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Folland J.B. — Real Analysis: Modern Techniques and Their Applications

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Название: Real Analysis: Modern Techniques and Their Applications

Автор: Folland J.B.

Аннотация:

This book covers the subject matter that is central to mathematical analysis: measure and integration theory, some point set topology, and rudiments of functional analysis. Also, a number of other topics are developed to illustrate the uses of this core material in important areas of mathematics and to introduce readers to more advanced techniques. Some of the material presented has never appeared outside of advanced monographs and research papers, or been readily available in comparative texts. About 460 exercises, at varying levels of difficulty, give readers practice in working with the ideas presented here.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 2-nd edition

Год издания: 1999

Количество страниц: 386

Добавлена в каталог: 09.12.2006

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Предметный указатель

$F_{\sigma}$ and $F_{\sigma \delta}$ sets      22
$G_{\delta}$ and $G_{\delta \sigma}$ sets      22
derivative      246
norm      181 184
space      181 184
space, intrinsic      363
space, weak      198
$T_0, \dots , T_4$ space      116 123
$\sigma$ -algebra      21
$\sigma$ -algebra of countable or co-countable sets      21
$\sigma$ -algebra, Borel      22
$\sigma$ -algebra, generated by a family of functions      44
$\sigma$ -algebra, generated by a family of sets      22
$\sigma$ -algebra, product      22
$\sigma$ -compact space      133
$\sigma$ -field      21
$\sigma$ -finite measure      25
$\sigma$ -finite set      25
$\sigma$ -finite signed measure      88
$\sigma$ -ring      24
a.e.      26
a.s.      314
Abel mean      261
Abel summation      261
Absolute continuity of a function      105
Absolute continuityof a measure      88
Absolute convergence      152
Accumulation point      114
Adjoint      160 177
Alaoglu's theorem      169
Alexandroff compactification      132
Algebra of functions      139
Algebra of sets      21
Algebra, Banach      154
Almost every(where)      26
Almost sure(ly)      314
Almost uniform convergence      62
Approximate identity      245
Arcwise connected set      124
Arzela — Ascoli theorem      137
Axiom of Choice      6
Axiomof countability      116
Axiomof separation      116
Baire category theorem      161
Baire classes      83
Baire set      215
Ball      13
Banach algebra      154
Banach space      152
Banach — Tarski paradox      20
Base for a topology      115
Bessel's inequality      175
Bijective mapping      4
Binomial distribution      320
Bochner integral      179
Bochner — Riesz means      279
Bolzano — Weierstrass property      15
Borel $\sigma$ -algebra      22
Borel isomorphism      83
Borel measurable function      44
Borel measure      33
Borel set      22
Borel — Cantelli lemma      321
Borel's normal number theorem      330
Boundary      114
bounded linear map      153
Bounded set      15
Bounded variation      102
Cantor function      39
Cantor set      38
Cantor — Lebesgue function      39
Caratheodory's theorem      29
Cartesian product      3—4
Cauchy net      167
Cauchy sequence      14
Cauchy sequence in measure      61
Cauchy — Riemann equation      308
Central limit theorem      326
Cesaro mean      262
Cesaro summation      262
Characteristic function      46
Characteristic function of a probability distribution      314
Chebyshev's inequality      193
Chi-square distribution      320
Closed graph theorem      163
Closed linear map      163
Closed set      13 114
Closure      13 114
Closure operator      119
Cluster point      118 126
Coarser topology      114
Cofinal net      127
Cofinite topology      113
Commutator subgroup      346
Compact set      16 128
Compact space      128
Compact space, countably      130
Compact space, locally      131
Compact space, sequentially      130
Compactification      144
Compactification, one-point      132
Compactification, Stone — Cech      144
Compactly supported function      132
Complement      3
Complete measure      26
Complete metric space      14
Complete orthonormal set      175
Complete topological vector space      167
Completely regular algebra      146
Completely regular space      123
Completion of a $\sigma$ -algebra      27
Completion of a measure      27
Completion of a normed vector space      159
Complex measure      93
composition      3
Condensation of singularities      165
Conditional expectation      93
Conjugate exponent      183
Connected component      119
Connected set      118
Constant-coefficient operator      273
Content      72
Continuity      14 119
Continuity of linear maps on $C_C^{\infty}$       282
Continuity of measures      26
Continuity, absolute      88 105
Continuity, Hoelder      138
Continuity, Lipschitz      108
Continuity, uniform      14 238 340
Continuous measure      106
Continuum      8
Continuum Hypothesis      17
Convergence in $C_C^{\infty}$       282
Convergence in       54
Convergence in measure      61
Convergence in probability      314
Convergence of a filter      147
Convergence of a net      126
Convergence of a sequence      14 116
Convergence of a series      152
Convergence, absolute      152
Convergence, almost uniform      62
Convex function      109
Convolution of distributions      285 292
Convolution of functions      239
Convolution of measures      270
Coordinate      4
Coordinate map      4
Countable additivity      24
Countable ordinals      10

Countable set      7
Countably compact space      130
Counting measure      25
Cover      15
cube      71 143
Daniell integral      81
Decomposable measure      92
Decreasing function      12
Decreasing rearrangement      199
DeMorgan's laws      3
Dense set      13 114
density      362
Density of a distribution      284
Density of a set      100
Density, Radon — Nikodym      91
Density, Riemannian volume      362
Denumerable set      7
Derivative,       246
Diameter      15
Diffeomorphism      74
Difference of sets      3
Differential operator      273
Dirac measure      25
Directed set      125
Dirichlet kernel      264
Dirichlet problem      274
disconnected set      118
Discrete      113
Discrete measure      106
Discrete topology      113
Disjoint sets      2
distribution      282
Distribution function      33 197 315
Distribution of a random variable      315
Distribution, homogeneous      289
Distribution, joint      315
Distribution, periodic      297
Distribution, tempered      293
Domain      4
Dominated Convergence Theorem      54
Dual space      157
Egoroff's theorem      62
Elementary family      23
Elliptic differential operator      307 311
Elliptic regularity theorem      307
Embedding      120
entropy      325
Equicontinuity      137
Equivalence class      3
Equivalence relation      3
Equivalent metric      16
Equivalent norm      152
Essential range      187
Essential supremum      184
Event      314
Eventually      126
Expectation      314
Expectation, conditional      93
Extended integrable function      86
Extended real number system      10
Fatou's lemma      52
Fejer kernel      269
Field of sets      21
Filter      147
Finer topology      114
Finite intersection property      128
Finite measure      25
Finite signed measure      88
Finitely additive measure      25
First category      161
First countable space      116
First uncountable ordinal      10
Fourier integral      253
Fourier inversion theorem      251
Fourier series      248
Fourier transform      248—249
Fourier transform of a measure      272
Fourier transform of a tempered distribution      295
Fourier — Stieltjes transform      272
Fractional integral      77
Frechetspace      167
Frequently      126
Fubini — Tonelli theorem      67—68 229
Fubini's theorem      67—68 229
Function      3
Fundamental theorem of calculus      106
Gamma distribution      320
Gamma function      58
Gauge      82
Gauss kernel      260
Gaussian distribution      325
Generalized Cantor set      39
Gibbs phenomenon      268
Gram — Schmidt process      175
Graph of a linear map      162
H-interval      33
Haar measure      341
Hahn decomposition      87
Hahn decomposition theorem      86
Hahn — Banach theorem      157—158
Hardy — Littlewood maximal function      96
Hardy's inequalities      196
Hausdorff dimension      351
Hausdorff maximal principle      5
Hausdorff measure      350
Hausdorff space      117
Hausdorff — Young inequality      248
Heat equation      275
Heine — Borel property      15
Heisenberg's inequality      255
Henstock — Kurzweil integral      82
Hermite function      256
Hermite operator      256
Hermite polynomial      257
Hilbert space      172
Hilbert's inequality      196
Hoelder continuity      138
Hoelder's inequality      182 196
Homeomorphism      119
Homogeneous distribution      289
Hull of an ideal      142
Ideal in an algebra      142
Identically distributed random variables      315
IFF      1
Image      3
Image measure      314
Increasing function      12
Independent events      315
Independent random variables      315
Indicator function      46
Indiscrete topology      113
Inequality, Bessel's      175
Inequality, Chebyshev ' s      193
Inequality, Hardy's      196
Inequality, Hausdorff — Young      248
Inequality, Heisenberg's      255
Inequality, Hilbert's      196
Inequality, Hoelder's      182 196
Inequality, Jensen's      109
Inequality, Kolmogorov's      322
Inequality, Minkowski's      183 194
Inequality, Schwarz      172
Inequality, triangle      151
Inequality, Wirtinger's      254
Inequality, Young's      240—241
Infimum      9—10
Initial segment      9
Injective mapping      4
Inner measure      32
Inner product      171

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