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Авторизация |
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Hewitt E., Stromberg K. — Real and abstract analysis: a modern treatment of the theory of functions of a real variable |
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Предметный указатель |
Diameter of a set 67
Differentiability of monotone functions 264
Differentiable function 257
Differentiation on a net 373
Differentiation theorem, Lebesgue’s 264
Differentiation, term by term 267
Dimension of , orthogonal 255
Dimension of a Hilbert space, orthogonal 247
Dimension of a vector space, algebraic 31
Dimension of a vector space, linear 31
Dimension, Hausdorff 145
Dini derivates 257
Dini’s theorem 205
Dirac measure 120
Dirichlet kernel 292
Discontinuities of a function, set of 78
Discontinuous measures, purely 334
Discrete metric 59
Discrete topology 56
Disjoint sets 3
Dissection, measurable 164
Distance between two sets 78
Distance from a point to a set 77
Distance function 59
Divergent Fourier series 300
Domain of a relation 7
Dominated convergence theorem, Lebesgue’s 172 174 185
Doob, J. L. 373
Dual space 211
Dunford, N. 210
Egorov’s theorem 158
Endpoints 54
Enumeration of a set 22
Equivalence relation 8
Equivalent sets 19
Essential supremum 346
Essentially bounded functions 346
Euclidean metric 59
Euclidean n-space 13
Evaluation functional 114
EXP 51
Expansions of real numbers 46 47
Exponential function 51
Extended real numbers 54
Extending a measure 162
Extension of a relation 8
Extension of Lebesgue measure, invariant 359
Extension of the Riemann integral, Lebesgue integral as an 184
Extension theorem, Hahn — Banach 212
Extension theorem, Hopf’s 142
Extension theorem, Kren’s 220
Extension theorem, Tietze’s 99
Extensions of Lebesgue measure 137 359
Extensions of measures 141 148
Family of finite character 13
Family, monotone 380
Fatou’s Lemma 124 172
Fejr — Lebesgue theorem 294
Fejr, L. 292
Fejr’s kernel 292 408
Field 34
Field of complex numbers 48
Field, Archimedean ordered 37
Field, automorphisms of the real 48
Field, characteristic of a 34
Field, complete Archimedean ordered 44
Field, completion of an ordered 38
Field, non — Archimedean ordered 46
Field, ordered 35
Field, real number 46
Filter 358
Filter, fixed 358
Filter, free 358
Finite character, family of 13
Finite intersection property 62
Finite measure space 127
Finite set 21
Finite variation, function of 266
Finitely additive measures 126 354
Finitely additive measures, integral for 355
Finitely additive measures, regular 364
Finitely additive signed measure, Jordan decomposition of 338
Finjte measure 127
First category 68
First mean value theorem for integrals 420
Fixed filter 358
Fixed-point theorem, Banach’s 78
Flett, T. M. 428
Fourier coefficients 237 249
Fourier integrals, summability of 404
Fourier series 245 291
Fourier series, Abel summability of 301
Fourier series, arithmetic means for a 292
Fourier series, Cesro summability for 293
Fourier series, divergent 300
Fourier series, uniform summability of 300
Fourier transform 249 401
Fourier transform on 411
Fourier transform, inversion of 409
Fourier transform, uniqueness theorem for the 408
Frchet compact space 62
Free filter 358
Fubini, G. 267
Fubini’s Theorem 384 386
Function 9
Function of bounded variation 266
Function of finite variation 266
Function, absolutely continuous 282
Function, additive on 49
Function, Baire 163
Function, Banach indicatrix of a 271
Function, Borel measurable 149 187
Function, bounded 83
Function, choice 12
Function, constant 82
Function, continuous 73
Function, continuous, that vanishes at infinity 86
Function, continuous, that vanishes in a neighborhood of infinity 86
Function, continuous, with compact support 86
Function, convex 202 271
Function, differentiable 257
Function, distance 59
Function, exponential 51
Function, indefinite integral of a 272
Function, integrable 173
Function, Lebesgue measurable 149
Function, Lebesgue points for a 278
Function, Lebesgue set for a 278
Function, Lebesgue’s singular 113
Function, locally integrable 186
Function, locally null 123
Function, lower semicontinuous 88
Function, measurable 149 154
Function, monotone 105
Function, nondecreasing 105
Function, nonincreasing 105
Function, normalized nondecreasing 112 330
Function, null 123
Function, onto 10
Function, oscillation 78
Function, reflection of a 110
Function, Riemann — Stieltjes integrable 106
Function, section of a 379
Function, set of discontinuities of a 78
Function, simple 159
Function, step 198
Function, strictly decreasing 105
Function, strictly increasing 105
Function, summable 173
Function, symmetric derivatives of a 271
| Function, total variation of 266 270 272
Function, translate of a 110
Function, uniformly continuous 87
Function, upper semicontinuous 88
Function, value of a 9
Functional, evaluation 114
Functional, linear 211
Functional, multiplicative linear 114 365
Functional, nonnegative linear 114 220
Functional, sublinear 212
Functionals on , multiplicative linear 417
Functionals on Hilbert spaces, bounded linear 248 255
Functions, 288 303
Functions, convolution of 396
Functions, differentiability of monotone 264
Functions, Hermite 243
Functions, integral representation of convex 300
Functions, lattice of 82
Functions, normalized Hermite 245
Functions, pointwise limits of continuous 79
Functions, separating family of 93
Functions, singular 278
Functions, uniformly integrable sequences of 204
Fundamental theorem of the integral calculus for Lebesgue integrals 285
Gauss’s kernel 408
Generalization of monotone convergence theorem 118
Generalized B. Levi theorem 118
Generalized Hlder’s inequality 200
Generated by a family, -algebra 132
Geometric interpretation of the complex field 50
Gram — Schmidt orthonormalization process 240
Greatest lower bound in an ordered field 44
Group 32
Group algebra of 399
Hlder’s inequality for 191
Hlder’s inequality for 190
Hlder’s inequality, generalized 200
Hahn decomposition 305
Hahn decomposition theorem 306
Hahn — Banach theorem 212
Half-open intervals 54
Halmos, P. 1
Hamel basis 18
Hamilton, Sir W. R. 102
Hardy — Littlewood maximal theorem 424 425 426
Hardy, G. H. 258 422
Hausdorff dimension 145
Hausdorff maximality principle 14
Hausdorff measure 145
Hausdorff space 56
Heine — Borel theorem 66
Heine — Borel — Bolzano — Weierstrass theorem 66
Hermite functions 243
Hermite functions, completeness of the 416
Hermite functions, normalized 245
Hermite polynomials 243
Hewitt, E. 137 250 378 445
Hewitt, M. 32
Hilbert parallelotope 254
Hilbert space 196 235
Hilbert space, bounded linear functionals on a 248 255
Hilbert space, orthogonal dimension of a 247
Hilbert space, pre 195
Hilbert space, projections in a 253
Hirschman Jr., I. I. 250
Holladay, J. C. 103
Hopf’s extension theorem 142
Huntington, E. V. 5
Ideals in 365
Ideals in 417
Ideals in a ring 33
Identity, Parseval’s 245 246 250
Identity, polar 235
Image sets, measures on 161
Images of measures, continuous 180
Imaginary part of a complex number 48
Increasing function, strictly 105
Indefinite integral of a function 272
Indefinite integral, derivative of an 275
Independence, linear 17
Indicatrix of a function, Banach 271
Indiscrete topology 56
Induction, transfinite 17
Inequalities, Clarkson’s 225 227
Inequalities, Hlder’s 190 191
Inequalities, Jensen’s 202
Inequality, Bessel’s 239
Inequality, Bunyakovski’s 190
Inequality, Cauchy — Bunyakovski — Schwarz 234
Inequality, Cauchy’s 190
Inequality, generalized Hlder’s 200
Inequality, Minkowski’s 191
Inequality, Schwarz’s 190 234
Inequality, Young’s 189
Infimum in 69
Infimum in an ordered field 44
Infinite product measures 432
Infinite product measures, absolute continuity of 453
Infinite product measures, singular measures induced by 456
Infinite product measures, singularity of 453
Infinite product spaces, iterated integrals on 439
Infinite products of measure spaces 429
Infinite set 21
Infinite set, countably 22
Initial segment 28
Initial segment of a well ordered set 16
Inner product space 195 234
Integers 2
Integers, positive 2
Integrability, Riemann 105 183
Integrable function 173
Integrable function, locally 186
Integrable function, Riemann — Stieltjes 106
Integral as a set function 175
Integral of a function, indefinite 272
Integral relative to a complex measure 312
Integral representation of convex functions 300
Integral, absolute continuity of the 176
Integral, Denjoy 299
Integral, derivative of an indefinite 275
Integral, iterated 385
Integral, Lebesgue 164
Integral, Lebesgue integral as an extension of the Riemann 184
Integral, linearity of the 169 171 174
Integral, Riemann 107
Integral, Riemann — Stieltjes 106
Integrals as areas 392
Integrals for finitely additive measures 355
Integrals on infinite product spaces, iterated 439
Integrals on the completion of a measure space 186
Integrals, change of variable in 342
Integrals, first mean value theorem for 420
Integrals, fundamental theorem of the integral calculus for Lebesgue 285
Integrals, integration by parts for Lebesgue — Stieltjes 419
Integrals, second mean value theorem for 420
Integrals, summability of Fourier 404
Integration by parts 287
Integration by parts for Lebesgue — Stieltjes integrals 419
Integration by substitution 342
Integration, term by term 171 175
Interior of a set 56
Intersections of sets 2
Interval, bounded 54
Interval, closed 54
Interval, open 54
Interval, subdivision of an 105
Interval, unbounded 54
Intervals, component 69
Intervals, Half-open 54
Invariant extension of Lebesgue measure 359
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