Dunford N., Schwartz J., Bade W.G. — Linear operators. Part 2 :: Электронная библиотека попечительского совета мехмата МГУ

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Dunford N., Schwartz J., Bade W.G. — Linear operators. Part 2

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Название: Linear operators. Part 2

Авторы: Dunford N., Schwartz J., Bade W.G.

Аннотация:

In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping V ↦ W between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication. An important special case is when V = W, in which case the map is called a linear operator, or an endomorphism of V. Sometimes the definition of a linear function coincides with that of a linear map, while in analytic geometry it does not.
A linear map always maps linear subspaces to linear subspaces (possibly of a lower dimension); for instance it maps a plane through the origin to a plane, straight line or point.
In the language of abstract algebra, a linear map is a homomorphism of modules. In the language of category theory it is a morphism in the category of modules over a given ring.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 1963

Количество страниц: 986

Добавлена в каталог: 18.02.2014

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Предметный указатель

Characterizations, of $L_{p}$       (394—396)
Characterizations, of Hilbert space      (393—394)
Characterizations, of the space of continuous functions      (394—397)
Charzynski, Z.      91
Chevalley, C.      79
Chiang, T.P.      928
Christian, R.R.      233 382 543 927
Clarkson, J.A.      235 384 393 396 397 473
Clifford, A.H.      82
Closed curve, positive orientation of      (225)
Closed graph theorem      II.2.4 (57)
Closed graph theorem, remarks on      (83—85)
Closed linear manifold spanned by a set      II.1.4 (50)
Closed operator, definition      II.2.3 (57)
Closed orthonormal system, definition      IV.14.1 (357)
Closed orthonormal system, study of      IV.14
Closed set, definition      I.4.3 (10)
Closed set, definition, properties      I.4.4—5 (10)
Closed sphere      II.4.1 (70)
Closed unit sphere      II.3.1 (59)
Closure of a set, criterion to be in      I.7.2 (27)
Closure of a set, definition      I.4.9 (11)
Closure of a set, properties of the closure operation      I.4.10—11 (11—12)
Closure of a symmetric operator, definition      XII.4.7 (1226)
Closure theorems      XI.4 (978—1001)
Closure theorems, Wiener $L_{1}$       XI.4.7 (986)
Closure theorems, Wiener $L_{1}$ , as a Tauberian theorem      XI.5.C (1003)
Closure theorems, Wiener $L_{1}$ , generalization of      XI.4.21 (996)
Cluster point, of a set      I.7.8 (29)
Coddington, E.A.      1433 1434 1498 1503 1587 1590 1591 1592
Cohen, I.S.      400
Cohen, L.W.      543 729
Collatz, L.      610 928
Collins, H.S.      466
Commutator of two operators, definition      X.9 (934)
Compact operator, criteria for and properties of      VI.9.30—35 (515)
Compact operator, definition      VI.5.1 (485)
Compact operator, elementary properties      VI.5
Compact operator, ideals of      (552—553)
Compact operator, in $L_{\infty}$       VI.9.51—57 (517—519)
Compact operator, in C      VI.9.45 (516)
Compact operator, remarks concerning      (539) (609—611)
Compact operator, representation of      (547—551)
Compact operator, representation of, into C(S)      VI.7.1 (490)
Compact operator, representation of, on $L_{1}$       VI.8.11 (507)
Compact operator, representation of, on C(S)      VI.7.7 (486)
Compact operator, spectral theory of      VII.4 VII.5.35 VII.8.2
Compact space, conditional compactness      I.5.5 (17)
Compact space, criteria for compactness      I.5.6 (17) 1.7.9 1.7.12
Compact space, definition      I.5.5 (17)
Compact space, metric spaces      I.6.13 (21—22) I.6.18—19
Compact space, properties      I.5.6—10 (17—18)
Compact space, sequential compactness, definition      I.6.10 (21)
Compact space, weak sequential compactness, conditions for in special B-spaces      IV.15
Compact space, weak sequential compactness, definition      II.3.25 (67)
Compact space, weak sequential compactness, in reflexive spaces      II.3.28 (68)
Complement, and projections      (553—554)
Complement, of a set      (2)
Complement, orthocomplement      IV.4.8 (249)
Complement, orthogonal      II.4.17 (72)
Complemented lattice      (43)
Complete and $\sigma$ -complete lattice      (43)
Complete metric space, compact      I.6.15 (22)
Complete metric space, definition      I.6.5 (19)
Complete metric space, properties      I.6.7 (20) I.6.9
Complete normed linear space      see "B-space"
Complete orthonormal set, in Hilbert space      IV.4.8 (250)
Complete partially ordered space, definition      I.3.9 (8)
Completely regular space, compactification of      IV.6.22 (276) IX.2.16
Completely regular space, definition      IV.6.21 (276) IX.2.15
Completion of a normed linear space      (89)
Complex numbers, extended      (3)
Complex vector space      (38) (49)
Conditional compactness, definition      I.5.5 (17) see
Cone, definition      V.9.9 (451)
Confluent hypergeometric function      XIII.8 (1526)
Conjugate space, definition      II.3.7 (61)
Conjugate space, representation for special spaces      IV.15
Conjugations, in groups      (85)
Conjugations, in Hilbert space      XII.4.17 (1231)
Connected set in n-space      (230)
Connected space      I.4.12 (12)
Continuity of functionals and topology      V.3.8—9 (420—421) V.3.11—12
Continuity of functionals and topology, criteria for existence of continuous linear functionals      V.7.3 (430)
Continuity of functionals and topology, in bounded $\mathfrak{X}$ -topology      V.5.6 (428)
Continuity of functionals and topology, non-existence in $L_{p}$ , 0<p<1      V.7.37 (438)
Continuous (or $\mu$ -continuous set functions), criterion for      III.14.13 (181)
Continuous (or $\mu$ -continuous set functions), definition      III.4.12 (131)
Continuous (or $\mu$ -continuous set functions), derivative of      III.12.6 (214)
Continuous (or $\mu$ -continuous set functions), relation with absolutely continuous functions      (338)
Continuous (or $\mu$ -continuous set functions), relation with integrable functions      III.10
Continuous functions      see also "Absolutely continuous functions"
Continuous functions, as a B-space, additional properties      IV.15
Continuous functions, as a B-space, definition      IV.2.14 (240)
Continuous functions, as a B-space, remarks concerning      (373—386)
Continuous functions, as a B-space, study of      IV.6
Continuous functions, characterizations of C-space      (396—397)
Continuous functions, criteria and properties      I.4.16—18 (13—14) I.6.8 I.7.4
Continuous functions, criteria for the limit to be continuous      I.7.7 (29) IV.6.11
Continuous functions, definition      I.4.15 (18)
Continuous functions, density in TM and $L_{p}$       III.9.14 (170) IV.8.19
Continuous functions, existence of non-differentiable continuous functions      I.9.6 (33)
Continuous functions, existence on a normal space      I.5.2 (15)
Continuous functions, extension of      I.5.3—4 (15—17) 1.6.17
Continuous functions, on a compact space      I.5.8 (18) I.5.10
Continuous functions, representation as a C-space, almost periodic functions      IV.7.6 (285)
Continuous functions, representation as a C-space, bounded functions      IV.6.18—22 (274—277)
Continuous functions, special C-spaces      (397—398)
Continuous functions, uniform continuity      I.6.16—18 (23—24)
Continuous functions, uniform continuity, of almost periodic functions      IV.7.4 (283)
Convergence of filters      I.7.10 (30)
Convergence of functions      IV.15
Convergence of functions, $\mu$ -uniform, criteria for      III.6.2—3 (145) III.6.12
Convergence of functions, $\mu$ -uniform, definition, III.6.1      (145)
Convergence of functions, almost everywhere, criteria for      III.6.12—13 (149—150)
Convergence of functions, almost everywhere, definition      III.1.11 (100)
Convergence of functions, almost everywhere, properties      III.6.14-17 (150-151)
Convergence of functions, in $L_{p}$ criteria for      III.3.6—7 (122—124) III.6.15 III.9.5 IV.8.12—14 (388)
Convergence of functions, in measure (or in $\mu$ -measure), counter examples concerning      III.9.4 (169) III.9.33
Convergence of functions, in measure (or in $\mu$ -measure), definition      III.2.6 (104)
Convergence of functions, in measure (or in $\mu$ -measure), properties      III.2.7—8 (104—105) III.6.2—3 III.6.13
Convergence of functions, quasi-uniform, definition      IV.6.10 (268)
Convergence of functions, quasi-uniform, properties      IV.6.11—12 (268—269) IV.6.30—31
Convergence of functions, uniform, definition      I.7.1 (26)
Convergence of functions, uniform, properties      I.7.6 7 (28—29)
Convergence of sequences, generalized      I.7.1—7 (26—29)
Convergence of sequences, in a metric space      I.6.5 (19)
Convergence of sequences, in special spaces      IV.15
Convergence of sequences, weak convergence in a B-space      II.3.25 (67)
Convergence of series in a B-space, absolute      (93)
Convergence of series in a B-space, unconditional      (92)
Convergence of sets, definition      (126—127)
Convergence of sets, measurable sets in $\Sigma(\mu)$       III.7.1 (158)
Convergence of sets, properties      III.9.48 (174)
Convergence of sets, remarks on      (389—392)
Convergence of sets, set functions      III.7.2—4 (158—160) IV.8.8 IV.9.4—5 IV.9.15 IV.10.6 IV.15
Convergence theorems      IV.15
Convergence theorems, Alexandroff theorem on convergence of measures      IV.9.15 (316)
Convergence theorems, Arzela theorem on continuous limits      IV.6.11 (268)
Convergence theorems, Banach theorem for operators into space of measurable functions      IV.11.2—3 (332—333)
Convergence theorems, Egoroff theorem on a.e. and $\mu$ -uniform convergence      III.6.12 (149)
Convergence theorems, Fatou theorem on limits of integrals      III.6.19 (152) III.9.35
Convergence theorems, for functions of an operator, by inverting sequences      VIII.2.12 (650)
Convergence theorems, for functions of an operator, examples of      VII.8
Convergence theorems, for functions of an operator, in finite dimensional spaces      VII.1.9 (559) see
Convergence theorems, for functions of an operator, in general spaces      VII.3.13 (571) VII.3.23 VII.5.32
Convergence theorems, for functions of an operator, study of      VII.7
Convergence theorems, for kernels      III.12.10—12 (219—222)
Convergence theorems, for linear operators in F- and B-spaces      II.1.17—18 (54—55) 11.3.6 (80—82)
Convergence theorems, Lebesgue dominated convergence theorem      III.3.7 (124) III.6.16 IV.10.10

Convergence theorems, Moore theorem on interchange of limits      I.7.6 (28)
Convergence theorems, Vitali theorem for integrals      III.8.6 (122) III.6.15 III.9.45 IV.10.9
Convergence theorems, Vitali — Hahn — Saks theorem for measures      III.7.2—4 (158—160)
Convergence theorems, Weierstrass theorem on analytic functions      (228)
Convex combination      V.2.2 (414) see "Convex "Convex
Convex function, definition      VI.10.I (520)
Convex function, study of      VI.10
Convex hull      V.2.2 (414)
Convex set      II.4.1 (70)
Convex set, definition      V.1.1 (410)
Convex set, study of      V.1—2
Convex space, locally      V.2.9 (417) (471)
Convex space, strictly      V.11.7 (458)
Convex space, uniformly, defined      II.4.27 (74)
Convex space, uniformly, remarks on      (471—474)
Convexity theorem of M. Riesz      VI.10.11 (525)
Convexity theorem of M. Riesz, applications of      VI.11
Convolution of functions, as an operator in $L_{1}(R)$       XI.3.3 (954)
Convolution of functions, definition      VIII.1.23 (633)
Convolution of functions, inequalities concerning      VI.11.6—12 (528—529)
Convolution of functions, properties      VIII.1.24—25 (634—635) XI.3.1
Convolution of measures      VIII.2.3 (643)
Cooke, R.G.      80 926 927
Cooper, J.L.B.      927 932 1258 1273
Correspondence      see "Function"
Coset, definition      (35)
Countably additive set function      see also "Set function"
Countably additive set function, countable additivity of the integral      III.6.18 (152) IV.10.8
Countably additive set function, definition      III.4.1 (126)
Countably additive set function, extension of      III.5
Countably additive set function, integration with respect to      III.6 IV.10
Countably additive set function, properties      IV.9 IV.15
Countably additive set function, spaces of      III.7 IV.2.16—17
Countably additive set function, study of      III.4
Countably additive set function, uniform countable additivity      III.7.2 (158) III.7.4 IV.8.8—9 IV.9.1
Countably additive set function, weak countable additivity, definition      (318)
Countably additive set function, weak countable additivity, equivalence with strong      IV.10 (818)
Covering of a topological space, definition      I.5.5 (17)
Covering of a topological space, Heine — Borel covering theorem      (17)
Covering of a topological space, in the sense of Vitali, definition      III.12.2 (212)
Covering of a topological space, in the sense of Vitali, Vitali covering theorem      III.12.3 (212)
Covering of a topological space, Lindeloef covering theorem      (12)
Cronin, J.      92
Cross product      see "Product"
Curve      see "Jordan curve" "Rectifiable
D'Alembert, J.      1582
da Silvas Dias, C.L.      399
Daniell, P.J.      381—382
Darboux, G.      1588
Davis, H.T.      80
Day, M.M.      82 233 393—394 398 463 729
Dcrwker, Y.N.      723—724 729
De Morgan, rules of      (2)
Decomposition of measures and spaces, Hahn decomposition      III.4—10 (129)
Decomposition of measures and spaces, Jordan decomposition, for finitely additive set functions      III.1.7 (98)
Decomposition of measures and spaces, Jordan decomposition, for measures      III.4.7 (128) III.4.11
Decomposition of measures and spaces, Lebesgue decomposition      III.4.14 (132)
Decomposition of measures and spaces, Saks decomposition      IV.9.7 (308)
Decomposition of measures and spaces, Yosida — Hewitt decomposition      (293)
Deficiency indices and spaces, definition      XII.4.9 (1226)
Delsarte, J.      1626
Dense convex sets      V.7.27 (437)
Dense linear manifolds      V.7.40—41 (438—439)
Dense set, definition      I.6.11 (21)
Dense set, density of continuous functions in TM and $L_{p}$       III.9.17 (170) IV.8.19
Dense set, density of simple functions in $L_{p}, 1\leq p<\infty$       III.3.8 (125)
Dense set, nowhere dense set      I.6.11 (21)
Density of the natural embedding of a B-space $\mathfrak{X}$ into $\mathfrak{X}^{**}$ in the $\mathfrak{X}^{*}$ topology      V.4.5—6 (424—425)
Derivative, chain rule for      III.13.1 (222)
Derivative, existence of      III.12.6 (214)
Derivative, of a set function      III.12.4 (212)
Derivative, of functions      III.12.7—8 (216—217) III.13.3 III.18.6
Derivative, of Radon — Nicodym      (132)
Derivative, properties      IV.15
Derivative, references for differentiation      (233)
Derivative, space of differentiable functions      IV.2.23 (242)
Determinant, definition      (44—45)
Determinant, elementary properties of      I.13
Diagonal process      (23)
Diameter of a set, definition, I.0.1      (19)
Diametral point      V.11.14 (459)
Dieudonne, J.A.      82 84 94 235 387—388 389 391 395 399—400 402 460 462—463 465 466 539 541
Differentiability of the norm, remarks on      (471—473) (474)
Differential calculus      see also "Derivative"
Differential calculus, Frechet differential      (92)
Differential calculus, in a B-space      (92—93)
Differential equations, solutions of systems of      (561) VII.2.19 VII.5.16 VII.5.27
Differential equations, stability of      VII.2.20—29 (564—565)
Differential operator, boundary condition at an end point for      XIII.2.29 (1304)
Differential operator, boundary condition for      XIII.2.17 (1297)
Differential operator, boundary form for      XIII.2.1 (1287)
Differential operator, boundary matrix for      XIII.2.1 (1287)
Differential operator, boundary value for      XIII.2.17 (1297)
Differential operator, bounded below      XIII.7.20 (1451) XIII.9.c
Differential operator, branching point of      XIII.7.62 (1490)
Differential operator, characteristic equation of, at iofinity      XIII.8 (1527)
Differential operator, complete set of boundary values for      XIII.2.17 (1297)
Differential operator, determining set for      XIII.5.22 (1374)
Differential operator, essential spectrum of      XIII.10.E (1607)
Differential operator, finite below $\lambda$       XIII.7.25 (1455)
Differential operator, first characteristics of      XIII.8 (1527)
Differential operator, formal      XIII.1.1 (1280)
Differential operator, formal adjoint of      XIII.2.1 (1287)
Differential operator, formally positive      XIII.7.6 (1439)
Differential operator, formally selfadjoint      XIII.2.1 (1287)
Differential operator, Green's formula for      XIII.2.4 (1288)
Differential operator, in $L_{p}(I)$       XIII.9.E (1549)
Differential operator, indicial equation of      XIII.8 (1504)
Differential operator, irregular formal      XIII.1 (1280)
Differential operator, mixed boundary condition      XIII.2.29 (1304)
Differential operator, nonselfadjoint      XIII.9.13 (1540)
Differential operator, real boundary value for      XIII.2.29 (1304)
Differential operator, regular formal, XIII.1      (1280)
Differential operator, regular, irregular, singular points for      XIII.8 (1504)
Differential operator, separated boundary conditions      XIII.2.29 (1304)
Differential operator, singular boundary value of second order for      XIII.10.D (1604)
Differential operator, Stokes lines of      XIII.8 (1527)
Differential operator, Sturm — Liouville      XIII.2 (1291) XIII.9.F
Differentiation theorems      VIII.9.13—14 (719—720) see
Dimension of a Hilbert space, as a criterion for Isometric isomorphism      IV.4.16 (254)
Dimension of a Hilbert space, definition      IV.4.15 (254)
Dimension of a Hilbert space, invariance of      IV.4.14 (258)
Dimension of a linear space, definition      (36)
Dimension of a linear space, invariance of      I.14.2 (44)
Dimension of a linear space, of a B-space      (91—92)
Dines, L.L.      466
Dini, U.      360 383 1583
Dirac, P.A.M.      402 1585 1648 1680
Direct product, of B-spaces      (89—90)
Direct sum, of B-spaces      (89—90)
Direct sum, of Hilbert spaces      IV.4.17 (256)
Direct sum, of linear manifolds in a linear space      (87)
Direct sum, of linear spaces      (87)
Directed set, definition      I.7.1 (26)
Disconnected, extremally      (398)
Disconnected, totally      (41) see
Disjoint family of sets, definition      (2)
Distinguish between points, definition      IV.6.15 (272)
Distributions      XIV.8
Distributions, carrier or support of      XIV.8.11 (1650)
Distributions, definition      XIV.3.2 (1645)
Ditkin, V.A.      1161
Divisor of zero      IX.1.27 (861)
Dixmier, J.      94 398 538 886 935
Dixon, A.C.      1588
Doeblin, W.      730
Domain, in complex variables      (224)
Domain, of a function      (2)
Dominated Convergence Theorem      III.8.7 (124) III.6.16 IV.10.10
Dominated ergodic theorem, k-parameter continuous case in $L_{1}, 1<p<\infty$       VIII.7.10 (694)

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