEMENTARY PROPERTIES OF HOLOMORPHI

  • Page 232 and 233:

    ELEMENTARY PROPERTIES OF HOLOMORPHI

  • Page 234 and 235:

    ELEMENTARY PROPERTIES OF HOLOMORPHI

  • Page 236 and 237:

    ELEMENTARY PROPERTIES OF HOLOMORPHI

  • Page 238 and 239:

    ELEMENTARY PROPERTIES OF HOLOMORPHI

  • Page 240 and 241:

    ELEMENTARY PROPERTIES OF HOLOMORPlD

  • Page 242 and 243:

    ELEMENTARY PROPERTIES OF HOLOMORPHI

  • Page 244 and 245:

    ELEMENTARY PROPERTIES OF HOLOMORPHI

  • Page 246 and 247:

    CHAPTERELEVENHARMONIC FUNCTIONSThe

  • Page 248 and 249:

    HARMONIC FUNCTIONS 233The Poisson I

  • Page 250 and 251:

    HARMONIC FUNCTIONS 235Note: This th

  • Page 252 and 253:

    HARMONIC FUNCTIONS 237The Mean Valu

  • Page 254 and 255:

    HARMONIC FUNCTIONS 239Hence

  • Page 256 and 257:

    HARMONIC FUNCTIONS 241The regions n

  • Page 258 and 259:

    HARMONIC FUNCTIONS 243Hence, settin

  • Page 260 and 261:

    HARMONIC FUNCTIONS 245(a) If P. is

  • Page 262 and 263:

    HARMONIC FUNCTIONS 247PROOF To say

  • Page 264 and 265:

    HARMONIC FUNCTIONS 249As before, Lo

  • Page 266 and 267:

    HARMONIC FUNCTIONS 251If {u.} is a

  • Page 268 and 269:

    CHAPTERTWELVETHE MAXIMUM MODULUS PR

  • Page 270 and 271:

    THE MAXIMUM MODULUS PRINCIPLE 255Th

  • Page 272 and 273:

    THE MAXIMUM MODULUS PRINCIPLE 257Fi

  • Page 274 and 275:

    THE MAXIMUM MODULUS PRINCIPLE 259Fi

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    THE MAXIMUM MODULUS PRINCIPLE 26112

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    THE MAXIMUM MODULUS PRINCIPLE 263No

  • Page 280 and 281:

    THE MAXIMUM MODULUS PRINCIPLE 265Su

  • Page 282 and 283:

    APPROXIMATIONS BY RATIONAL FUNCTION

  • Page 284 and 285:

    APPROXIMATIONS BY RATIONAL FUNCTION

  • Page 286 and 287:

    APPROXIMA nONS BY RA nONAL FUNCTION

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    APPROXIMA TIO~S BY RATIONAL FUNCTIO

  • Page 290 and 291:

    APPROXIMATIONS BY RATIONAL FUNCTION

  • Page 292 and 293:

    APPROXIMATIONS BY RATIONAL FUNCTION

  • Page 294 and 295:

    CONFORMAL MAPPING 279Here /'(zo) =

  • Page 296 and 297:

    CONFORMAL MAPPING 281Let us discuss

  • Page 298 and 299:

    CONFORMAL MAPPING 283Under these co

  • Page 300 and 301:

    CONFORMAL MAPPING 28514.9 Remarks T

  • Page 302 and 303:

    CONFORMAL MAPPING 287Divide Eqs. (5

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    CONFORMAL MAPPING 289PROOF Iff= I/(

  • Page 306 and 307:

    CONFORMAL MAPPING 291Theorem 14.18(

  • Page 308 and 309:

    CONFORMAL MAPPING 293Put y(t) = .JR

  • Page 310 and 311:

    CONFORMAL MAPPING 29518 Suppose 0 i

  • Page 312 and 313:

    CONFORMAL MAPPING 297(d) Let ex be

  • Page 314 and 315:

    ZEROS OF HOLOMORPHIC FUNCTIONS 299I

  • Page 316 and 317:

    ZEROS OF HOLOMORPHIC FUNCTIONS 301c

  • Page 318 and 319:

    ZEROS OF HOLOMORPHIC FUNCTIONS 303I

  • Page 320 and 321:

    ZEROS OF HOLOMORPlDC FUNCTIONS 305a

  • Page 322 and 323:

    ZEROS OF HOWMORPHIC FUNCTIONS 307Je

  • Page 324 and 325:

    ZEROS OF HOLOMORPIDC FUNCTIONS 309a

  • Page 326 and 327:

    ZEROS OF HOWMORPHIC FUNCTIONS 311is

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    ZEROS OF HOLOMORPHIC FUNCTIONS 313t

  • Page 330 and 331:

    ZEROS OF HOLOMORPHIC FUNCTIONS 315i

  • Page 332 and 333:

    ZEROS OF HOLOMORPHIC FUNCTIONS 317P

  • Page 334 and 335:

    CHAPTERSIXTEENANALYTIC CONTINUATION

  • Page 336 and 337:

    ANALYTIC CONTINUATION 321integers.

  • Page 338 and 339:

    ANALYTIC CONTINUATION 323the radius

  • Page 340 and 341:

    ANALYTIC CONTINUATION 325There are

  • Page 342 and 343:

    ANALYTIC CONTINUATION 32716.15 Tbeo

  • Page 344 and 345:

    ANALYTIC CONTINUATION 329We claim t

  • Page 346 and 347:

    ANALYTIC CONTINUATION 331PROOF Let

  • Page 348 and 349:

    ANALYTIC CONTINUATION 333P(f1' 01)

  • Page 350 and 351:

    CHAPTERSEVENTEENThis chapter is dev

  • Page 352 and 353:

    HP-SPACES 337some r > 0 we have D(z

  • Page 354 and 355:

    HP-SPACES 339PROOF Note first thatI

  • Page 356 and 357:

    For each z E U, 11 - e-ilz I and P(

  • Page 358 and 359:

    log I 9 I are 0 a.e. (Theorem 11.22

  • Page 360 and 361:

    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 351Let us recall that eve

  • Page 368 and 369:

    5 Suppose fe HP, qJ e H(U), and qJ(

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    ffI' -SPACES 355exists (and is fini

  • Page 372 and 373:

    ELEMENTARY TIIEORY OF BANACH ALGEBR

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    ELEMENTARY THEORY OF BANACH ALGEBRA

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    ELEMENTARY THEORY OF BANACH ALGEBRA

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    ELEMENTARY THEORY OF BANACH ALGEBRA

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    ELEMENTARY THEORY OF BANACH ALGEBRA

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    ELEMENTARY THEORY OF BANACH ALGEBRA

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    ELEMENTARY THEORY OF BANACH ALGEBRA

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    CHAPTERNINETEENHOLOMORPHIC FOURIER

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    HOLOMORPHIC FOURIER TRANSFORMS 373a

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    HOLOMORPHIC FOURIER TRANSFORMS 3751

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    HOLOMORPHIC FOURIER TRANSFORMS 377i

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    HOLOMORPHIC FOURIER TRANSFORMS 379C

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    HOLOMORPHIC FOURIER TRANSFORMS 381T

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    HOLOMORPHIC FOURIER TRANSFORMS 383F

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    HOLOMORPHIC FOURIER TRANSFORMS 385h

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    UNIFORM APPROXIMATION BY POLYNOMIAL

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    UNIFORM APPROXIMATION BY POLYNOMIAL

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    UNIFORM APPROXIMATION BY pOLYNOMIAL

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    UNIFORM APPROXIMATION BY POLYNOMIAL

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    APPENDIXHAUSDORFF'S MAXIMALITY THEO

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    NOTES AND COMMENTSChapter 1The firs

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    NOTES AND COMMENTS 399Chapter 6Sec.

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    NOTES AND COMMENTS 401orthonormal s

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    NOTES AND COMMENTS 403CbapterlOSee

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    406 BIBLIOGRAPHY24. F. Riesz and B.

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    408 LIST OF SPECIAL SYMBOLS AND ABB

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    410 INDEXCartesian product, 7,160Ca

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    412 INDEXHalmos, P. R., 398,403Hard

  • Page 429 and 430:

    414 INDEXOuter regular set, 47Overc

  • Page 431:

    416 INDEXUniform continuity, 51Unif

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