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

$L_{1}(R)$ as a B-algebra      XI.3.2 (953)
$L_{p}(S,\Sigma,\mu)$ , 0<p<1, definition      III.9.29 (171)
$L_{p}(S,\Sigma,\mu)$ , 0<p<1, properties      III.9.29—31 (171)
$L_{p}(S,\Sigma,\mu), 1\leq p<\infty$ , characterizations of      (394—396)
$L_{p}(S,\Sigma,\mu), 1\leq p<\infty$ , completeness of      III.6.6 (146) III.9.10
$L_{p}(S,\Sigma,\mu), 1\leq p<\infty$ , criteria for convergence in      III.3.6—7 (122—124) III.6.15 IV.15
$L_{p}(S,\Sigma,\mu), 1\leq p<\infty$ , definition      III.4.4 (121)
$L_{p}(S,\Sigma,\mu), 1\leq p<\infty$ , remarks on      (387—388)
$L_{p}(S,\Sigma,\mu), 1\leq p<\infty$ , separable manifolds in      III.8.5 (168) III.9.6
$L_{p}(S,\Sigma,\mu), 1\leq p<\infty$ , study of      III.3 III.6 IV.8 IV.15
$L_{\infty}(S,\Sigma,\mu)$ , definition      IV.2.19 (244)
$L_{\infty}(S,\Sigma,\mu)$ , study of      IV.8 IV.15
a.e.      see "Almost everywhere"
Abdelhay, J.      397
Abel summability, of series      II.4.42 (76)
Abel, N.H.      76 352 383
Abelian group      (34)
Absolute convergence, in a B-space      (93)
Absolutely continuous functions, definition      IV.2.22 (242)
Absolutely continuous functions, set function      see "Continuous set function" "Set
Absolutely continuous functions, space of, additional properties      IV.15 (378)
Absolutely continuous functions, space of, definition      IV.2.22 (242)
Absolutely continuous functions, space of, remarks concerning      (392)
Absolutely continuous functions, space of, study of      IV.12.8 (338)
Accumulation, point of      I.4.1 (10)
Adams, C.R.      393
Additive set function      see "Set function"
Adjoint element, in an algebra with involution      (40) see
Adjoint of an operator, between B-spaces      VI.2
Adjoint of an operator, compact operator      VI.5.2 (485) VI.5.6 VII.4.2
Adjoint of an operator, continuity of operation      VI.9.12 (513)
Adjoint of an operator, criterion for      VI.9.13—14 (513)
Adjoint of an operator, in Hilbert space      VI.2.9 (479) VI.2.10
Adjoint of an operator, remarks on      (538)
Adjoint of an operator, resolvent of      VII.3.7 (568)
Adjoint of an operator, spectra of      VII.3.7 (568) VII.5.9—10 VII.5.23
Adjoint of an operator, weakly compact operator      VI.4.7—8 (484—485)
Adjoint space, definition      II.3.7 (61)
Adjoint space, representation for special spaces      IV.15
Affine mapping, definition      (456)
Affine mapping, fixed points of      V.10.6 (456)
Agmon, S.      1161
Agnew, R.P.      87
Ahiezer, N.I.      936 927 929 1269 1270 1272 1273 1274 1276 1588 1589 1590 1592
Ahlfors, L.V.      48
Akilov, G.P.      554
Alaoglu, L.      235 424 462 463 729
Alexandroff theorem, on C(S) convergence of bounded additive set functions      IV.9.15 (316)
Alexandroff theorem, on countable additivity of regular set functions on compact spaces      III.5.13 (138)
Alexandroff, A.D.      138 233 316 380—381 390
Alexandroff, P.      47 467
Alexiewicz, A.      82 83 234 235 392 543
Algebra, algebraic preliminaries      I.10—13
Algebra, B*-algebra      IX.3
Algebra, B*-algebra, *-equivalences of B*-algebras      IX.3.4 (875)
Algebra, B*-algebra, *-homomorphism in      IX.3.4 (875)
Algebra, B*-algebra, *-isomorphism of      IX.3.4 (875)
Algebra, B*-algebra, as an algebra of continuous functions      IX.3.7 (876)
Algebra, B*-algebra, non-commutative      IX.5 (884—886)
Algebra, B*-algebra, spectrum of      IX.3.4 (875)
Algebra, B-algebra      IX
Algebra, B-algebra, as an algebra of continuous functions      IX.2.9 (870)
Algebra, B-algebra, as an operator algebra      IX.1 (860)
Algebra, B-algebra, generating set for      IX.2.10 (870—871)
Algebra, B-algebra, ideal in      IX.1 (865—866)
Algebra, B-algebra, quotient      IX.1 (866)
Algebra, B-algebra, radical of      IX.2.5 (869)
Algebra, B-algebra, semi-simple      IX.2.5 (869)
Algebra, B-algebra, structure space of      (869)
Algebra, boolean      see also "Field of sets"
Algebra, Boolean, definition      (43)
Algebra, Boolean, representation of      (44)
Algebra, commutative      IX.1.1 (860) IX.2
Algebra, definition      (40)
Algebra, of sets      see "Field of sets"
Algebra, quotient      (40)
Almost everywhere (or $\mu$ -almost everywhere) definition for additive scalar set functions      III.1.11 (100)
Almost everywhere (or $\mu$ -almost everywhere) definition for additive scalar set functions, definition for vector-valued set functions      IV.10.6 (322)
Almost periodic functions, definition      IV.2.25 (242)
Almost periodic functions, space of, additional properties      IV.15 (379)
Almost periodic functions, space of, definition      IV.2.25 (242)
Almost periodic functions, space of, remarks concerning      (386—387)
Almost periodic functions, space of, study of      IV.7
Almost uniform (or $\mu$ -uniform convergence), definition      III.6.1 (145) see
Altman, M.S.      94 609 610
Ambrose, W.      1013 1160 1274
Analytic continuation      (230)
Analytic function (vector-valued), between complex vector spaces      VI.10.5 (522)
Analytic function (vector-valued), definition      (224)
Analytic function (vector-valued), properties      III.14
Analytic function (vector-valued), space of, definition      IV.2.24 (242)
Analytic function (vector-valued), space of, properties, IV      15
Annihilator of a set      II.4.17 (72)
Anzai, H.      886
Arens' lemma      IX.3.5 (875—876)
Arens, R.F.      381 382 384 385 396—397 399 466 875 884 886
Arnous, E.      1274
Aronszajn, N.      87 91 234 394 610 928 980 1120
Artemenko, A.      387 392
Arzela theorem, on continuity of limit function      IV.6.11 (268)
Arzela theorem, on continuity of limit function, remarks concerning      (383)
Arzela, C.      266 268 382 383
Ascoli — Arzela theorem, on compactness of continuous functions      IV.6.7 (266)
Ascoli — Arzela theorem, remarks concerning      (382)
Ascoli, G.      266 382 460 466
Atkinson, F.V.      610 611 1615
Atom, in a measure space      IV.9.6 (308)
Audin, M.      611
Automorphisms, in groups      (35)
B*-algebras      IX.3 (874—879) see
B-algebra      see "Algebra"
B-space (or Banach space), basic properties of      II
B-space (or Banach space), definition      II.3.2 (59)
B-space (or Banach space), integration      III
B-space (or Banach space), special B-spaces      IV
B-space (or Banach space), special B-spaces, properties      IV.15
Babenko, K.I.      94 1183
Bade, W.G.      538 612 728 928 1269
Baire category theorem      I.6.8 (20)
Baire, R.      20
Baker, H.F.      1588
Banach limits, existence and properties      II.4.22—23 (73)
Banach theorem, on convergence of measurable functions      IV.11.2—3 (332—334)
Banach — Stone theorem, on equivalence of C-spaces      V.8.8 (442)
Banach — Stone theorem, on equivalence of C-spaces, remarks on      (396—397 466)
Banach, S.      59 62 73 80 81 82—84 85 86 89 91—93 94 234 232 380 385—386 392 442 462—463 465—466 472 538 539 609
Barankin, E.W.      1163
Bari, N.K.      94
Bartle, R.G.      85 92 233 383 386 389 392 539—540 543
Base (or basis)      see also "Hamel base"
Base (or basis), in a B-space, criterion for compactness with      IV.5.5 (260)
Base (or basis), in a B-space, definition      II.4.7 (71)
Base (or basis), in a B-space, properties      II.4.8—12 (71)
Base (or basis), in a B-space, remarks on      (93—94)
Base (or basis), in a linear space      see "Hamel base"
Base (or basis), orthogonal and orthonormal bases in Hilbert space, definition      IV.4.11 (252)
Base (or basis), orthogonal and orthonormal bases in Hilbert space, existence of      IV.4.12 (252)
Base for a topology, criterion for      I.4.7 (11)
Base for a topology, definition      I.4.6 (10)
Base for a topology, theorems concerning countable bases      I.4.14 (12) I.6.12 I.6.19
Basic separation theorem concerning convex sets      V.1.12 (412)
Bellman, R.      1550
Bendixon, I.      1080
Bennett, A.A.      85
Berezanskii, Yu.M.      1587 1626
Berkowitz, J.      1543 1580 1591 1592 1594 1595 1599 1604
Bernoulli, D.      1581 1582
Bernstein theorem, concerning cardinal numbers      I.14.2 (46)
Bernstein, F.      46
Berri, R.      395

Besicovitch, A.S.      386
Bessel equation      XIII.8 (1535)
Bessel, F.W.      977 1348 1349 1535
Beurling, A.      361 930 978 1100 1161 1162
Bieberbach, L.      48
Bilateral Laplace and Laplace — Stieltjes transforms, definitions      VIII.2.1 (642)
Bilinear functional      II.4.4 (70)
Biorthogonal system, in a B-space      II.4.11 (71)
Birkhoff, G.      48 90 93 232 235 393—394 395 729
Birkhoff, G.D.      470 658—658 729 1497 1583 1586 1589 1592
Birnbaum, Z.W.      400
Blumenthal, L.M.      393
Boas, R.P., Jr.      94 473 1266
Bocher, M.      1583 1588 1589
Bochner moment problem      XII.8.3 (1254)
Bochner, S.      232—233 235 283 315 386 390 395 540 543 552 883 1160 1254 1273 1274
Bodiou, G.      1264
Bogoliouboff, N.      730
Bohnenblust, H.F.      86 94 393 394 395—396 554
Bohr, H.      281 386—387 399 949 1149
Bohr, H., theorem concerning almost periodic functions      XI.2.4 (949)
Boltzmann, L.      657
Bonnesen, T.      471
Bonsall, F.F.      88
Boolean algebra      see also "Boolean ring"
Boolean algebra, definition      (43)
Boolean algebra, properties      (44)
Boolean algebra, representation of      (441
Boolean ring, definition      (40)
Boolean ring, representation of      I.12.1 (41)
Borel field of sets, definition      III.5.10 (187)
Borel function      X.1 (891)
Borel measurable function      X.1 (891)
Borel measure (or Borel — Lebesgue measure), construction of      (139) III.13.8
Borel — Stieltjes measure      (142)
Borel, E.      132 139 142 1588
Borg, G.      1501 1622
Borsuk, K.      91
Botts, T.      387 460
Bound, of an operator      II.8.5 (60)
Bound, of an operator, in a partially ordered set      I.2.3 (4)
Bound, of an operator, in the (extended) real number system      (3)
Boundary condition, adjoint      XII.4.27 (1237)
Boundary condition, definition      XII.4.25 (1285) see boundary
Boundary condition, linearly independent      XII.4.25 (1235)
Boundary condition, symmetric      XII.4.25 (1286)
Boundary values, complete set of      XII.4.22 (1235) see
Boundary values, for an operator      XII.4.20 (1284)
Boundary, of a set      I.4.9 (11)
Bounded $\mathfrak{X}$ topology, continuous linear functionals      V.5.6 (428)
Bounded $\mathfrak{X}$ topology, system of neighborhoods for      V.5.4 (427)
Bounded function space, additional properties      IV.15
Bounded function space, definition      IV.2.13 (240)
Bounded function space, remarks concerning      (373)
Bounded function space, study of      IV.5
Bounded sets, in linear spaces      V.7.5 (436) V.7.7 V.7.8
Bounded strong operator topology, definition and properties      VI.9.9 (512)
Bounded variation of a function, additional properties      IV.15 (378)
Bounded variation of a function, criterion to be      IV.13.73 (350)
Bounded variation of a function, definition      III.5.15 (140)
Bounded variation of a function, generating Borel — Stieltjes measure      (142)
Bounded variation of a function, integral with respect to      IV.13.63 (349)
Bounded variation of a function, integration by parts      III.6.22 (154)
Bounded variation of a function, remarks on      (392—393)
Bounded variation of a function, right- and left-hand limits of      III.5.16 (140) III.6.21
Bounded variation of a function, set function, criteria for      III.4.4—5 (127—128) see
Bounded variation of a function, set function, definition      III.1.4 (97)
Bounded variation of a function, study of      IV.12
Bounded weak operator topology, definition and properties      VI.9.7—10 (512)
Bounded, essentially (or $\mu$ -essentially) definition      III.1.11 (101)
Bounded, operator, definition      II.3.5 (60)
Bounded, set in a linear topological space      II.1.7 (51)
Bounded, set in a linear topological space, criterion for boundedness in a B-space      II.3.3 (59)
Bounded, set in a linear topological space, remarks on      (80)
Bounded, totally bounded set, definition      I.6.14 (22)
Boundedness, in F-spaces      II.1.11 (52)
Boundedness, of a continuous function on a compact set      I.5.10 (18)
Boundedness, of a finite countably additive set function      III.4.4—7 (127—128)
Boundedness, of an almost periodic function      IV.7.3 (283)
Boundedness, principle of uniform boundedness in B-spaces      II.3.20—21 (66) (80—82)
Bounding point of a set, criteria for      V.1.8 (411) V.2.1
Bounding point of a set, definition      V.1.6 (410)
Bourbaki, N.      47 80 62 84 232 382 463 465 471
Bourgin, D.G.      383 462
Brace, J.W.      466
Bram, J.      932 933
Brauer, A.      1078
Brauer, R.      1149
Bray, H.E.      391
Brelot, M.      1268
Brillouin, L.      1592 1614
Brodsku, M.S.      471 1164
Brouwer fixed point theorem, proof of      (467)
Brouwer fixed point theorem, statement      (453)
Browder, F.E.      1269 1634 1635 1708 1746
Brown, A.      934 935
Buchheim, A.      697
Buniakowsky, V.      372
Burkhardt, H.      1589
Cafiero, F.      389 392
Calderon — Zygmund inequality      XI.7.11 (1063) XI.7.16
Calderon — Zygmund, convolution kernel of      XI.7.4 (1058)
Calderon — Zygmund, convolution product of      XI.7.6 (1054)
Calderon, A.P.      541 730 1063 1072 1077 1164 1165
Calkin, J.W.      553 1270 1273 1586 1589
Cameron, R.H.      406 407
Camp, B.H.      390
Canonical factorization of operators      XII.7
Cantor diagonal process      (23)
Cantor perfect set      V.2.13—14 (486)
Caratheodory theorem, concerning outer measures      III.5.4 (134)
Caratheodory, C.      48 134 232 729 1048
Cardinal numbers, Bernstein theorem      I.14.2 (46)
Cardinal numbers, comparability theorem      I.3.5 (3)
Carleman's inequality      XI.6.27 (1038)
Carleman, T.      536 627 1162 1163 1268 1260 1277
Cartan, E.      607 1148
Cartan, H.      80 1152 1160 1274
Cartesian product of sets, definition      I.3.11 (9)
Cartesian product of sets, properties      I.3.12—14 (9)
Cartesian product of topological spaces      I.8 (81)
Category theorem, of Baire      I.6.9 (20)
Cauchy integral formula      (227)
Cauchy integral formula, for functions of an operator, in a finite dimensional space      VII.1.10 (560)
Cauchy integral formula, for functions of an operator, in general space      VII.8.9 (568)
Cauchy integral formula, for unbounded closed operators      VII.9.4 (601)
Cauchy integral formula, remarks on      (607—609) (612)
Cauchy integral theorem      (225)
Cauchy problem      (613—614) (639—641)
Cauchy sequence, generalized      (28)
Cauchy sequence, in a metric space      I.6.5 (18—20)
Cauchy sequence, weak, criterion for in various spaces      IV.15
Cauchy sequence, weak, in a B-space      II.3.25 (67—68)
Cauchy, A.      372 382—383
Cayley, A.      1270 1271
Cech compactification theorem      IV.6.22 (276)
Cech compactification theorem, of a completely regular space      (279)
Cech, E.      279 385 872
Cesaro summability, of Fourier series      IV.14.44 (363)
Cesaro summability, of series      II.4.37 (75)
Cesaro, E.      75 352 363
Chang, S.H.      610 1168
Change of variables, for functions      III.8.4—5 (222—228)
Change of variables, for measures      III.10.8 (182)
Character group, definition      XI.3.13 (968)
Character, definition      XI.1.5 (944)
Characteristic function      (3)
Characteristic polynomial, definition      VII.2.1 (561) XI.6.9
Characteristic polynomial, properties      VII.2.1—4 (561—562) VII.5.17 VII.10.8
Characteristic value      (606)

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