McShane E.J., Botts T.A. — Real Analysis :: Электронная библиотека попечительского совета мехмата МГУ

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McShane E.J., Botts T.A. — Real Analysis

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Название: Real Analysis

Авторы: McShane E.J., Botts T.A.

Аннотация:

Real analysis (traditionally, the theory of functions of a real variable) is a branch of mathematical analysis dealing with the real numbers and real valued functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real numbers, and continuity, smoothness and related properties of real-valued functions.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1959

Количество страниц: 279

Добавлена в каталог: 12.04.2013

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Предметный указатель

-tuple      5
$T_{2}$ -space      40
$\sigma$ -algebra      173
$\sigma$ -ring of sets      173
Absolute continuity      102ff 108 109 177ff 188
Absolute convergence      136 145
Absolute value      15
Absolutely continuous functions      102ff
Absolutely continuous functions, of intervals      108 109
Absolutely continuous functions, of sets      177ff
Accumulation point      51
Additive function, of intervals      105
Additive function, of sets      174
Algebras, of sets      173
Almost everywhere      138
Antisymmetric relation      27
Antitone function      58
Approximately continuous function      189
Arc wise connected set      72
Archimedean property      20
Ascoli, G., Ascoli’s theorem      92ff
Associative laws      9 12 13
Axiom of Choice      29ff 230ff
Baire functions      147ff 158 242
Banach, Banach space      204
Banach, Banach — Steinhaus theorem      73
Banach, Hahn — Baoach theorem      214
Basic neighborhood      32 39
Basic neighborhood, deleted      32
Bernstein polynomials      88 243
Bessel, F. W., Bessel’s inequality      228
Bolzano — Weierstrass theorem      53
Boole, Q., Boolean algebra, $\sigma$ -algebra      173
Boole, Q., Boolean ring, $\sigma$ -ring      173
Borel, E., Borel set      154 158 159
Borel, E., Heine — Borel theorem      53
Bound      20 28
Bound, of linear functional      210
Bounded linear function      210
Bounded operator      241ff
Bounded set      20 53
Bounded variation      99ff 106
Buniakowski, V., Cauchy — Buniakowaki — Schwars inequality      162
Cardinality      5
Cardinally equivalent sets      4
Cartesian product      2 43
Cauchy, A., Cauchy — Buniakowski — Schwarz inequality      162
Cauchy, A., Cauchy — Schwarz inequality      59
Cauchy, A., criterion      37 61
Cauchy, A., directed function      61
Cauchy, A., sequence      61
Chain      29 30
Chain, rule      120
Character, finite      28
Characteristic function of set      77
Choice, axiom of      29 255ff
Choice, function      29 255ff
Closed interval      27
Closed set      46ff 83
Closed under operation      11
Closure      46
Cluster point, of a directed function      42 50
Cluster point, of a set      51
Commutative group      9
Commutative laws      13
Compact sets      50ff
Compact sets, image of      70
Compact sets, topological product of      52
Compactification, one-point      80
Complement, orthogonal      224
Complete lattice      28
Complete metric space      81 204 206
Complete ordered field      20ff
Complete orthonormal set      227
Component      43
Composite function      67ff 120
Composite product      13
Composite sum      12
Connected, arcwise      72
Continuity      67ff
Continuity, absolute      102ff 108 109 177ff
Continuity, approximate      189
Continuity, interval of      107
Continuity, modulus of      78
Continuity, of linear functionals      210
Continuity, uniform      77ff 97
Convergence      33ff
Convergence, almost everywhere      140
Convergence, at infinity      79
Convergence, dominated      140
Convergence, in $R^{*}$       62ff
Convergence, in norm      225
Convergence, monotone      137
Convergence, order      54
Convergence, pointwise      50
Convergence, regular      188
Convergence, strong      225
Convergence, unconditional      34ff
Convergence, uniform      82 84ff 225
Convergence, weak      225
Convergent directed function      33
Convergent sequence      34
Convergent Series      34
Convex set      222
Convolution      200 201
Countable set      5
Countably additive function      174
Countably infinite set      5
Covering, open      50
Cramer’s Rule      122
Criterion of Cauchy      61
Daniell, P. J.      127 133
Darboux, J. O., Darboux property      71 72
Darboux, J. O., upper and lower Darboux — Stieltjes integral      168
Decomposition, Hahn      175ff
Decomposition, Lebesgue      178ff
Deleted basic neighborhood      32
Deleted relative neighborhood      68
Dense set      21 47
Density, metric      188
Denumerably infinite set      5
Derivate      110 183
Derivate, of absolutely continuous functions      188
Derivate, of functions of bounded variation      191
Derivative      110ff
Derivative, first      117
Derivative, left      111
Derivative, of composite function      120 125
Derivative, of integral with respect to parameter      140
Derivative, of Lebesgue integral      183ff
Derivative, of order       117
Derivative, partial      118
Derivative, regular      184
Derivative, right      111
Determinant      121ff
Determinant, function      122
Diagonal matrix      241
Diameter      61
Difference, first      105
Difference, of two sets      2
Differentiability of the Lebesgue integral      183ff
Differentiable function      110 118
Differential      118
Differentiation under integral sign      140
Dini’s theorem      85
Directed function      33 50
Directed set      28 86
Directed set of functions      128
Direction      33 50
Distance function      59

Distributive law      10 13
Domain of a function      8
Dominated Convergence Theorem      140
Double limit      80ff
Downwardly directed set      28 33 128
Egoroff, D. F., Egoroff’s theorem      159
Elementary function (= step-function)      126 135
Elementary integral      128
Equicontinuity      83 92
Equivalence, classes      205
Equivalence, of functions      139
Equivalence, of topologies      40
Essential supremum      203
Essential upper bound      203
Essentially bounded function      203
Exponential function      230
Extended real number system      26
Extension of a function      94
Fatou, P., lemma of      140
Fields      10ff
Fields, complete ordered      20ff
Fields, isomorphism of      18
Fields, ordered      14ff
Finite additivity      174
Finite character      28 30
Finite set      4
First difference      105
Fischer, E., Riesz — Fischer theorem      232
Fourier, J., Fourier coefficients      230
Fourier, J., Fourier series      227ff
Fourier, J., Fourier transform      233ff
Fubini, G., Fubini’s theorem      143ff
Function      3
Function, -      126
Function, -      126
Function, absolutely continuous      102ff 108 177ff
Function, additive      105
Function, antitone      58
Function, Baire      147ff 242
Function, bounded linear      210
Function, characteristic      77
Function, choice      29
Function, composite      67
Function, countably additive      174
Function, determinant      122
Function, differentiate      110
Function, directed      33
Function, distance      59
Function, elementary      126
Function, equicontinuous family of      183
Function, essentially bounded      203
Function, exponential      230
Function, extension of a      94
Function, implicit      121
Function, inhomogeneous-linear      90
Function, inverse      4
Function, isotone      58
Function, linear      90
Function, Lipschitzian      77 78 80 97 103 112 113 210
Function, lower semicontinuous      74
Function, measurable      146ff 161
Function, monomial      72
Function, monotone      58 98ff
Function, nowhere differentiable      118
Function, of bounded variation      99ff
Function, of intervals      105ff
Function, polynomial      72
Function, quadratic      224ff
Function, semicontinuous      74
Function, simple      212
Function, singular      178
Function, spaces      202ff
Function, step      126
Function, strictly antitone      198
Function, strictly isotone      198
Function, subdirected      37 38
Function, summable      133 161
Function, uniformly continuous      78
Function, upper semicontinuous      74
Functional, linear      209ff
Functional, positive linear      216
Functional, quadratic      224
Generated ring, algebra, etc      173
Graph of a function      3
Greatest lower bound      20 28
Group      9
Group, commutative      9
H $\ddot{o}$ lder, O., H $\ddot{o}$ lder condition      78 97
H $\ddot{o}$ lder, O., H $\ddot{o}$ lder inequality      163
Hahn, H., Hahn decomposition      175
Hahn, H., Hahn — Banach theorem      214
Hausdorff, F., Hausdorff maximality principle      30 251ff
Hausdorff, F., Hausdorff space      40 41
Hausdorff, F., normal Hausdorff space      95
Heine — Borel theorem      53
Hermite, C., Hermitian matrix      241
Hermite, C., Hermitian operator      223
Hilbert, D., Hilbert space      219ff
Hilbert, D., operators on      241ff
Homeomorphism      72
Identity element of a group      10
Identity mapping      41
Implicit ftmctions theorem      121ff
Inductive definition      7 254
Inductive proof      6 253
Inequality, Bessel      228
Inequality, Cauchy — Buniakowski — Schwarz      59 162
Inequality, H $\ddot{o}$ lder      163
Inequality, Minkowski      59 164
Inequality, triangle      59
Infimum      20 28
Infinite (countably or denumerably)      5
infinity      26
Infinity, limit of function st      79
Infinity, vanishing at      79
Inhomogeneous linear function      90
Inner produet      219
Integral, Daniell — Stieltjes      133
Integral, defined from measure      160ff
Integral, differentiability of      183ff
Integral, elementary      128 135
Integral, iterated      143
Integral, Lebesgue and Lebesgue — Stieltjes      126ff
Integral, Radon — Stieltjes      133
Integral, Riemann and Riemann — Stielljes      165ff
Integration, by parts      191ff
Integration, of Fourier series      231
Interior      45
Interior, point      44
Intermediate-value theorem      71 72
Intersection      2
Intervals      27
Intervals, closed      27
Intervals, function of      105ff
Intervals, half open      27
Intervals, nonoverlapping      108
Intervals, of continuity      107 135 141
Intervals, open      27 32
Intervals, volume of      108
Inverse element      9
Inverse function      4
Isomorphism      18ff
Isotone function      58
Iterated integral      143
Iterated limit      80 81
Jacobi, C. G. J., Jacobian      196
L-function      126ff
Lattice      28
Lattice, complete      28
Least upper bound      20 28

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