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| Bartle R.G. — The Elements of Real Analysis |
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| Предметный указатель |
Abel summability 416
Abel's lemma on partial summation 397
Abel's test for convergence 398
Abel's test for uniform convergence 407
Abel's theorem 415
Abel, N.H. 397
Absolute convergence of a series 378
Absolute convergence of an integral 351
Absolute value in an ordered field 38
Absolute value of a complex number 96
Absolute value of a function 153
Accumulation point 73
Additive function 158
Alternating series 399
Angle between two vectors 62
Appell, P. 419
Approximation theorems 177—186
Archimedean field 40
Archimedes 40
Arithmetic mean 67 272
Arzela — Ascoli theorem 191
Arzela, C. 191
Ascoli, G. 191
Axiom of Choice 26
Baire's theorem 90
Baire, R. 90
Ball in a Cartesian space 64
Bernoulli's inequality 44
Bernoulli, J. 44
Bernstein's approximation theorem 182
Bernstein's theorem 414
Bernstein, S.N. 180
Bessel, F.W. 213
beta function 371
Bilinearity of the Riemann — Stieltjes integral 280
Binary operation 28
Binominal expansion 419
Bolzano intermediate value theorem 162
Bolzano — Weierstrass theorem for sequences 115
Bolzano — Weierstrass theorem for sets 76
Bolzano, B. 75
Bonnet, O. 305
Borel, E. 85
Bound, lower 47
Bound, upper 47
Boundary point 83 324
Bounded convergence theorem 288
Bounded set in a Cartesian space 75
Bounded variation 283
Bunyakovskii, V. 61
Cantor intersection theorem 88
Cantor set 51
Cantor, G. 25
Cartesian product 9
Cartesian spaces 59—69
Category 90
Cauchy condensation test 386
Cauchy convergence criteria 117 130 140 279 321 345 352 378 407
Cauchy mean value theorem 210
Cauchy principal value 343 345
Cauchy product 384
Cauchy root test 388
Cauchy sequences 115 121
Cauchy — Bunyakovskii — Schwarz inequality 61—62 297
Cauchy — Hadamard theorem 410
Cauchy, A.L. 61
Cesaro's method of summation 137
Cesaro's theorem 387
Cesaro, E. 137
Chain rule 235
Change of variable 305
Characteristic zero 32
Chebyshev's inequality 69
Chebyshev, P.L. 69
Choice, axiom of 26
Circumscribing contour theorem 90
class 1
Class C' 250
Closed set 71—72
Cluster point 73
Collection 1
Compact set 84
Compactness, preservation of 163
Comparison tests 346—347 388
Complete Archimedean field 45 121
Complex number system 94—97
Components of a vector 59
Conditional convergence 387
Conjugate of a complex number 94
Connected set 77
Connectedness, preservation of 162
Constraint 266
Content of a set 325
Content of an interval 316
Content, zero 317
Continuity 146—175 especially
Continuity of the inverse function 166
Continuity, one-sided 278
Continuity, uniform 166
contour 90
Contraction 169—170
Convergence in a metric space 110
Convergence of a sequence 100
Convergence of a sequence of functions 122
Convergence, absolute 351 378
Convergence, interval of 409
Convergence, radius of 409
Convergence, uniform 126 352 406—407 411
Convex function 224
Convex set 66
Coordinates of a vector 59
Correspondence 24
Countable set 23
Covering 84
Critical point 262
Curve, polygonal 80
Curve, space-filling 319
Cut 50
D'Alembert, J. 390
Darboux's theorem 219
Darboux, G. 211
de Moivre, A. 300
De Morgan's laws 9 11
De Morgan, A. 8
Dedekind, R. 50
Density of the rational elements 41
Denumerable set 23
Derivative 206—249 especially 228
Derivative, directional 225
Derivative, partial 226
Descartes, R. 9
Diagonal method 25—26 193
Difference of two functions 153
Difference of two sequences 99
Differentiable functions 227
Differentiation theorem for integrals 301
Differentiation theorem for power series 411
Dini, U. 194
Direct image 19
Directional derivative 225
Dirichlet's discontinuous function 150
Dirichlet, P.G.L. 150
Diriehlet's test for convergence 347 397
Diriehlet's test for uniform convergence 353 407
Discrete metric 67
Disjoint sets 5
Divergence of a sequence 100
Domain of a function 13
Dominated Convergence Theorem 359
Dot product 61
| Double limit 139
Double sequences 139 ff.
Double series 381 ff.
Element of a set 1
Equicontinuity 190
Euler, L. 248
Exponential function 56 174 220 420
Extension of a continuous function 187
Extreme point 66
Fejer, L. 138
Field 28—34 especially
Field, Archimedean 40
Field, ordered 34—45 especially
figure 337
Finite set 23
First mean value theorem 301 303
Fixed points 170—172
Flyswatter principle 109
Fresnel integral 349
Fresnel, A. 349
Frobenius, G. 420
Function 11—22 especially
Function, absolute value of 38
Function, additive 158
Function, Beta 371
Function, bounded variation 283
Function, Class C' 250
Function, composition of 16
Function, continuous 146
Function, convex 224
Function, derivative of 207 228
Function, differentiable 227
Function, direct image of 19
Function, domain of 13
Function, exponential 56 174 220 420
Function, Gamma 350 371
Function, harmonic 271
Function, homogeneous 248
Function, hyperbolic 223
Function, inverse 17—18
Function, inverse image of 20
Function, Laplace transform of 372
Function, linear 154
Function, logarithm 56 221 314
Function, non-differentiable 208
Function, piecewise linear 178
Function, polynomial 186
Function, positively homogeneous 248
Function, range of 13
Function, semi-continuous 202
Function, step 177
Function, trigonometric 222 315 420
Fundamental theorem of algebra 96
Fundamental theorem of integral calculus 302
Gamma function 350 371
Gauss, C.F. 96
Geometric mean 67 272
Geometric series 378
Global continuity theorem 160
Gradient 247
Hadamard, J. 410
Half-closed interval 39
Half-open interval 39
Hardy, G.H. 344
Harmonic function 271
Harmonic series 119 379
Heine — Borel theorem 84—94
Heine, E. 85
Hoelder's inequality 68—69 215 272
Hoelder, O. 68
Homogeneous function 248
Hyperbolic function 223
Hypergeometric series 403
Identity element of a field 28
Imaginary part of a complex number 94
Implicit function theorem 260 273
Improper integrals 341 ff.
Inequalities, basic properties of 34—37
Inequality, arithmetic-geometric 67 272
Inequality, Bernoulli 44
Inequality, C.—B.—S. (Cauchy — Bunyakovskii — Schwarz) 61—62 297
Inequality, Chebyshev 69
Inequality, Hoelder 68—69 215 272
Inequality, Minkowski 69 273
Inequality, triangle 38 64
Infimum 47
Infinite integral 344 ff.
Infinite product 403—405.
Infinite series 375—421
Infinite sets 23
Inner product 61 297
Integrability theorems 283 323—324
integral 274—374
Integral test, for series 393
Integral, infinite 344 ff.
Integral, iterated 328 ff.
Integral, lower 298—299
Integral, partial 343
Integral, transformation of 331 ff.
Integral, upper 298—299
integrand 277
Integration, by parts 282 304
Integrator 277
Interchange theorems, relating to continuity 175—176 306 355 406
Interchange theorems, relating to differentiation 217 243 307 356 406 411
Interchange theorems, relating to infinite integrals 355—356 358—367
Interchange theorems, relating to integration 285—289 306—308 355 406
Interchange theorems, relating to sequences 175—176 217 285—289 358—360
Interchange theorems, relating to series 406 411
Interior maximum 209
Interior point 73
Intermediate Value Theorem 162
Intersection of sets 4 7
Interval of convergence of a series 409
Interval, in a Cartesian space 74
Interval, in an ordered field 39
Interval, unit 4
Inverse function 17—19 166
Inverse image 20
Inversion mapping 97
Inversion theorems 252 256
Irrational elements of a field 33
Irrational powers of a real number 56
Isomorphism 46
Iterated integrals 328 ff.
Iterated limits 140 143
Iterated suprema 54
Jacobi, C.G.J. 232
Jacobian determinant 232
Jacobian theorem 334
Jerrard's series 378
Kronecker, L. 46
l'Hospital, G.F. 215
Lagrange identity 68
Lagrange method 267
Lagrange multiplier 266
Lagrange, J.-L. 68
Landau, E. 135
Laplace transform 372
Laplace, P.-S. 372
Least squares 271
Lebesgue covering theorem 89
Lebesgue integral 274
Lebesgue number 89
Lebesgue, H. 89
Leibniz's alternating series test 399
Leibniz's formula 230
Leibniz, G.W. 230
Length of a vector 61
Limit of a double sequence 139
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