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| Shilov G.E. — An introduction to the theory of linear spaces |
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| Предметный указатель |
Algebra of linear operators 90—96
Algebra of linear operators, left ideal of 90
Algebra of linear operators, null ideal of 95
Algebra of linear operators, right ideal of 90
Algebra of linear operators, two-sided ideal of 95
Algebraic complement see "Cofactor"
Annihilating polynomial 77 107 179 180
Basis minor theorem 22 51
Bender, C.D. 288
Bessel's inequality 158 264
Bilinear form(s) 113—134
Bilinear form(s), canonical basis for 125
Bilinear form(s), conjugate vectors of 125
Bilinear form(s), definition of 113
Bilinear form(s), general representation of 114
Bilinear form(s), matrix of 115
Bilinear form(s), matrix of, rank of 116
Bilinear form(s), matrix of, transformation of 115
Bilinear form(s), negative definite 132
Bilinear form(s), negative index of inertia of 127
Bilinear form(s), norm of 154
Bilinear form(s), positive definite 131—132
Bilinear form(s), positive index of inertia of 127
Bilinear form(s), rank of 127
Bilinear form(s), symmetric 115 126 195
Bilinear functional 114
Bolzano — Weierstrass theorem 272
Boron, L.F. 145 256
Bounded set 138
Brand, L. 248 266 272
Cauchy criterion 249ff.
Characteristic polynomials 106 184
Closed set 247
Closure of a set 247
Cofactor of a minor of a determinant 20
Cofactor of an element of & determinant 10
Completely continuous operators 271ff.
Completely continuous operators, convergence of 274
Completely continuous operators, definition of 273
Completely continuous operators, symmetric 275
Completely continuous operators, symmetric, eigenvalues and eigenvectors of 275—278
Completely continuous operators, symmetric, maximal vectors of 275
Conditional extrema 132
Conformal transformations 217
Conformal transformations, Liouvilie's theorem on 217
Convergence in the mean 246
Coordinate curves, net of 212
Coordinate transformations 97—112
Coordinate transformations, consecutive 101
Coordinate transformations, isometric 150
Coterminal fundamental sequences 253—256
Coterminal fundamental sequences, operations on classes of 253—254
Courant, R. 199 288
Cramer's rule 15—18 55—56
Curvature 212
Curvature, lines of 216—217
Curvature, normal 213
Curvature, principal 214
Determinant(s) 4—26
Determinant(s) with identical columns (rows) 7
Determinant(s), addition of columns (rows) of 9
Determinant(s), antisymmetry property of 7
Determinant(s), cofactors of see "Cofactors"
Determinant(s), definition of 4
Determinant(s), elements of 4
Determinant(s), evaluation of 12—15
Determinant(s), expansion with respect to columns (rows) of 10
Determinant(s), linear dependence between columns (rows) of 21—26
Determinant(s), linear property of 8
Determinant(s), quasi-triangular 20
Determinant(s), sign rule for 4 6
Determinant(s), terms of 4
Determinant(s), transpose of 6
Determinant(s), triangular 12
Determinant(s), Vandermonde 15
Directions, principal 214
Directions, principal, invariant (eigenrays) 187
Dirichlet function 257
Dirichlet problem 300
Dirichlet problem, exterior 300
Dirichlet problem, interior 300
Dupin indicatrix 213
Elementary operations 60—61
Ellipsoid 226 236
Energy of a system, kinetic 218
Energy of a system, potential 218
Euclidean spaces 135ff.
Euclidean spaces, definition of 136
Euclidean spaces, infinite-dimensional 244—303
Euclidean spaces, isomorphism of 143—144
Euclidean spaces, orthogonal and orthonormal bases in 142—143
Euler's formula 214
Faguet, M.K. 174
Fomin, S.V. 145
Fourier coefficients 143 263
Fredholm alternative 288—298
Fredholm integral see "Fredholm integral operator"
Fredholm integral operator 68 182 192 271
Fredholm integral operator, complete continuity of 274
Fredholm integral operator, eigenvalues and eigenvectors of 278—280 286—288
Fredholm integral operator, kernel of 68 272
Fredholm integral operator, kernel of, degenerate 275 286 291
Fredholm integral, general 96
Fredholm integral, general form of, in n dimensions 69—73
Fredholm integral, inverse 82—86
Fredholm integral, isogonal 151 171
Fredholm integral, isometric 148 163 171 180—181 191
Fredholm integral, matrix of 69
Fredholm integral, matrix of, determinant of 105
Fredholm integral, matrix of, transformation of 104
Fredholm integral, maximal vectors of 145 188—189
Fredholm integral, norm of 144—147 191 271
Fredholm integral, orthogonal 147—151
Fredholm integral, powers of 74—75
Fredholm integral, products of 73—74
Fredholm integral, projection 67 178 182
Fredholm integral, rank of 87
Fredholm integral, rotation 67 178 181
Fredholm integral, self-adjoint 153
Fredholm integral, similarity 67
Fredholm integral, square root of 191
Fredholm integral, symmetric see "Symmetric operators"
Fredholm integral, trace of 106
Fredholm integral, unit (identity) 67
Fredholm integral, zero 67
Fubini's theorem 260 278
Generalized coordinates 218
Generalized velocities 218
Gram determinant 167ff.
Green's formulas 299
Green's function 284
Hadamard's inequality 173—175
Harmonic function 298
Hilbert, D. 279 288
Homeomorphic figures 228
Homotopic figures 228
Hyperboloid, of one sheet 226 234 235
Hyperboloid, of two sheets 226
Hyperparallelepipeds, volume of 169ff.
Hyperplanes 45—47 59
Hyperplanes, dimension of 46
Improper Riemann integral 257
Inclusion relations 29
Infinite-dimensional Euclidean space(s) 244—303
Infinite-dimensional Euclidean space(s), basis in 263
Infinite-dimensional Euclidean space(s), compact sets of points in 272—273
Infinite-dimensional Euclidean space(s), complete 249—252
Infinite-dimensional Euclidean space(s), complete systems of orthonormal vectors in 263
Infinite-dimensional Euclidean space(s), complete, definition of 250
Infinite-dimensional Euclidean space(s), completion(s) of 253—256
| Infinite-dimensional Euclidean space(s), completion(s) of, isomorphism of 255
Infinite-dimensional Euclidean space(s), convergence of points (vectors) in 245
Infinite-dimensional Euclidean space(s), dense sets of elements of 248
Infinite-dimensional Euclidean space(s), fundamental sequences in 249ff.
Infinite-dimensional Euclidean space(s), fundamental sequences in, coterminal 253
Infinite-dimensional Euclidean space(s), orthogonal complements in 260—262
Infinite-dimensional Euclidean space(s), orthogonal expansions in 262—271
Infinite-dimensional Euclidean space(s), separable 269
Infinite-dimensional Euclidean space(s), separable, complete 269
Infinite-dimensional Euclidean space(s), unit sphere in 273
Integral equations with nonsymmetric kernels 288—298
Integral equations, inhomogeneous 281—283
Inversions 3 81
Jacobi's method 127—130
k-vectors 172
k-vectors, angles between 172
k-vectors, equality of 172
k-vectors, scalar product of 172
Kantorovich, L.V. 288
Kolmogorov, A.N. 145
Krasnosyelski, M.A. 163 171 177
Krein, M.G. 163 203
Kronecker — Capelli theorem 55
Krylov, V.I. 288
Lagrange's equations 218
Lagrange's method 199
Laplace's theorem 20
Least squares, method of 175—177
Lebesgue integrable function 257
Lebesgue integral 257ff.
Legendre polynomials 163—167 194 268
Legendre polynomials, Rodrigues' formula for 164
Leibniz' formula 166
Levitan, B.M. 286
Limit point 246
Linear forms 65—67
Linear forms, transformation of 103
Linear functionals 66
Linear manifolds, basis for 44
Linear manifolds, definition of 43
Linear manifolds, dimension of 45 51
Linear operator(s) 67ff.
Linear operator(s), addition of 73
Linear operator(s), adjoint of 153
Linear operator(s), bounded 271
Linear operator(s), commuting 183
Linear operator(s), completely continuous see "Completely continuous operators"
Linear operator(s), continuous 271
Linear operator(s), diagonal 65 178 182
Linear operator(s), differentiation 68 182
Linear operator(s), eigenspaces (characteristic subspaces) of 183ff.
Linear operator(s), eigenspaces (characteristic subspaces) of, dimension of 186
Linear operator(s), eigenvalues (characteristic values) of 181ff.
Linear operator(s), eigenvectors of 181ff.
Linear operator(s), equality of 73
Linear space(s) 27ff.
Linear space(s), basis of 36
Linear space(s), definition of 10
Linear space(s), dimension of 38
Linear space(s), infinite-dimensional 38
Linear space(s), infinite-dimensional, Euclidean see "Infinite-dimensional Euclidean space(s)"
Linear space(s), inverse of an element of 30—31
Linear space(s), isomorphism of 47
Matrices, adjugate 85
Matrices, augmented 54
Matrices, basis columns (rows) of 51
Matrices, coefficient 15
Matrices, determinant of 4
Matrices, diagonal 71
Matrices, elements of 3
Matrices, inverse 82—86
Matrices, minors of 11
Matrices, order of 3
Matrices, orthogonal 147—151
Matrices, products of 78ff.
Matrices, products of, determinant of 80—82
Matrices, products of, rank of 88—90
Matrices, rank of 22 50—54
Matrices, rank of, calculation of 61—64
Matrices, singular (degenerate) 83
Matrices, sums of 76
Matrices, transposed 79—80
McShane, E.J. 199
Mean square deviation 175—177
Measurable function 256
Measure 256
minimax 201
Minor(s) 11 18 50
Minor(s), basis 22 51
Minor(s), basis, calculation of 60—64
Minor(s), bordered 237
Minor(s), cofactor (algebraic complement) of 20
Minor(s), complementary 18
Minor(s), principal 106
Minor(s), principal, descending 128
Minor(s), product of 19
Modenov, P.S. 243
Multilinear forms 133—134
Multilinear forms, antisymmetric 133—134
Multilinear forms, symmetric 133
Natanson, I.P. 256 259
Natural (resonant) frequencies 219
Neumann problem 300
Neumann problem, exterior 300
Neumann problem, interior 300
Normal sections 213
Normal sections, curvature of 211—217
Null space of an operator 87
Null space of an operator, dimension of 87
One-to-one correspondence 47
Operators see "Linear operators"
Orthogonalization 156—177
Orthogonalization theorem 160
Orthonormal basis 142
Osculating plane 211
Paraboloids 230—233
Paraboloids, circular 231 235
Paraboloids, elliptic 231
Paraboloids, hyperbolic 232 235
Parallelogram lemma 261
Perpendiculars 156—160
Perpendiculars, dropped onto a hyperplane 159
Perpendiculars, dropped onto a subspace 156
Petrovsky, I.G. 284
Potential theory 298—303
Principal normal 211
Pythagorean Theorem 141 157 265
Quadratic form(s) 116—134 195—220
Quadratic form(s) in a Euclidean space 195—220
Quadratic form(s) on subspaces 201—207
Quadratic form(s), canonical basis for 118
Quadratic form(s), canonical coefficients of 119
Quadratic form(s), canonical form of 119
Quadratic form(s), comparable 207
Quadratic form(s), definition of 117
Quadratic form(s), extremal properties of 199—201
Quadratic form(s), index of inertia of 124
Quadratic form(s), index of inertia of, negative 124
Quadratic form(s), index of inertia of, positive 124
Quadratic form(s), invariants of 122
Quadratic form(s), law of inertia for 122—125
Quadratic form(s), nonsingular 125
Quadratic form(s), positive definite 125 130—132
Quadratic form(s), rank of 124
Quadratic form(s), reduction of, to canonical form 118ff.
Quadratic form(s), reduction of, to canonical form, uniqueness questions for 198—199
Quadratic form(s), simultaneous reduction of two 207—211
Quadratic form(s), simultaneous reduction of two, uniqueness questions for 210—211
Quadratic form(s), stationary values of 199
Quadric surfaces 221—243
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