Kirillov A.A., Gvishiani A.D., McFaden H.H. — Theorems and Problems in Functional Analysis :: Электронная библиотека попечительского совета мехмата МГУ

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Kirillov A.A., Gvishiani A.D., McFaden H.H. — Theorems and Problems in Functional Analysis

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Название: Theorems and Problems in Functional Analysis

Авторы: Kirillov A.A., Gvishiani A.D., McFaden H.H.

Аннотация:

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear operators acting upon these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 1982

Количество страниц: 355

Добавлена в каталог: 09.02.2014

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Предметный указатель

$\delta$ -algebra      12
$\delta$ -ring      12
$\sigma$ -algebra      12
$\sigma$ -ring      12
$\varepsilon$ -net      54
$\varepsilon$ -perpendicular      177
Absolute continuity      33
Absolute continuity of a signed measure      36
Alexandrov line      142
Algebra of sets      12
Algebra, irreducible      220
angle      86
Anti-linearity      90
Approximation dimension      55
Axiom of choice (Zermelo’s axiom)      5
Ball, unit      38
Base of neighborhoods of zero      50
Basis, Hamel      274
Canonical imbedding of objects in a category into a sum      146
Canonical projection of a product of elements in a category onto a factor      147
Cantor set      180
Cantor staircase      165
Category      8
Centered system of sets      180
Ces $\grave{a}$ ro means      316
Character on a group      104
Closure of an operator      125
Codimension      45
Cokernel      60
Commutative diagram      9
Commuting operators      127
Compact set      54
Complemented subspace      177
Completeness of a metric space      6
Completeness of an orthonormal system of vectors      88
Completion      6
Congruent pairs of subspaces      201
Conjugacy class of elements      204
Conjugate numbers      47
Continuous basis (coherent or overfilled system)      203
Convergence, almost everywhere      23
Convergence, in measure      23
Convergence, uniform      23
Convex body      173
Convex hull      171
Convolution of functions      96 100
Convolution of functions on the line      168
Countable monotonicity of a measure      14
Countable monotonicity of an outer measure      15
Countable topological basis      189
Cross-norms      49
Cross-norms, uniform      49
Cube, -dimensional      175
Cyclic vector      128
Diameter of a partition      28
Direct product of generalized functions      83
Direct sum of algebras      220
Disjoint union      12
Distribution, tempered      78
Doubly stochastic matrix      181
Dual category      10
Dual group      104
Enveloping algebra (associative hull)      146
Equation, heat conduction      217
Equation, integral, of the second kind with degenerate kernel      184
Equation, Volterra      184
Equicontinuous family of operators      51
Equimeasurable functions      157
EQUIVALENCE      3
Equivalence class      4
Equivalence of collections of seminorms      41
Equivalence of signed measures      36
Equivalence of well-ordered countable sets      141
Equivalent categories      11
Essential spectrum of an operator      229
Essential supremum      26
Extension of an operator      125
Extreme point      56
Extreme subset      180
Field of -adic numbers      144
Formula, Plancherel      112
Formula, Poisson      115
Formula, Stone’s      229
Fourier coefficients      107
Fourier transforms      107
Fourier transforms in the space       114
Fourier transforms of generalized functions      109
Fredholm alternative      66
Fredholm operator criterion      63
Free Abelian group      145
Free group      145
Free Lie algebra      146
Function, (complex, vector-valued), $\mathfrak{U}$ -measurable      22
Function, absolutely continuous      168
Function, almost periodic      198
Function, Bessel      208 324
Function, Borel      157
Function, characteristic      13
Function, continuous in the mean      186
Function, Dirac ( $\delta$ -function)      78 79
Function, Dirichlet      159
Function, essentially bounded      26
Function, Euler      141
Function, generalized, homogeneous, with degree of homogeneity ( $\lambda$ , $\varepsilon$ )      195
Function, generalized, positive-definite      213
Function, generalized, regular      78
Function, generalized, with compact support      78
Function, generating      107
Function, Haar      199
Function, Heaviside      163
Function, integrable      26
Function, jump      163
Function, M $\ddot{o}$ bius      140
Function, M $\ddot{o}$ bius, classical      234
Function, measurable ( $\,mu$ -measurable)      22 158
Function, positive-definite      210
Function, quasi-periodic      216
Function, Rademacher      199
Function, simple      24
Function, Steklov      212
Function, test      78
Function, Walsh      199
Function, weakly measurable      176
Functional operator calculus      116
Functor, covariant (contravariant)      10
Generalized (nonunitary) character      209
Gibbs phenomenon      317
Graph of an operator      124
Group algebra      96
Half-ring      12
Hausdorff criterion      54
Heine — Borel property      71
Hermitian conjugate matrix      91
Hermitian dual space      90
Hermitian form      92
Hermitian symmetry      85
Hilbert basis      89
Hilbert dimension      89
Homotopically equivalent sets      279
Hyperplane      45
Idempotent      220
Identity, Cayley’s      325
Identity, Euler      182
Identity, Hilbert      333
Identity, Sokhotskii      193
Index of an operator      60
Indicatrix, Banach      165
Inductive limit      148
Inequality, Bessel’s      88
Inequality, Cauchy — Bunyakovskii      85

Inequality, H $\ddot{o}$ lder’s      47
Inequality, Minkowski integral      185
Integer lattice      96
Integer, -adic      144
Integral sum, Lebesgue      132
Integral sum, Riemann      227
Integral sum, Riemann — Stieltjes      28
Integral, Fourier      107 110
Integral, Lebesgue      27
Integral, Lebesgue, of a simple function      25
Integral, Riemann — Stieltjes      29
Integral, Riemann, with respect to a projection-valued measure      227
Integral, weak      176
Integral, with respect to a projection-valued measure      132
Interlacing principle      202
Inverse Fourier transformation      109
Isomorphic objects of a category      8
Jordan block      220
Kernel      59
Largest (smallest) element      4
Lemma, Fatou’s      32
Lemma, Zorn’s      5
Length of a word      241
Lexicographical order      141
Limit, of a sequence of sets      151
Limit, supremum (infimum) of a sequence of sets      151
Linear manifold      45
Luzin -property      157
Majorant (minorant)      4
Mapping, bounded      189
Mapping, sequentially continuous      189
Mapping, sequentially continuous, in the variables jointly      190
Maximal (minimal) element      4
Measure      13
Measure, $\sigma$ -finite      17
Measure, countably additive ( $\sigma$ -additive)      13
Measure, inner      152
Measure, Lebesgue      18
Measure, outer      14
Measure, projection-valued      131
Measure, quasi-invariant, with respect to translations      167
Measure, Wiener      22
Measure, with a countable base      185
Measures, disjoint      17
Minkowski functional      39
Mixed area of a pair of sets      273
Mixed volume of a pair of sets      273
Morphism, functorial (natural transformation of functors)      10
Morphism, identity      8
Morphism, of a category      8
Natural domain      126
Net (direction)      4
Newton’s Formulas      326
Norm      38
Norm of an operator      43
Norm, -adic      143
Object of a category      8
One-parameter group      134
Opening between subspaces      201
Operator, adjoint      43
Operator, almost invertible      62
Operator, closed      125
Operator, compact (completely continuous)      57
Operator, continuous linear      50
Operator, creation (annihilation)      183 213
Operator, essentially selfadjoint      125
Operator, Fredholm      60
Operator, Hermitian adjoint      91
Operator, Hilbert — Schmidt      184 202
Operator, Laplace      217
Operator, normal      91
Operator, nuclear      203
Operator, occupation number      213
Operator, of finite rank      57
Operator, of fractional integration (fractional differentiation)      208
Operator, particle number      213
Operator, positive      91
Operator, regular      219
Operator, selfadjoint (Hermitian)      91 125
Operator, unbounded      124
Operator, unitary      91
Operator, Volterra      222
Operator, with simple spectrum      128
Order of a generalized function      80 81
Orthogonal complement      86
Orthogonal projection      91
Orthogonal reflection      200
Orthogonality      86
Orthogonalization process      89
Orthonormal system of vectors      88
Parallelogram law      87 199
Parseval’s equality      88
Partial isometry      202
Partition of unity      190
Point, Lebesgue      157
Point, of density of a set      154
Polar decomposition of an operator      202
Polynomials, Bernoulli      198
Polynomials, Legendre, Tchebycneff, Laguerre, Hermite      303
Pontryagin duality principle      105
Positive linear functional      187
Positive semidefiniteness      85
Principal value of an integral in the Cauchy sense      192
Product of objects in a category      147
Product of partially ordered sets      140
Product, torsion (of groups)      147
Projective limit      148
Quadratic form      92
Quaternions      84
Quotient set      4
Range of an operator      59
Reflexivity      3
Region      69
Relation      3
Relation, equivalence      3
Relation, partial-order      4
Relation, total order      4
Relation, transposed      3
Relations, composition of      3
Representation of the algebra       228
Resolvent of an operator      117
Resolvent set      119
Reversing arrows      11
Ring of subsets      12
Ring, generated by a collection      13
Scalar product      84
Segment (interval)      38
Semicontinuity of a measure      152
Seminorm      38
Sequence, $\delta$ -shaped      205
Sequence, convex      200
Sequence, exact      60
Sequence, exact, at a term      60
Sequence, positive-definite      212
Sequence, rapidly converging      27
Sequence, semi-exact      60
Set      5
Set, $\mu$ -measurable      15
Set, absorbing      38
Set, balanced      38
Set, bounded above (below)      4
Set, bounded above (below), in a polynormed space      71
Set, bounded above (below), in an LTS      171
Set, convex      38
Set, directed      4
Set, Egorov      158
Set, Lebesgue (of a function)      22
Set, Lebesgue measurable      15
Set, Lebesgue measurable, with respect to a $\sigma$ -finite measure      17
Set, level      159

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