Measure, Integration & Real Analysis, 2021a

More documents

Recommendations

Info

Notation Index 405 tA, 23 T C , 309 T k , 288 U ⊥ , 229 U f , 133 U( f , [a, b]), 4 U( f , P, [a, b]), 2 V ′ , 180 V ′′ , 183 V, 286 V C , 234 ˜X, 392 ‖(x 1 ,...,x n )‖ ∞ , 136 ∫ ∫ f (x, y) dν(y) dμ(x), 126 X Y Z, 5 |z|, 155 z, 158 Z(μ), 202 Z + , 5 Measure, Integration & Real Analysis, by Sheldon Axler
Index Abel summation, 344 absolute value, 155 absolutely continuous, 270 additivity of outer measure, 47–48, 50, 60 adjoint of a linear map, 281 a. e., 90 Agnesi, Maria Gaetana, 73 algebra, 120 algebraic multiplicity, 324 almost every, 90 almost surely, 382 Apollonius’s identity, 223 approximate eigenvalue, 310 approximate identity, 346 approximation of Borel set, 48, 52 approximation of function by continuous functions, 98 by dilations, 100 by simple functions, 96 by step functions, 97 by translations, 100 area paradox, 24 area under graph, 134 average, 386 Axiom of Choice, 22–23, 176, 183, 236 bad Borel set, 113 Baire Category Theorem, 184 Baire’s Theorem, 184–185 Baire, René-Louis, 184 Banach space, 165 Banach, Stefan, 146 Banach–Steinhaus Theorem, 189, 192 basis, 174 Bayes’ Theorem, 391 Bayes, Thomas, 391 Bengal, vi Bergman space, 253 Bessel’s inequality, 241 Bessel, Friedrich, 242 Borel measurable function, 33 Borel set, 28, 36, 137, 252 Borel, Émile, 19 Borel–Cantelli Lemma, 383 bounded, 136 linear functional, 173 linear map, 167 bounded below, 192 Bounded Convergence Theorem, 89 Bounded Inverse Theorem, 188 Bunyakovsky, Viktor, 219 Measure, Integration & Real Analysis, by Sheldon Axler Cantor function, 58 Cantor set, 56, 278 Carleson, Lennart, 352 Cauchy sequence, 151, 165 Cauchy, Augustin-Louis, 152 Cauchy–Schwarz inequality, 218 chain, 176 characteristic function, 31 Chebyshev’s inequality, 106, 389 Chebyshev, Pafnuty, 106 Clarkson’s inequalities, 209 Clarkson, James, 209 closed, 136 ball, 148 half-space, 235 set, 149 Closed Graph Theorem, 189 closure, 149 compact operator, 312 complete metric space, 151 complex conjugate, 158 complex inner product space, 212 complex measure, 256 complex number, 155 406
Page 1 and 2:
Measure, Integration & Real Analysi
Page 3 and 4:
About the Author Sheldon Axler was
Page 5 and 6:
viii Contents 2D Lebesgue Measure 4
Page 7 and 8:
x Contents 6E Consequences of Baire
Page 9 and 10:
xii Contents 11 Fourier Analysis 33
Page 11 and 12:
Preface for Instructors You are abo
Page 13 and 14:
xvi Preface for Instructors • Cha
Page 15 and 16:
Acknowledgments I owe a huge intell
Page 17 and 18:
2 Chapter 1 Riemann Integration 1A
Page 19 and 20:
4 Chapter 1 Riemann Integration m (
Page 21 and 22:
6 Chapter 1 Riemann Integration whe
Page 23 and 24:
8 Chapter 1 Riemann Integration 6 S
Page 25 and 26:
10 Chapter 1 Riemann Integration Ca
Page 27 and 28:
12 Chapter 1 Riemann Integration EX
Page 29 and 30:
14 Chapter 2 Measures 2A Outer Meas
Page 31 and 32:
16 Chapter 2 Measures The next resu
Page 33 and 34:
18 Chapter 2 Measures Outer Measure
Page 35 and 36:
20 Chapter 2 Measures Now we can pr
Page 37 and 38:
≈ 7π Chapter 2 Measures Clearly
Page 39 and 40:
24 Chapter 2 Measures 10 Prove that
Page 41 and 42:
26 Chapter 2 Measures We have shown
Page 43 and 44:
28 Chapter 2 Measures Borel Subsets
Page 45 and 46:
30 Chapter 2 Measures Inverse image
Page 47 and 48:
32 Chapter 2 Measures The definitio
Page 49 and 50:
34 Chapter 2 Measures 2.42 Definiti
Page 51 and 52:
Page 53 and 54:
38 Chapter 2 Measures EXERCISES 2B
Page 55 and 56:
40 Chapter 2 Measures 23 Suppose f
Page 57 and 58:
42 Chapter 2 Measures • Suppose X
Page 59 and 60:
44 Chapter 2 Measures For convenien
Page 61 and 62:
46 Chapter 2 Measures 5 Suppose (X,
Page 63 and 64:
48 Chapter 2 Measures Thus |A ∪ G
Page 65 and 66:
50 Chapter 2 Measures where the equ
Page 67 and 68:
52 Chapter 2 Measures Lebesgue Meas
Page 69 and 70:
54 Chapter 2 Measures In practice,
Page 71 and 72:
56 Chapter 2 Measures 2.74 Definiti
Page 73 and 74:
58 Chapter 2 Measures Now we can de
Page 75 and 76:
60 Chapter 2 Measures EXERCISES 2D
Page 77 and 78:
62 Chapter 2 Measures 2E Convergenc
Page 79 and 80:
64 Chapter 2 Measures Proof Suppose
Page 81 and 82:
66 Chapter 2 Measures Luzin’s The
Page 83 and 84:
68 Chapter 2 Measures For each inte
Page 85 and 86:
Page 87 and 88:
72 Chapter 2 Measures 9 Suppose F 1
Page 89 and 90:
74 Chapter 3 Integration 3A Integra
Page 91 and 92:
76 Chapter 3 Integration 3.6 Exampl
Page 93 and 94:
78 Chapter 3 Integration 3.11 Monot
Page 95 and 96:
80 Chapter 3 Integration Now we can
Page 97 and 98:
82 Chapter 3 Integration The condit
Page 99 and 100:
84 Chapter 3 Integration The inequa
Page 101 and 102:
86 Chapter 3 Integration 12 Show th
Page 103 and 104:
88 Chapter 3 Integration 3B Limits
Page 105 and 106:
90 Chapter 3 Integration 3.27 Defin
Page 107 and 108:
92 Chapter 3 Integration Suppose (X
Page 109 and 110:
94 Chapter 3 Integration Proof Supp
Page 111 and 112:
96 Chapter 3 Integration 3.42 Examp
Page 113 and 114:
98 Chapter 3 Integration in other w
Page 115 and 116:
100 Chapter 3 Integration 7 Let λ
Page 117 and 118:
102 Chapter 4 Differentiation 4A Ha
Page 119 and 120:
104 Chapter 4 Differentiation Suppo
Page 121 and 122:
106 Chapter 4 Differentiation EXERC
Page 123 and 124:
108 Chapter 4 Differentiation 4B De
Page 125 and 126:
110 Chapter 4 Differentiation Deriv
Page 127 and 128:
112 Chapter 4 Differentiation The n
Page 129 and 130:
114 Chapter 4 Differentiation Proof
Page 131 and 132:
Chapter 5 Product Measures Lebesgue
Page 133 and 134:
118 Chapter 5 Product Measures 5.3
Page 135 and 136:
120 Chapter 5 Product Measures Mono
Page 137 and 138:
122 Chapter 5 Product Measures Now
Page 139 and 140:
124 Chapter 5 Product Measures The
Page 141 and 142:
126 Chapter 5 Product Measures 5.23
Page 143 and 144:
128 Chapter 5 Product Measures EXER
Page 145 and 146:
Page 147 and 148:
132 Chapter 5 Product Measures As y
Page 149 and 150:
Page 151 and 152:
136 Chapter 5 Product Measures 5C L
Page 153 and 154:
Page 155 and 156:
140 Chapter 5 Product Measures Volu
Page 157 and 158:
142 Chapter 5 Product Measures This
Page 159 and 160:
144 Chapter 5 Product Measures EXER
Page 161 and 162:
Chapter 6 Banach Spaces We begin th
Page 163 and 164:
148 Chapter 6 Banach Spaces The mat
Page 165 and 166:
150 Chapter 6 Banach Spaces The def
Page 167 and 168:
152 Chapter 6 Banach Spaces Entranc
Page 169 and 170:
154 Chapter 6 Banach Spaces 14 Supp
Page 171 and 172:
156 Chapter 6 Banach Spaces For b a
Page 173 and 174:
158 Chapter 6 Banach Spaces We now
Page 175 and 176:
160 Chapter 6 Banach Spaces 6.28 Ex
Page 177 and 178:
162 Chapter 6 Banach Spaces EXERCIS
Page 179 and 180:
164 Chapter 6 Banach Spaces Sometim
Page 181 and 182:
166 Chapter 6 Banach Spaces 6.40 De
Page 183 and 184:
168 Chapter 6 Banach Spaces 6.45 Ex
Page 185 and 186:
170 Chapter 6 Banach Spaces EXERCIS
Page 187 and 188:
172 Chapter 6 Banach Spaces 6D Line
Page 189 and 190:
174 Chapter 6 Banach Spaces Discont
Page 191 and 192:
176 Chapter 6 Banach Spaces The not
Page 193 and 194:
178 Chapter 6 Banach Spaces If V is
Page 195 and 196:
180 Chapter 6 Banach Spaces Then A
Page 197 and 198:
182 Chapter 6 Banach Spaces 2 Suppo
Page 199 and 200:
184 Chapter 6 Banach Spaces 6E Cons
Page 201 and 202:
186 Chapter 6 Banach Spaces Because
Page 203 and 204:
188 Chapter 6 Banach Spaces The nex
Page 205 and 206:
190 Chapter 6 Banach Spaces 6.86 Pr
Page 207 and 208:
192 Chapter 6 Banach Spaces A linea
Page 209 and 210:
194 Chapter 7 L p Spaces 7A L p (μ
Page 211 and 212:
196 Chapter 7 L p Spaces What we ca
Page 213 and 214:
198 Chapter 7 L p Spaces 7.11 Examp
Page 215 and 216:
200 Chapter 7 L p Spaces 6 Suppose
Page 217 and 218:
202 Chapter 7 L p Spaces 7B L p (μ
Page 219 and 220:
204 Chapter 7 L p Spaces L p (μ) I
Page 221 and 222:
206 Chapter 7 L p Spaces Duality Re
Page 223 and 224:
208 Chapter 7 L p Spaces EXERCISES
Page 225 and 226:
210 Chapter 7 L p Spaces 18 Suppose
Page 227 and 228:
212 Chapter 8 Hilbert Spaces 8A Inn
Page 229 and 230:
214 Chapter 8 Hilbert Spaces As we
Page 231 and 232:
216 Chapter 8 Hilbert Spaces The ne
Page 233 and 234:
218 Chapter 8 Hilbert Spaces The or
Page 235 and 236:
220 Chapter 8 Hilbert Spaces The pr
Page 237 and 238:
222 Chapter 8 Hilbert Spaces 9 The
Page 239 and 240:
224 Chapter 8 Hilbert Spaces 8B Ort
Page 241 and 242:
226 Chapter 8 Hilbert Spaces In the
Page 243 and 244:
228 Chapter 8 Hilbert Spaces 8.37 o
Page 245 and 246:
230 Chapter 8 Hilbert Spaces The re
Page 247 and 248:
232 Chapter 8 Hilbert Spaces 8.45 r
Page 249 and 250:
234 Chapter 8 Hilbert Spaces Suppos
Page 251 and 252:
236 Chapter 8 Hilbert Spaces 16 Sup
Page 253 and 254:
238 Chapter 8 Hilbert Spaces • Fo
Page 255 and 256:
240 Chapter 8 Hilbert Spaces • Su
Page 257 and 258:
242 Chapter 8 Hilbert Spaces Proof
Page 259 and 260:
244 Chapter 8 Hilbert Spaces 8.61 D
Page 261 and 262:
246 Chapter 8 Hilbert Spaces A mome
Page 263 and 264:
248 Chapter 8 Hilbert Spaces 8.73 E
Page 265 and 266:
250 Chapter 8 Hilbert Spaces Riesz
Page 267 and 268:
252 Chapter 8 Hilbert Spaces 10 (a)
Page 269 and 270:
254 Chapter 8 Hilbert Spaces 23 Pro
Page 271 and 272:
256 Chapter 9 Real and Complex Meas
Page 273 and 274:
Page 275 and 276:
Page 277 and 278:
Page 279 and 280:
Page 281 and 282:
Page 283 and 284:
Page 285 and 286:
Page 287 and 288:
≈ 100e Chapter 9 Real and Complex
Page 289 and 290:
Page 291 and 292:
Page 293 and 294:
Page 295 and 296:
Chapter 10 Linear Maps on Hilbert S
Page 297 and 298:
282 Chapter 10 Linear Maps on Hilbe
Page 299 and 300:
Page 301 and 302:
Page 303 and 304:
Page 305 and 306:
Page 307 and 308:
Page 309 and 310:
Page 311 and 312:
Page 313 and 314:
Page 315 and 316:
Page 317 and 318:
Page 319 and 320:
Page 321 and 322:
Page 323 and 324:
Page 325 and 326:
Page 327 and 328:
Page 329 and 330:
≈ 100π Chapter 10 Linear Maps on
Page 331 and 332:
Page 333 and 334:
Page 335 and 336:
Page 337 and 338:
Page 339 and 340:
Page 341 and 342:
Page 343 and 344:
Page 345 and 346:
Page 347 and 348:
Page 349 and 350:
Page 351 and 352:
Page 353 and 354:
Page 355 and 356:
340 Chapter 11 Fourier Analysis 11A
Page 357 and 358:
342 Chapter 11 Fourier Analysis Hil
Page 359 and 360:
344 Chapter 11 Fourier Analysis 11.
Page 361 and 362:
346 Chapter 11 Fourier Analysis 11.
Page 363 and 364:
348 Chapter 11 Fourier Analysis Sol
Page 365 and 366:
350 Chapter 11 Fourier Analysis Fou
Page 367 and 368:
352 Chapter 11 Fourier Analysis In
Page 369 and 370: 354 Chapter 11 Fourier Analysis 12
Page 371 and 372: 356 Chapter 11 Fourier Analysis Now
Page 373 and 374: 358 Chapter 11 Fourier Analysis 11.
Page 375 and 376: 360 Chapter 11 Fourier Analysis 11.
Page 377 and 378: 362 Chapter 11 Fourier Analysis 11
Page 379 and 380: 364 Chapter 11 Fourier Analysis (b)
Page 381 and 382: 366 Chapter 11 Fourier Analysis Not
Page 383 and 384: 368 Chapter 11 Fourier Analysis Con
Page 385 and 386: 370 Chapter 11 Fourier Analysis Our
Page 387 and 388: 372 Chapter 11 Fourier Analysis The
Page 389 and 390: 374 Chapter 11 Fourier Analysis Fou
Page 391 and 392: 376 Chapter 11 Fourier Analysis whe
Page 393 and 394: 378 Chapter 11 Fourier Analysis 3 S
Page 395 and 396: Chapter 12 Probability Measures Pro
Page 397 and 398: 382 Chapter 12 Probability Measures
Page 415 and 416: Photo Credits • page vi: photos b
Page 417 and 418: Bibliography The chapters of this b
Page 419: 404 Notation Index ∑ ∞ k=1 g k,
Page 423 and 424: 408 Index Hahn Decomposition Theore
Page 425 and 426: 410 Index random variable, 385 rang
show all

Measure, Integration & Real Analysis, 2021a

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?