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Shilov G.E. — An introduction to the theory of linear spaces
Shilov G.E. — An introduction to the theory of linear spaces

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Название: An introduction to the theory of linear spaces

Автор: Shilov G.E.

Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1961

Количество страниц: 307

Добавлена в каталог: 28.10.2012

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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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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