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| Авторизация |
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| Kolmogorov A.N., Fomin S.V. — Introductory real analysis |
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| Предметный указатель |
Functional (s) 108 123
Functional (s), addition of 183
Functional (s), additive 123
Functional (s), bounded linear see “Bounded linear functional”
Functional (s), conjugate-homogeneous 123
Functional (s), conjugate-linear 124
Functional (s), continuous 175
Functional (s), continuous linear see “Continuous linear functionals”
Functional (s), convex 130 134
Functional (s), extension of 132
Functional (s), homogeneous 123
Functional (s), linear 124 175
Functional (s), Minkowski 131
Functional (s), null space of 125
Functional (s), product of, with a number 183
Functional (s), separation of sets by 136
Fundamental functions see “Test functions”
Fundamental parallelepiped 98
Fundamental sequence see “Cauchy sequence”
Fundamental space see “Test space”
General measure theory 269 ff.
Generalized function(s) 124 206
Generalized function(s) and differential equations 211—214
Generalized function(s) of several variables 214—215
Generalized function(s) on the circle 216
Generalized function(s), complex 215
Generalized function(s), convergence of 209
Generalized function(s), definition of 208
Generalized function(s), derivative of 210
Generalized function(s), operations on 209—210
Generalized function(s), product of, with a number 209
Generalized function(s), product of, with an infinitely differentiable function 210
Generalized function(s), regular 208
Generalized function(s), singular 208
Generalized function(s), sum of 209
Goedel, K. 209
Graph 238
Greatest lower bound (in a partially ordered set) 30
Gurevich, B.L. 350 351
Hahn decomposition 345
Hahn — Banach theorem 132 180
Hahn — Banach theorem, complex version of 134 181
Hamel basis 128
Hausdorff axiom of separation 85
Hausdorff space 85
Hausdorff’s maximal principle 28
Heine — Borel theorem 92
Helly’s convergence theorem 370
Helly’s selection principle 372
Hereditary property 87
Hilbert cube 98
Hilbert space(s) 155 ff.
Hilbert space(s), complex 165
Hilbert space(s), countably 173
Hilbert space(s), isomorphic 155 165
Hilbert space(s), linear manifold in 156
Hilbert space(s), linear manifold in, closed 156
Hilbert space(s), subspace(s) of 156
Hilbert space(s), subspace(s) of, (mutually) orthogonal 158
Hilbert space(s), subspace(s) of, direct sum of orthogonal 159
Hilbert space(s), subspace(s) of, orthogonal complement of 157
Hilbert — Schmidt theorem 248
Hilbert, D. 155
Hoelder’s inequality 41
Hoelder’s inequality, homogeneity of 42
Hoelder’s integral inequality 45
Homeomorphic mapping 44 89
Homeomorphic spaces 44 89
Homeomorphism 44 89
Hyperplane 127
Ideal, two-sided 252
Image of a set 5
Image of an element 5
Infimum 51
Infinite set 10
Initial section 25
Inner measure 258 276
Integrable function 294 296 308
Integral part 8
Interior 128
Interior point 50
Intersection of sets 2
Into mapping 5
Invariant subspace 238
Inverse function 5
Invisible point from the left 319
Invisible point from the right 319
Isolated point 47
Isometry 44
Isomorphism 21 120 155 165
Isomorphism theorem 155 165
Isomorphism, conjugate-linear 194 234
Jordan decomposition 346
Jordan extension 281
Jordan measurable set 281
Jordan measure 281
Jump 315
Jump function 315 341
Kelley, J.L. 87 90 92 97
Kernel 74
Lattice 30
Least upper bound (in a partially ordered set) 30
Lebesgue decomposition 341 351 363
Lebesgue extension 277 279
Lebesgue integral 293 ff.
Lebesgue integral as a set function 343—351
Lebesgue integral of a general measurable function 296 308
Lebesgue integral of a simple function 294
Lebesgue integral, absolute continuity of 300—301
Lebesgue integral, indefinite 313 ff.
Lebesgue integral, over a set of infinite measure 308
Lebesgue integral, vs. Riemann integral 293—294 309—310
Lebesgue — Stieltjes integral 364
Lebesgue — Stieltjes integral, vs. Riemann — Stieltjes integral 368
Lebesgue-integrable function see “Integrable function”
Lebesgue’s bounded convergence theorem 303
Lebesgue’s theorem on differentiation of a monotonic function 321
Lebesgue’s theorem on integration of the derivative of an absolutely continuous function 340
Left-hand limit 315
Levi’s theorem 305
Limit of a sequence in a metric space 47
Limit of a sequence in a topological space 84
Limit point 47 79
Limit point, complete 97
Linear closure 140
linear combination 120
Linear dependence 120
Linear functional 175 ff.
Linear functional, bounded see “Bounded linear functional”
Linear functional, continuous see “Continuous linear functionals”
Linear hull 122
Linear independence 121
Linear manifold 140 156
Linear operator 221
Linear operator, bounded 223
Linear operator, bounded, norm of 224
Linear operator, bounded, spectral radius of 239
Linear operator, closed 237
Linear operator, completely continuous see “Completely continuous operators”
Linear operator, graph of 238
Linear space(s) 118 ff.
Linear space(s), basis in 121
Linear space(s), basis in, Hamel 128
Linear space(s), closed segment in 128
Linear space(s), complex 119
Linear space(s), countably normed 171
Linear space(s), dimension of 121
Linear space(s), dimension of, algebraic 128
Linear space(s), finite-dimensional 121
Linear space(s), functionals on see “Functional”
| Linear space(s), infinite-dimensional 121
Linear space(s), isomorphic 120
Linear space(s), linearly dependent elements of 120
Linear space(s), linearly independent elements of 121
Linear space(s), n-dimensional 121
Linear space(s), normed see “Normed linear spaces”
Linear space(s), open segment in 128
Linear space(s), real 119
Linear space(s), subspace 121
Linear space(s), subspace, proper 121
Linear space(s), topological see “Topological linear space”
Linearly ordered set see “Ordered set”
Lipschitz condition 55
Locally integrable function 208
Lower limit 111
Lower semicontinuous function 110
Luzin’s theorem 293
Mapping 5 ff.
Mapping, continuous 44 87
Mapping, contraction 66
Mapping, fixed point of 66
Mapping, into 5
Mapping, natural 191
mapping, one-to-one 5
Mapping, onto 5
Mapping, order-preserving 21
Mathematical expectation 366
Mathematical induction 28
Mean (value) 366
Mean square deviation 384
Measurable function 284 ff.
Measurable function, integration of 294 296 308
Measurable set(s) 259 ff 267
Measurable set(s), decreasing sequence of 266
Measurable set(s), increasing sequence of 267
Measurable set(s), Jordan 281
Measure space 294
Measure(s) 254 ff.
Measure(s) of a plane set 259 276
Measure(s) of a rectangle 255
Measure(s) of an elementary set 256
Measure(s) on a semiring 270
Measure(s),  -finite 308
Measure(s), additivity of 255 263
Measure(s), complete 280
Measure(s), continuity of 267
Measure(s), countably  additive 266 272
Measure(s), direct product of 354
Measure(s), extension(s) of 271 275—283
Measure(s), inner 258 276
Measure(s), Jordan 281
Measure(s), Lebesgue 259 276 279
Measure(s), outer 258 276
Measure(s), product 354
Measure(s), signed 344
Measure(s), Stieltjes see “Stieltjes measure”
Measure(s), with a countable base 382
Method of successive approximations 66 67
Metric see “Distance”
Metric space(s) 37 ff.
Metric space(s), complete 56
Metric space(s), completion of 62
Metric space(s), continuous curves in 112—113
Metric space(s), continuous curves in, length of 114 115
Metric space(s), continuous curves in, sequence of 115
Metric space(s), continuous mapping of 44
Metric space(s), convergence in 47
Metric space(s), incomplete 56
Metric space(s), isometric 44
Metric space(s), isometric mapping of 44
Metric space(s), real functions on 108
Metric space(s), real functions on, equivalent continuous 113
Metric space(s), real functions on, uniformly continuous 109
Metric space(s), relatively compact subsets of 101
Metric space(s), separable 48
Metric space(s), subspace of 43
Metric space(s), total boundedness of 97—99
Metric space(s), total boundedness of, compactness and 99—101
Metrizable space 90
Minkowski functional 131
Minkowski’s inequality 41
Minkowski’s integral inequality 45
Monotonic function 314
n-dimensional (vector) space 119
n-dimensional simplex 137
n-dimensional simplex, k-dimensional face of 137
n-dimensional simplex, vertices of 137
n-Space 119
Negative set 344
Neighborhood 46 79
Neighborhood base 83
Neighborhood base, at zero 168
Nested sphere theorem 60
Noncomparable elements 21
Nondecreasing function 314
Nonincreasing function 314
Nonmeasurable set 268
Norm(s) 138 142 163
Norm(s) of a bounded linear functional 177
Norm(s) of a bounded linear operator 224
Norm(s), comparable 172
Norm(s), compatible 171
Norm(s), equivalent 141 172
Norm(s), properties of 138
Norm(s), stronger 172
Norm(s), weaker 172
Normal space 86
Normed linear space(s) 138
Normed linear space(s), bounded subset of 141
Normed linear space(s), complete 140
Normed linear space(s), complete set in 140
Normed linear space(s), conjugate space of 184
Normed linear space(s), direct product of 238
Normed linear space(s), subspaces of 140
Null space 125
One-to-one correspondence 5 10 13
One-to-one function 5
Onto mapping 5
Open ball see “Open sphere”
Open set(s) 50
Open set(s) in a topological space 78
Open set(s) on the real line 51
Open set(s), component of 55
Open set(s), unions and intersections of 50
Open sphere 45
Open sphere, center of 46
Open sphere, radius of 46
Operator(s) 221 ff.
Operator(s), adjoint 232
Operator(s), adjoint, in Hilbert space 234
Operator(s), continuous 221
Operator(s), degenerate 240
Operator(s), domain (of definition) of 221
Operator(s), eigenvalue of 235
Operator(s), eigenvector of 235
Operator(s), identity (or unit) 222
Operator(s), inverse 228
Operator(s), invertible 228
Operator(s), linear see “Linear operator”
Operator(s), product of 225
Operator(s), projection 223
Operator(s), resolvent of 236
Operator(s), self-adjoint 235
Operator(s), spectrum of 235
Operator(s), sum of 225’
Operator(s), with a number 225
Operator(s), zero 222
Order type see “Type”
Order-preserving mapping 21
Ordered product 23
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