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

Set function, regular, definition      III.5.11 (137)
Set function, regular, properties      III.5.12—14 (137—138) III.9.19—22 IV.13.75 IV.6.1—3
Set function, relativization or restrictions of      III.8
Set function, singular      III.4.12 (131)
Set function, spaces of, as conjugate spaces      IV.5.1 (258) IV.5.3 IV.6.2—3 IV.8.16
Set function, spaces of, definitions      (160—162) IV.2.15—17 IV.6.1
Set function, spaces of, remarks on      (389—390)
Set function, spaces of, study of      III.7 IV.9—10 IV.15
Set function, variation of      III.1.4 (97)
Set(s), $\lambda$ -set      III.5.1 (133)
Set(s), $\sigma$ -field of      III.4.2 (126)
Set(s), Borel      III.5.10 (137)
Set(s), convergence of      (120—127) III.9.48
Set(s), field of      III.1.3 (86)
Set(s), in $\Sigma(\mu)$       III.7.1 (158)
Set(s), Lebesgue      III.4.2 (218)
Set(s), open      see "Open"
Shapiro, J.M.      406
Shiffman, M.      88
Shohat, J.A.      1274 1276
Sikorski, R.      610
Silberstein, J.P.O.      610
Silov, G.      384 385 883 884 1161
Silverman, L.L.      75
Simple function(s), definition      III.2.9 (105)
Simple function(s), density in $L_{p}, 1\leq p<\infty$ of      III.3.8 (125) III.8.3 III.9.46
Simple Jordan curve      (225)
Sin, D.      1588
Singer, I.M.      935
Singular element in a B-algebra      IX.1.2 (861)
Singular element in a ring      (40)
Singular element in a ring, non-singular operator      (45)
Singular set function, definition      III.4.12 (131)
Singular set function, derivatives of      III.12.6 (214)
Singular set function, Lebesgue decomposition theorem      III.4.14 (132)
Singularity of an analytic function      (229)
Sirohov, M.F.      395
Sirvint, G.      383 386 538—540 541 548
Skorohod, A.      94
Smiley, M.F.      394 395
Smith, K.T.      610 927 930 1120
Smithies, F.      548 610 1082 1083 1162
Smulian, and Eberlein theorem on weak compactness      V.6.1 (430)
Smulian, and Krein      see "Krein — Smulian theorem"
Smulian, criterion for $\Gamma$ -compactness      (464)
Smulian, criterion for weak compactness      V.6.2 (433)
Smulian, V.L.      392 395 429 430 433 434 461 463—464 465—466 472—473 612
Snol, E.      1562 1563 1587 1591 1596 1600 1601 1610
Sobczyk, A.      86 393—394 553—554
Sobolev, S.L.      1680 1686
Solomyak, M.Z.      612
Soukhomlinoff, G.A.      86
Space      IV
Space, B- and F-, elementary properties of      II
Space, B- and F-, list of special spaces      IV.2
Space, B- and F-, study of      IV
Space, Banach      see "B-space"
Space, Cech compactification of      IV.6.27 (279)
Space, compact      I.5.5 (17)
Space, complete      I.6.5 (19)
Space, complete normed linear      see "B-space"
Space, completely regular      IV.6.21 (276)
Space, complex linear      (38) (49)
Space, conjugate      II.3.7 (61)
Space, connected      I.4.12 (12)
Space, dimension of      (86)
Space, direct sum of      (88)
Space, extremally disconnected      (398)
Space, F-space      II.1.10 (51)
Space, factor      (38)
Space, fixed point property of      V.10.1 (458)
Space, Hausdorff      I.5.1 (15)
Space, linear topological      II.1.1 (49)
Space, locally compact      I.5.5 (17)
Space, locally convex topological linear      V.2.9 (417)
Space, measure      III.4.3 (126)
Space, metric      I.6.1 (18)
Space, normal      I.5.1 (15)
Space, normal structure of      V.11.14 (459)
Space, normed or normed linear      II.3.1 (59)
Space, product      I.8.1 (32)
Space, real linear      (38) (49)
Space, reflexive      II.3.22 (66)
Space, regular      I.5.1 (15)
Space, separable      I.6.11 (21)
Space, subspace      (36)
Space, subspace spanned      (36)
Space, topological      I.4.1 (10)
Space, total, of functionals      V.3.1 (418)
Space, totally disconnected      (41)
Span, in a linear space      (36) II.1.4
Sparre Andersen, E.      235
Spectral asymptotics      XIII.10.G (1614)
Spectral measure      X.1 (888)
Spectral measure, countably additive      X.1 (889)
Spectral measure, self adjoint      X.1 (892)
Spectral multiplicity theory, definition      X.5 (913)
Spectral radius, definition      VII.8.5 (567)
Spectral radius, of an element in a B-algebra      IX.1.2 (861)
Spectral radius, properties      VII.3.4 (567) VII.5.11—18
Spectral representation, definition      X.5.1 (909) XII.3.4 see
Spectral set, definition      VII.3.17 (572)
Spectral set, of a bounded measurable function      XI.4.10 (988)
Spectral set, of von Neumann      X.9 (933)
Spectral set, properties      VII.3.19—21 (574—575)
Spectral synthesis, problem of      XI.4 (987)
Spectral theorem, for a B*-algebra      X.2.1 (395)
Spectral theorem, for a formally self adjoint differential operator      XIII.5.1 (1333)
Spectral theorem, for a normal operator      X.2.4 (897)
Spectral theorem, for a self adjoint differential operator with compact resolvent      XIII.4.2 (1331)
Spectral theorem, for an unbounded operator      XII.2 (1191)
Spectral theory, for compact operators      VII.4
Spectral theory, in a finite dimensional space      VII.1
Spectrum, $\Sigma$ -simple function      X.1 (891)
Spectrum, continuous      VII.5.1 (580) X.8.1
Spectrum, essential, of a closed operator      XIII.6.1 (1398)
Spectrum, in a finite dimensional space      VII.1.2 (556)
Spectrum, in a general space      VII.3.1 (566)
Spectrum, isolated point of      VII.3.15 (571)
Spectrum, of a B*-algebra      IX.3.4 (875)
Spectrum, of an element in a B-algebra      IX.1.2 (861)
Spectrum, of an element of a sub B-algebra      IX.1 (865)
Spectrum, of an unbounded operator      (599)
Spectrum, of special bounded operators      VII.5.2—15 (580—581)
Spectrum, of special unbounded operators      VII.10.1—3 (604—605)
Spectrum, point      VII.5.1 (580) X.3.1
Spectrum, residual      VII.5.1 (580) X.3.1
Sphere, closed      II.4.1 (70)
Sphere, closed unit      II.8.1 (59)
Sphere, in a metric apace      I.6.1 (19)
Sreider, Y.      392
Stability of a system of differential equations      VII.2.23 (564)
Stasevskaya, V.V.      1588 1626
Steinhaus, H.      80—81 94 387—388
Stekloff, W.      1583
Stepanoff, W.      729
Stewart, F.M.      233
Stickelberger, L.      607
Stieltjes moment problem      XII.2 (1253)
Stieltjes, T.J.      132 142 929 1250 1253 1269
Stokes, G.G.      388 1527
Stone space, definition      (898)
Stone theorems on representation of Boolean rings and algebras      I.12.1 (41) (44)
Stone — Cech compactification theorem      IV.6.22 (276) IX.2.16
Stone — Cech compactification theorem, remarks on      (385)
Stone — Weierstrass theorem      IV.6.16 (272)
Stone — Weierstrass theorem, complex case      IV.6.17 (274)
Stone — Weierstrass theorem, remarks on      (383—385)
Stone, and Banach      see also "Banach — Stone theorem"
Stone, M.H.      41 48 80 85 272 279 382 383—385 393 396 398 442 460 466 606 608 726 872 884 926 927 928 929 1243 1268 1269 1270 1272 1278 1274 1276 1277 1586 1588 1590 1591 1616 1619

Strictly convex B-space, definition      VII.7 (458)
Strong operator topology, definition      VI.1.2 (475)
Strong operator topology, properties      VI.9.1—5 (511) VI.9.11—12
Strong topology, in a normed space      I.3.1 (59) (419)
Structure space of a B-algebra      IX.2.7 (869)
Sturm — Liouville operator      XIII.2 (1291) XIII.9.F
Sturm, C.      1291 1462 1531 1582 1583
Subadditive function, definition      (618)
Subbase for a topology      I.4.6 (10)
Subbase for a topology, criterion for      I.4.8 (11)
Subspace, of a linear space      (36) see
Summability, general principle of      XIII.9.J2 (1577)
Summability, of Fourier series      IV.14.34—51 (331—364)
Summability, of integrals      IV.13.78—101 (351—356)
Summability, of series      II.4.31—54 (74—78)
Summability, regular methods      II.4.35 (75)
Summability, special types of, Abel      II.4.42 (76)
Summability, special types of, Cesaro      II.4.37 (75) II.4.39 IV.14.44
Summability, special types of, Noerlund      II.4.38 (75)
Summability, special types of, Poisson      IV.14.47 (363)
Sunouchi, G.      238 234 391 543 552
Support function, definition      V.1.7 (410)
Supremum, limit superior of a sequence of sets      (126)
Supremum, limit superior of a set of real numbers      (4)
Supremum, of a set of real numbers      (3)
Sylvester, J.J.      606—607
Symmetric difference      (41) (96)
Symmetric operator, definition      X.4.1 (906) XII.1.7
Symmetric subspace, definition      XII.4.4 (1225)
Szasz, O.      384
Tagamlitzki, Y.      396 478
Takahashi, T.      388 400
Taldykin, A.T.      610
Tamarkin, J.D.      80 234—285 388 542 543 610 1118 1162 1269 1274 1276 1583
Tangent function, definition      V.9.2 (446)
Tangent function, examples      V.11.9—13 (458—459)
Tangent function, properties      V.9.1 (445) V.9.3 V.11.10—11
Tangent functionals, definition      V.9.4 (447)
Tarski fixed-point theorem      I.3.10 (8)
Tarski, A.      3
Tauber, A.      78 1007
Taylor expansion for analytic functions      (228)
Taylor, A.E.      92 238 399 540 543 552 554 606 608 612
Taylor, B.      1582
Tchebichef, P.L.      1512
Tchebicheff polynomial      (369)
Teichmueller, O.      48 927
Thorin, G.O.      541 1183
Tietze extension theorem      I.5.3—4 (15—17)
Tietze, H.      15
Tingley, A.J.      406
Titchmarsh — Kodaira theorem      XIII.5.18 (1364)
Titchmarsh, E.C.      48 612 1160 1364 1586 1587 1590 1591 1592 1614 1616 1618
Titov, N.S.      93
Toeplitz, O.      75 72 80 85 399 539 609 926 928 936 1269
Tomita, M.      473
Tonelli theorem      III.11.14 (194)
Tonelli, L.      194
Topology, $\mathfrak{X}$ and $\mathfrak{X}^{**}$ topologies in $\mathfrak{X}^{*}$       (419)
Topology, base and subbase for      I.4.8 (10)
Topology, basic definitions      I.4.1 (10)
Topology, bounded $\mathfrak{X}$ topology      V.5.3 (427)
Topology, functional or $\Gamma$ topology      V.3.2 (419)
Topology, functional or $\Gamma$ topology, study of      V.3
Topology, linear spaces      see "Operator topology"
Topology, metric or strong, in a B-space      (419)
Topology, metric or strong, in a B-space, study of      I.6
Topology, metric, definition      I.6.1 (18)
Topology, norm or strong, in a normed linear space      II.3.1 (59)
Topology, of real numbers      (11)
Topology, of real numbers, study of      I.4—8
Topology, product, definition      I.8.1 (32)
Topology, topological group, definition      II.1.1 (49)
Topology, topological space, definition      I.4.1 (10)
Topology, topological space, study of      I.4—8
Topology, weak* topology      (462)
Topology, weak, in a B-space      (419)
Tornhelm, L.      884
Total boundedness, in a metric space      I.6.14 (22)
Total differential      (92)
Total disconnectedness      (41)
Total family of functions      II.2.6 (58)
Total measurability, definition      III.2.10 (106) see
Total space of functionals, definition      V.3.1 (418)
Total variation of a function      III.5.13 (140)
Total variation of a set function      III.1.4 (97) see
Totally ordered set      I.2.2 (4)
Trace, of a finite matrix      VI.9.28 (515) XI.6.8
Trace, of a matrix, definition      VI.9.28 (515)
Trace, of two operators      XI.6.17 (1026)
Transfinite closure of a manifold      (462)
Transformation      see also "Operator"
Transformation, measure preserving      (687)
Transformation, metrically transitive      (667)
Translate of a function, definition      (283)
Translation by a vector      (36)
Translation number      IV.7.2 (282)
Tseng, Y.Y.      94
Tsuji, M.      338 927
Tukey, J.W.      460—461
Tulajkov, A.      388
Tychonoff theorem, on fixed points      V.10.5 (456) (470)
Tychonoff theorem, on product spaces      I.8.5 (82)
Tychonoff, A.      32 372 456 470
Udin, A.I.      396
Ulam, S.      91 1152
Ultrafilter, definition      I.7.10 (30)
Ultrafilter, properties      I.7.11—12 (3O)
Unbounded operators, exercises on      VII.10
Unbounded operators, in Hilbert space      XII
Unbounded operators, remarks on      (612)
Unbounded operators, study of      VII.9
Unconditional convergence of a series      (92)
Uniform boundedness principle, discussion of      (30—82)
Uniform boundedness principle, for measures      IV.9.8 (309)
Uniform boundedness principle, in B-spaces      II.3.20—21 (66)
Uniform boundedness principle, in F-spaces      II.1.11 (52)
Uniform continuity, criterion for      I.6.18 (24)
Uniform continuity, definition      I.6.16 (23)
Uniform continuity, extension of a function      I.6.17 (23)
Uniform continuity, of an almost periodic function      IV.7.4 (283)
Uniform convergence, $\mu$ -uniform convergence, criteria for      III.6.2—3 (145) III.6.12
Uniform convergence, $\mu$ -uniform convergence, definition      III.6.1 (145)
Uniform convergence, as a criterion for limit interchange      I.7.6 (28)
Uniform convergence, definition      I.7.1 (26)
Uniform convergence, remarks concerning      (382—383)
Uniform convexity, definition      II.4.27 (74)
Uniform convexity, properties      II.4.28—29 (74)
Uniform convexity, remarks on      (471—474)
Uniform countable additivity      see "Countably additive"
Uniform ergodic theory      VIII.8
Uniform ergodic theory, remarks on      (730)
Uniform operator topology, definition      VI.1.1 (475)
Uniform operator topology, properties      VI.9.11—12 (512—513)
Unit sphere in a normed space, compactness and finite dimensionality of      IV.3.5 (245)
Unit sphere in a normed space, definition      II.3.1 (59)
Unit, adjunction of in a B-algebra      IX.1 (860)
Unit, of a group      (34)
Unitary equivalence of operators      X.5.12 (919)
Unitary operator      X.4.1 (906)
Upper bound for an operator      XII.5.1 (1240)
Urysohn theorems, for normal spaces      I.5.2 (15)
Urysohn theorems, metrization      I.6.19 (24)
Urysohn, P.      15 24
van Dantzig, D.      79 91
van Kampen, E.R.      1160
Variation, of a $\mu$ -continuous set function      (131)
Variation, of a countably additive set function      III.4.7 (128)
Variation, of a function      III.5.15 (140) see "Total
Variation, of a regular set function      III.5.12 (137)

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