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

Masani, P.R.      233
Maslow, A.      233
Matrix      (44)
Matrix, characteristic polynomial of      VII.2.1—4 (561—562)
Matrix, exercises on      VII.2
Matrix, hermitian      (561)
Matrix, Jordan canonical form for      VII.2.17 (563)
Matrix, normal      VII.2.14 (563)
Matrix, of a projection      VI.9.27 (514)
Matrix, study of      VII.1
Matrix, trace of      VI.9.28 (515)
Mautner, F.I.      1269 1634 1708
Maximal element, Hausdorff maximality theorem      I.2.6 (6)
Maximal element, in a partially ordered space      I.2.4 (4)
Maximal element, relative to a normal operator      X.5.6 (913)
Maximal ergodic lemma, discrete case      VIII.6.7 (676)
Maximal ergodic lemma, k-parameter case      VIII.7.11 (697)
Maximal ergodic lemma, one-parameter continuous case      VIII.7.6 (690)
Maximal ergodic lemma, remarks on      (729)
Maximal ideal, definition and existence in a ring      (39)
Maximum modulus theorem      (230—231)
Maxwell, J.C.      1749
Mazur theorem, on the convex hull or a compact set      V.2.6 (416)
Mazur, S.      80 81—82 88 91—92 392 400 416 460 461—462 466 472 883
McShane, E.J.      84 232—233 382 387 927
Mean ergodic theorem      (728—729)
Mean ergodic theorem, continuous case in $L_{1}$       VIII.7.4 (689)
Mean ergodic theorem, continuous case in $L_{p}$       VIII.7.10 (694)
Mean ergodic theorem, continuous case in B-space      VIII.7.1—3 (687—689)
Mean ergodic theorem, discrete case in $L_{1}$       VIII.5.5 (662)
Mean ergodic theorem, discrete case in $L_{p}$       VIII.5.9 (667)
Mean ergodic theorem, discrete case in B-space      VIII.5.1—4 (661—662)
Mean Fubini — Jessen theorem      III.11.24 (207)
Measurable function, conditions for (total) measurability      III.2.21 (116) III.6.9—11 III.6.14 III.9.9 III.9.11 III.9.18 III.9.24 III.6.37 III.9.44 III.13.11
Measurable function, definition      III.2.10 (106)
Measurable function, extensions of the notion of measurability      (118—119) (322)
Measurable function, properties      III.2.11—12 (106)
Measurable function, space of (totally), $\Sigma$ -measurable function      IV.2.12 (240) (891)
Measurable function, space of (totally), as a topological linear space      III.9.7 (169) III.9.28
Measurable function, space of (totally), criterion for completeness      III.6.5 (146)
Measurable function, space of (totally), definition      III.2.10 (106) IV.2.27
Measurable function, space of (totally), properties      III.2.11—12 (106)
Measurable function, space of (totally), remarks concerning      (392)
Measurable function, space of (totally), study of TM      IV.11 IV.15
Measurable set, definition      III.4.3 (126)
Measure      see also "Set function"
Measure space, $\sigma$ -finite      III.5.7 (136)
Measure space, as a metric space      III.7.1 (158) III.9.6
Measure space, decomposition of      see also "Decomposition"
Measure space, definition      III.4.3 (126)
Measure space, finite      III.4.3 (126)
Measure space, Lebesgue extension of      III.5.18 (143)
Measure space, positive      III.4.3 (126)
Measure space, product, of finite number of $\sigma$ -finite measure spaces      (188)
Measure space, product, of finite number of finite measure spaces      III.11.3 (186)
Measure space, product, of infinite number of finite measure spaces      III.11.21 (205)
Measure, Borel or Borel — Lebesgue      (189)
Measure, Borel — Stieltjes      (142)
Measure, Borel — Stieltjes, decomposition of      see "Decomposition"
Measure, Borel — Stieltjes, determined by a function      (142) (144)
Measure, Borel — Stieltjes, differentiation of      III.12
Measure, change of      III.10.8 (182) X.I
Measure, definition      III.4.8 (126)
Measure, Haar      V.11.22—23 (460)
Measure, Hausdorff $\alpha$ -      III.9.47 (174)
Measure, invariant      VI.9.38—44 (516)
Measure, Lebesgue and Lebesgue — Stieltjes      (143)
Measure, Lebesgue extension of      III.5.18 (143)
Measure, of hypersurface of unit sphere in $E^{n}$       XI.7 (1048—1049)
Measure, outer      III.5.3 (133)
Measure, positive matrix      XIII.5.12 (1349)
Measure, product      III.11
Measure, Radon      (142)
Measure, regular vector-valued      IV.13.75 (350)
Measure, restriction of      (166)
Measure, spaces of      III.7 IV.2.15—17 IV.9 IV.15 (388—391)
Measure, vector-valued, study of      IV.10 (301)
Measure-preserving transformation      (667)
Medvedev, Yu.T.      392
Mercer, T.      1088
Mertens, F.      77
Metric spaces, complete      I.6.5 (19)
Metric spaces, definition      I.6.1 (18)
Metric spaces, properties      I.6
Metric(s)      I.6.1 (18)
Metric(s), invariant, in a group      (90—91)
Metric(s), invariant, in a linear space      II.1.10 (51)
Metric(s), topology in normed linear space      II.3.1 (59) (419)
Metrically transitive transformation      (667)
Metrizability      see also "Metrization"
Metrizability, and dimensionality      V.7.9 (436) V.7.34—35
Metrizability, and separability      V.5.1—2 (426) V.6.3 V.7.15
Metrization, of a measure space      III.7.1 (158)
Metrization, of a regular space      I.6.19 (24)
Metrization, of the set of all functions      III.2.1 (102)
Michael, E.      462 538
Michael, E.A.      886 931
Michal, A.D.      79 92 610 888
Mihlin, C.G.      1181
Mikusinski, J.G.      395
Miller, D.S.      392 724 729
Milman      see "Krein — Milman theorem"
Milman, D.P.      94 440 463 466 471 473 1587 1591
Milne, W.E.      1592 1614
Mimura, Y.      541 928
Minimax principle      X.4 (908) XIII.9.D
Minkowski inequality      III.3.3 (120)
Minkowski inequality, conditions for equality      III.9.43 (173)
Minkowski Inequality, generalizations of      VI.11.13—18 (530—531)
Minkowski, H.      120 372 471
Miranda, C.      88 470
Mirkil, H.      1161
Miyadera, I.      727
Mohr, E.      1590
Molcanov, A.M.      1563 1587 1591
Monna, A.F.      233 400
Montroll, E.W.      406
Moore theorem, concerning interchange of limits      I.7.6 (28)
Moore, E.H.      28 80 608
Moore, R.L.      48
Morse, A.P.      87 235 393
Moser, J.      612
Moses, H.E.      1622
Moskovitz, D.      466
Muentz, C.H.      384
Multiplicative linear functional      IV.6.23 (277) see
Multiplicity of eigenvalues      X.4 (907)
Munroe, M.E.      93 232 235
Murray, F.J.      554 884 886 1269
Myers, S.B.      382 397
Nachbin, L.      93 395 397 398 554
Nagata, J.      385
Nagumo, M.      79 94 394 608 609 726 883
Naimark, M.A.      876 883 884 886 932 1149 1180 1260 1261 1270 1274 1587 1588 1589 1590 1591 1593 1596 1597 1607 1608 1609 1610 1611
Nakamura, M.      233 391 395 539
Nakano, H.      80 395 471 927 928 929 1269 1273 1274
Nakayama, T.      927
Nathan, D.S.      726
Natural domain of existence, of an analytic function      (230)
Natural embedding of a B-space      II.3.18 (66)
Natural homomorphism onto factor space      (38) (39)
Neighborhood, $\varepsilon$ -, in a metric space      I.6.1 (18)
Neighborhood, fundamental family of      I.4.6 (10)
Neighborhood, of a point or set      I.4.1 (10)
Nemyckii, V.      1587
Neumann, C.      608 1346 1349
Neumark, M.      396
Newburgh, J.D.      612
Newton, R.G.      1626
Nicolescu, M.      388

Niemytsky, V.      470
Nikodym      see also "Radon — Nykodym theorem"
Nikodym, boundedness theorem      IV.9.8 (309)
Nikodym, O.M.      93 160 176 181—182 234—235 309 387 390 392
Nikol'skii, V.N.      94 611
Nikovic, I.A.      1163
Nilpotent element      (40)
Nilpotent, topological nilpotent in B-algebra      IX.2.5 (869)
Noerfund, N.E.      75
Non-singular linear transformation      (45)
Norm, differentiability of      (471—473) (474)
Norm, existence of      (91)
Norm, in a B-space      II.3.1 (59)
Norm, in a conjugate space      II.3.5 (60)
Norm, in an F-space      II.1.10 (51)
Norm, in Hilbert space      IV.2.26 (242)
Norm, in special spaces      IV.2
Norm, inequalities on $L_{p}$ -norma      VI.11.30—37 (535—536)
Norm, of an operator      II.3.5 (60)
Norm, topology      II.3.1 (50)
Normal operator, in a finite dimensional space      VII.2.14 (563)
Normal operator, in Hilbert space      X.1 (887)
Normal operator, real and imaginary parts of      X.4 (906)
Normal structure, definition      V.11.14 (459)
Normal structure, properties      V.11.15—18 (459)
Normal subgroup      (85)
Normal topological space, compact Hausdorff space      I.5.9 (18)
Normal topological space, definition      I.5.1 (15)
Normal topological space, metric space      I.6.3 (19)
Normal topological space, properties      I.5.2—4 (15—17)
Normal topological space, regular space with countable base      (24)
Normed (normed linear space)      see also "B-space"
Normed (normed linear space), definition      II.3.1 (59)
Normed (normed linear space), study of      II.3
Nowhere dense      I.6.11 (21)
Null function      see also "Null set"
Null function, criterion for      III.6.8 (147)
Null function, definition      III.2.3 (103)
Null set      see also "Null function"
Null set, additional properties of      III.9.2 (169) III.9.8 III.9.16
Null set, criterion for      III.6.7 (147)
Null set, definition      III.1.11 (100)
O'Neill, B.      474
Ogasawara, T.      395 927
Ohira, K.      894
Ono, T.      88 400 884
Open set, criterion for      I.4.2 (10)
Open set, definition      I.4.1 (10)
Operational calculus      X.1 (890)
Operational calculus, for functions of an infinitesimal generator      VIII.2.6 (645)
Operational calculus, for unbounded closed operators      VII.9.5 (602)
Operational calculus, in finite dimensional space      VII.1—5 (558)
Operational calculus, in general complex B-space      VII.3.10 (568)
Operational calculus, remarks on      (607—609)
Operator topologies      VI.1
Operator topologies, bounded strong      VI.9.9 (512)
Operator topologies, bounded weak      VI.9.7—10 (512)
Operator topologies, continuous linear functionals in      VI.1.4 (477)
Operator topologies, properties      VI.9.1—12 (511—513)
Operator topologies, remarks on      (538)
Operator topologies, strong      VI.1.2 (475)
Operator topologies, strongest      (538)
Operator topologies, uniform      VI.1.1 (475)
Operator topologies, weak      VI.1.3 (476)
Operator, adjoint of      VI.2
Operator, bound of      II.3.5 (60)
Operator, bounded      XII.1 (1185)
Operator, closed      II.2.3 (57) XII.1
Operator, compact, definition      VI.5.1 (485)
Operator, compact, study of      VI.5 VII.4
Operator, continuity of, discussion      (82—83)
Operator, continuity of, in B-spaces      II.3.4 (59)
Operator, continuity of, in F-spaces      II.1.14—16 (54)
Operator, definition      (36)
Operator, equality of      XII.1 (1185)
Operator, extensions of      VI.2.5 (478) (554) XII.1
Operator, finite below $\lambda$       XIII.7.25 (1455)
Operator, functions of      see "Calculus"
Operator, graph of      II.2.3 (57) XII.1
Operator, hermitian      IV.13.72 (350) (561)
Operator, ideals of      (552—553) (611)
Operator, identity      (37)
Operator, in a finite dimensional space      (44)
Operator, inverse of      XII.1.2 (1187)
Operator, limits of, in B-spaces      II.3.6 (60)
Operator, limits of, in F-spaces      II.1.17—18 (54—55)
Operator, matrix of      (44)
Operator, non-singular      (45)
Operator, norm of      II.8.5 (60)
Operator, normal      VII.2.14 (583) IX.3.14
Operator, perturbation of      VII.6
Operator, polynomials in      VII.1.1 (556)
Operator, product of      (37) XII.1.1
Operator, projection      (37) VI.3.1
Operator, projection, study of      VI.3
Operator, quasi-nilpotent      VII.5.12 (581)
Operator, range of      VI.2.8 (470)
Operator, range of, with closed range      VI.6
Operator, representation of, in $L_{1}$       VI.8
Operator, representation of, in C      VI.7
Operator, representation of, in other spaces      (542—552)
Operator, resolvent      VII.3.1 (566)
Operator, resolvent, study of      VII.3
Operator, self adjoint      IX.3.14 (879) XII.1.7
Operator, spectral radius of      VII.3.5 (567)
Operator, spectrum of      VII.3.I (566)
Operator, sum of      (37) XII.1.1
Operator, symmetric      X.4.1 (906) XII.1.7
Operator, unbounded      VII.9 XII
Operator, unbounded, adjoint of      XII.1.4 (1188)
Operator, unbounded, spectrum and resolvent set of      XII.1 (1187)
Operator, weakly compact, definition      VI.4.1 (482)
Operator, weakly compact, study of      VI.4
Operator, zero      (37)
Order of a pole      (230)
Order of a zero      (230)
Order of an operator      VII.3.15 (57)
Ordered representation, definition      X.5.9 (916) XII.3.15
Ordered representation, equivalence of      X.5.9 (916) XII.3.15
Ordered representation, measure of      X.5.9 (916) XII.3.15
Ordered representation, multiplicity of      X.5.9 (916) XII.3.15
Ordered representation, multiplicity sets of      X.5.9 (916) XII.3.15
Ordered set, definition      I.2.2 (4)
Ordered set, directed set      I.7.1 (26)
Ordered set, partially      I.2.1 (4)
Ordered set, study of      I.2
Ordered set, totally      I.2.2 (4)
Ordered set, well      I.2.8 (7)
Orientation, of a closed curve      (225)
Origin, of a linear space      II.3.I (59)
Orihara, M.      395
Orlicz, W.      80 81—82 83 93 94 235 387 388 391—392 400 543
Orthocomplement of a set in Hilbert space, definition      IV.4.3 (249)
Orthocomplement of a set in Hilbert space, properties      IV.4.4 (249) IV.4.18
Orthogonal complement of a set in a normed space      II.4.17 (72)
Orthogonal complement of a set in a normed space, remarks on      (93)
Orthogonal elements and manifolds in Hilbert space      IV.4.3 (249)
Orthogonal projections in Hilbert spaces      IV.4.8 (250)
Orthogonal series, exercises on      VI.11.43—47 (537)
Orthogonal series, study of      IV.14
Orthonormal basis in Hilbert space      IV.4.11 (252)
Orthonormal basis in Hilbert space, cardinality of, IV      4.14 (253)
Orthonormal basis in Hilbert space, criteria for      IV.4.13 (253)
Orthonormal basis in Hilbert space, existence of      IV.4.12 (252)
Orthonormal set in Hilbert space, closed set      IV.14.1 (357)
Orthonormal set in Hilbert space, complete set      IV.4.8 (250)
Orthonormal set in Hilbert space, definition      IV.4.3 (250)
Orthonormal set in Hilbert space, properties      IV.4.9—16 (251—254)
Outer measure      III.5.8 (133)
Owchar, M.      406

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