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Kolmogorov A.N., Fomin S.V. — Introductory real analysis
Kolmogorov A.N., Fomin S.V. — Introductory real analysis

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Название: Introductory real analysis

Авторы: Kolmogorov A.N., Fomin S.V.

Аннотация:

Comprehensive, elementary introduction to real and functional analysis. Self-contained, readily accessible to those with background in advanced calculus. Cover basic concepts and introductory principles in set theory, metric spaces, topological and linear spaces, linear functionals and linear operators, much more. 350 problems.


Язык: en

Рубрика: Математика/Анализ/Функциональный анализ/

Серия: Посвящена 110-летию со дня рождения Колмогорова Андрея Николаевича

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: revised english edition

Год издания: 1975

Количество страниц: 403

Добавлена в каталог: 09.04.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$T_{1}$-space      85
$T_{2}$-space      85
$\delta$-algebra      35
$\delta$-ring      35
$\sigma$-additivity      see “Countable additivity”
$\sigma$-algebra      35
$\sigma$-finite measure      308
$\sigma$-ring      35
$\varepsilon$-neighborhood      46
$\varepsilon$-net      98
Absolutely continuous charge      347
Absolutely continuous function      336
Absolutely summable sequence      185
Adjoint operator      232
Adjoint operator in Hilbert space      234
Aleph null      16
Alexandroff, P.S.      90 97
Algebra of sets      31
Algebraic dimension      128
Algebraic number      19
Almost everywhere      288
Angle between vectors      143
Arzela’s theorem      102
Arzela’s theorem, generalization of      107
Axiom of Choice      27
Axiom of countability, first      93
Axiom of countability, second      82
Axiom of separation, first      85
Axiom of separation, Hausdorff      85
Axiom of separation, second      85
B-algebra      see “Borel algebra”
B-set      see “Borel set”
Baire’s theorem      61
Banach space      140
Banach, S.      138 229 238
Base      81
Base, countable      382
Base, neighborhood (local)      83
Basis      121
Basis, dual      185
Basis, Hamel      128
Basis, orthogonal      143
Basis, orthonormal      143
Bessel’s inequality      150 165
Bicompactum      96
Binary relation      see “Relation”
Birkhoff, G.      28
Bolzano — Weierstrass theorem      101
Borel algebra      35
Borel algebra, irreducible      36
Borel algebra, minimal      36
Borel closure      36
Borel sets      36
Bounded linear functional      177
Bounded linear functional, norm of      177
Bounded real function      110
Bounded set      65 141 169
Bounded set, locally      169
Bounded set, strongly      197
Bounded set, weakly      197
Cantor function      335
Cantor set      52
Cantor set, points of the first kind of      53
Cantor set, points of the second kind of      53
Cantor — Bernstein theorem      17
Cantor, G.      29
cardinal number      24
Cartesian product      see “Direct product”
Cauchy criterion      56
Cauchy sequence      56
Cauchy — Schwarz inequality      38
Chain      28
Chain, maximal      28
Characteristic function      349
Charge      344
Charge, absolutely continuous      347
Charge, concentrated, on a set      346
Charge, continuous      346
Charge, density of      350
Charge, discrete      347
Charge, negative      344
Charge, negative variation of      346
Charge, positive      344
Charge, positive variation of      346
Charge, Radon — Nikodym derivative of      350
Charge, singular      347
Charge, total variation of      346
Chebyshev’s inequality      299
Choice function      27
Classes      6
Classes, equivalence      8
Closed ball      see “Closed sphere”
Closed graph theorem      238
Closed set(s)      49
Closed set(s) in a topological space      79
Closed set(s) on the real line      51
Closed set(s), unions and intersections of      49
Closed sphere(s)      46
Closed sphere(s), center of      46
Closed sphere(s), nested (or decreasing) sequence of      59
Closed sphere(s), radius of      46
Closure      46 79
Closure operator      46
Closure operator, properties of      46
Codimension      122
Cohen, P.J.      29
Compact space      92
Compact space, countably      95
Compact space, locally      97
Compactness      92
Compactness, countable      95
Compactness, relative      97
Compactness, relative countable      97
Compactum      92 96
Compactum, metric      96
Complement of a set      3
Complete limit point      97
Complete measure      280
Completely continuous operator(s)      239 ff.
Completely continuous operator(s) in Hilbert space      246—251
Completely continuous operator(s), basic properties of      243—246
Completely regular space      92
Completion (of a metric space)      62
Component (of an open set)      55
Conjugate space      185
Conjugate space of a normed linear space      184
Conjugate space, second      190
Conjugate space, strong topology in      190
Conjugate space, third      190
Conjugate space, weak topology in      200
Conjugate space, weak* topology in      202
Connected set      55
Connected space      84
Contact point      46 79
Continuity      44 87
Continuity from the left      315
Continuity from the right      315
Continuity, uniform      109
Continuous charge      346
Continuous linear functional(s)      175 ff.
Continuous linear functional(s), order of      182
Continuous linear functional(s), sufficiently many      181
Continuum      16
Continuum, power of      16
Contraction mapping(s)      66 ff.
Contraction mapping(s) and differential equations      71—72
Contraction mapping(s) and integral equations      74—76
Contraction mapping(s) and systems of differential equations      72—74
Contraction mapping(s), principle of      66
Convergence almost everywhere      289
Convergence in measure      292
Convergence in the mean      379
Convergence in the mean square      385
Convergent sequence in a metric space      47
Convergent sequence in a topological space      84
Convex body      129
Convex functional      130 134
Convex hull      130
Convex set      129
Convexity      128
Countability of rational numbers      11
Countable additivity      266 272
Countable base      382
Countable set      10
Countably compact space      95
Countably Hilbert space      173
Countably normed (linear) space      171
Countably normed (linear) space, complete      173
Cover      83
Cover, closed      83
Cover, open      83
Covering      see “Cover”
Curve(s) in a metric space      112—113
Curve(s) in a metric space, length of      114 115
Curve(s) in a metric space, sequence of      115
Curve(s), rectifiable      332
Decomposition of a set into classes      6—9
Delta function      124 208
Dense set      48
Dense set, everywhere      48
Dense set, nowhere      48 61
density      350
Derived numbers      318
Derived numbers, left-hand lower      318
Derived numbers, right-hand upper      318
Diameter of a set      65
difference between sets      3
Differentiation of a monotonic function      318—323
Differentiation of an integral with respect to its upper limit      323—326
DIMENSION      121
Dimension, algebraic      128
Dini’s theorem      115
Direct product      238 352
Direct product of measures      354
Directed set      29
Dirichlet function      289 291 301
Discontinuity point of the first kind      315
Discrete charge      347
Discrete space      38
Disjoint sets      2
Disjoint sets, pairwise      2
Distance, between a point and a set      54
Distance, between two sets      55
Distance, properties of      37
Distance, symmetry of      37
Domain (of definition)      4 5 221
Domain (open connected set)      71
Egorov’s theorem      290
Eigenvalue      235
Eigenvector      235
Elementary set      255
Elementary set, measure of      256
Empty set      2
Equicontinuous family of functions      102
Equivalence classes      8
Equivalence relation      7
Equivalent functions      288
Equivalent sets      13
Essential supremum      311
Essentially bounded function      310
Euclidean ?-space      38 144
Euclidean space(s)      138 142
Euclidean space(s), characterization of      160
Euclidean space(s), complete      153
Euclidean space(s), complete, norm of vector in      164
Euclidean space(s), complete, orthogonal elements of      164
Euclidean space(s), components of elements of      149
Euclidean space(s), norm in      142
Euclidean space(s), separable      146
Euler lines      105
Exhaustive sequence of sets      308
Extension of a functional      132
Extension of a measure      271 277 279
Extension of a measure, Jordan      281
Factor space      122
Fatou’s theorem      307
Field      37
Finite expansion      33
Finite function      208
Finite set      10
First axiom of countability      83
First axiom of separation      85
Fixed point      66
Fixed point theorem      66
Fourier coefficients      149 152 165
Fourier series      149 165
Fractional part      8
Fraenkel, A.A.      25 27
Fredholm equation      74
Fredholm equation, homogeneous      74
Fredholm equation, kernel of      74
Fredholm equation, nonhomogeneous      74
Friedman, A.      212
Fubini’s Theorem      359
Function space      39 108
Function(s)      4 ff.
Function(s) of bounded variation      328—332
Function(s), absolutely continuous      336
Function(s), Borel-measurable      284
Function(s), bounded (real)      110 207
Function(s), Cantor      335
Function(s), characteristic      349
Function(s), continuous      44 79
Function(s), continuous, from the left      315
Function(s), continuous, from the right      315
Function(s), continuous, uniformly      109
Function(s), delta      124 208
Function(s), domain (of definition of)      4 5
Function(s), equivalent      288
Function(s), essentially bounded      310
Function(s), finite      207
Function(s), general      5
Function(s), generalized      see “Generalized functions”
Function(s), generating      362
Function(s), infinitely differentiable      169
Function(s), integrable      294 296 308
Function(s), integrable, locally      208
Function(s), inverse      5
Function(s), jump      315 341
Function(s), jump of      315
Function(s), left-hand limit of      315
Function(s), lower limit of      111
Function(s), lower semicontinuous      110
Function(s), measurable      284 ff.
Function(s), monotonic      314
Function(s), nondecreasing      314
Function(s), nonincreasing      314
Function(s), one-to-one      5
Function(s), oscillation of      111
Function(s), range of      4 5
Function(s), real      4
Function(s), right-hand limit of      315
Function(s), simple      286
Function(s), singular      341
Function(s), step      316
Function(s), summable      294 296 308
Function(s), test      208
Function(s), uniformly continuous      109
Function(s), upper limit of      111
Function(s), upper semicontinuous      110
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