Real and complex analysis

More documents

Recommendations

Info

DIFFERENTIATION 1452- n - 1
146 REAL AND COMPLEX ANALYSISfor any n and any disjoint collection of segments (lXI' PI), ... , (IXn, Pn) in Iwhose lengths satisfynL (P; - lXi) < fJ. (2);= ISuch anfis obviously continuous: simply take n = 1.In the following theorem, the implication (b)-+ (c) is probably the mostinteresting. That (a) -+ (c) without assuming monotonicity off is the contentof Theorem 7.20.7.18 Tbeorem Let I = [a, b], let f: 1-+ RI be continuous and nondecreasing.Each of the following three statements about f implies the other two:(a) fis AC on I.(b) fmaps sets of measure 0 to sets of measure O.(c) fis differentiable a.e. on I,f' E E, andf(x) - f(a) = IX !'(t) dt (IX :s; X :s; b). (1)Note that the functions constructed in Example 7.16(b) map certain compactsets of measure 0 onto the whole unit interval!Exercise 12 complements this theorem.PROOF We will show that (a)-+ (b)-+ (c)-+ (a).Let Wi denote the u-algebra of all Lebesgue measurable subsets of RI.Assumefis AC on I, pick E c: I so that E E Wi and m(E) :;: O. We have toshow thatf(E) E Wi and m(f(E)) = O. Without loss of generality, assume thatneither a nor b lie in E.Choose E > O. Associate fJ > 0 to f and E, as in Definition 7.17. There isthen an open set V with m(V) < fJ, so that E c: V c: I. Let (IX;, PJ be thedisjoint segments whose union is V. Then L (P; - IXI) < fJ, and our choice of fJshows that thereforeL (f(P;) - f(IX;)) :s; E. (2);[Definition 7.17 was stated in terms of finite sums; thus (2) holds for everypartial sum of the (possibly) infinite series, hence (2) holds also for the sum ofthe whole series, as stated.]Since E c: V,f(E) c: U [f(IX;),f(P;)]. The Lebesgue measure of this unionis the left side of (2). This says that f(E) is a subset of Borel sets of arbitrarilysmall measure. Since Lebesgue measure.is complete, it follows thatf(E) E Wiand m(f(E)) = O.We have now proved that (a) implies (b).
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
Page 52 and 53:
POSITIVE BOREL MEASURES 372.6 Theor
Page 54 and 55:
POSITIVE BOREL MEASURES 39will mean
Page 56 and 57:
POSITIVE BOREL MEASURES 41(b) I-'(K
Page 58 and 59:
POSITIVE BOREL MEASURES 43Put V = U
Page 60 and 61:
POSITIVE BOREL MEASURES 45PROOF If
Page 62 and 63:
POSITIVE BOREL MEASURES 47Since hi
Page 64 and 65:
POSITIVE BOREL MEASURES 49PROOF Put
Page 66 and 67:
POSITIVE BOREL MEASURES 51(c) m is
Page 68 and 69:
POSITIVE BOREL MEASURES 532.21 Rema
Page 70 and 71:
POSITIVE BOREL MEASURES 55If T is o
Page 72 and 73:
POSITIVE BOREL MEASURES 57Sincethe
Page 74 and 75:
POSITIVE BOREL MEASURES S913 Is it
Page 76 and 77:
CHAPTERTHREELP-SPACESConvex Functio
Page 78 and 79:
If-SPACES 63(4) becomesexp U (Xl +
Page 80 and 81:
I!'-SPACES 65Hence the left side of
Page 82 and 83:
H-SPACES 67Suppose j, g, and h are
Page 84 and 85:
I!-SPACES 69The simple functions pl
Page 86 and 87:
l!'-SPACES 71Given / E Co(X) and
Page 88 and 89:
If-SPACES 73(b) Prove that equality
Page 90 and 91:
I!'-SPACES 75(b) If 1 :5 P < 00 and
Page 92 and 93:
ELEMENTARY HILBERT SPACE THEORY 774
Page 94 and 95:
ELEMENTARY HILBERT SPACE THEORY 79I
Page 96 and 97:
ELEMENTARY HILBERT SPACE THEORY 81P
Page 98 and 99:
ELEMENTARY HILBERT SPACE THEORY 834
Page 100 and 101:
ELEMENTARY IDLBERT SPACE THEORY 8S4
Page 102 and 103:
ELEMENTARY HILBERT SPACE THEORY 87(
Page 104 and 105:
ELEMENTARY HILBERT SPACE THEORY 89W
Page 106 and 107:
ELEMENTARY HILBERT SPACE THEORY 91T
Page 108 and 109:
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 128 and 129: EXAMPLFS 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
Page 140 and 141: 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 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
Page 188 and 189: 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
Page 206 and 207: 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:
Page 216 and 217:
Page 218 and 219:
Page 220 and 221:
ELEMENTARY PROPERTIES OF HOLOMORPlD
Page 222 and 223:
Page 224 and 225:
Page 226 and 227:
Page 228 and 229:
Page 230 and 231:
Page 232 and 233:
Page 234 and 235:
Page 236 and 237:
Page 238 and 239:
Page 240 and 241:
ELEMENTARY PROPERTIES OF HOLOMORPlD
Page 242 and 243:
Page 244 and 245:
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
Page 276 and 277:
THE MAXIMUM MODULUS PRINCIPLE 26112
Page 278 and 279:
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:
Page 286 and 287:
APPROXIMA nONS BY RA nONAL FUNCTION
Page 288 and 289:
APPROXIMA TIO~S BY RATIONAL FUNCTIO
Page 290 and 291:
Page 292 and 293:
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
Page 304 and 305:
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
Page 328 and 329:
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
Page 362 and 363:
X n , the one-dimensional spaces sp
Page 364 and 365:
So there exists a qJ E Y such that
Page 366 and 367:
HP-SPACES 351Let us recall that eve
Page 368 and 369:
5 Suppose fe HP, qJ e H(U), and qJ(
Page 370 and 371:
ffI' -SPACES 355exists (and is fini
Page 372 and 373:
ELEMENTARY TIIEORY OF BANACH ALGEBR
Page 374 and 375:
ELEMENTARY THEORY OF BANACH ALGEBRA
Page 376 and 377:
Page 378 and 379:
Page 380 and 381:
Page 382 and 383:
Page 384 and 385:
Page 386 and 387:
CHAPTERNINETEENHOLOMORPHIC FOURIER
Page 388 and 389:
HOLOMORPHIC FOURIER TRANSFORMS 373a
Page 390 and 391:
HOLOMORPHIC FOURIER TRANSFORMS 3751
Page 392 and 393:
HOLOMORPHIC FOURIER TRANSFORMS 377i
Page 394 and 395:
HOLOMORPHIC FOURIER TRANSFORMS 379C
Page 396 and 397:
HOLOMORPHIC FOURIER TRANSFORMS 381T
Page 398 and 399:
HOLOMORPHIC FOURIER TRANSFORMS 383F
Page 400 and 401:
HOLOMORPHIC FOURIER TRANSFORMS 385h
Page 402 and 403:
UNIFORM APPROXIMATION BY POLYNOMIAL
Page 404 and 405:
Page 406 and 407:
UNIFORM APPROXIMATION BY pOLYNOMIAL
Page 408 and 409:
Page 410 and 411:
APPENDIXHAUSDORFF'S MAXIMALITY THEO
Page 412 and 413:
NOTES AND COMMENTSChapter 1The firs
Page 414 and 415:
NOTES AND COMMENTS 399Chapter 6Sec.
Page 416 and 417:
NOTES AND COMMENTS 401orthonormal s
Page 418:
NOTES AND COMMENTS 403CbapterlOSee
Page 421 and 422:
406 BIBLIOGRAPHY24. F. Riesz and B.
Page 423 and 424:
408 LIST OF SPECIAL SYMBOLS AND ABB
Page 425 and 426:
410 INDEXCartesian product, 7,160Ca
Page 427 and 428:
412 INDEXHalmos, P. R., 398,403Hard
Page 429 and 430:
414 INDEXOuter regular set, 47Overc
Page 431:
416 INDEXUniform continuity, 51Unif
show all

Real and complex analysis

Create successful ePaper yourself

Delete template?

Save as template?