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Helemskii A.Ya. — Lectures and Exercises on Functional Analysis, Vol. 233
Helemskii A.Ya. — Lectures and Exercises on Functional Analysis, Vol. 233

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Название: Lectures and Exercises on Functional Analysis, Vol. 233

Автор: Helemskii A.Ya.

Аннотация:

This book contains a unique exposition intended to serve as an introduction to functional analysis. Topics covered include normed spaces and bounded operators, Banach spaces, polynormed spaces and distributions, compact operators, $C^*$ algebra, spectral theorems, Fourier transform, and more. A distinguishing feature of the book is the wide use of the language and elementary constructions of category theory, which are explained in the opening chapter of the book. Among nonstandard topics discussed in the book are the theory of Banach tensor products, basics of quantum functional analysis, and Borel operator calculus. General definitions and main results are supplemented with many examples and exercises. Prerequisites for the main part of the book include standard undergraduate courses in algebra and analysis. It is suitable for graduate students and researchers interested in functional analysis.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2006

Количество страниц: 468

Добавлена в каталог: 09.04.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$l_{p}$-sum      63
$l_{p}$-sum of operators      95
$\delta$-function      285
$\lambda$-dilation      3
$\varepsilon$-net      201
$\varepsilon$-perpendicular      206
*-algebra      348
*-homomorphism      348
*-homomorphism, isometric      350
*-isomorphism      348
*-isomorphism, contraction      350
*-isomorphism, isometric      350
*-isomorphism, topological      350
*-representation      348
*-subalgebra      349
*-subset      349
Absolute value of an operator      370
Adherent point      10
Adjoint associativity law      180
Adjointness formula      329
Admissible interval      376
Alexandroff theorem      194
Algebra      304
Algebra, Banach      311
Algebra, Banach, semisimple      323
Algebra, Banach, star-      349
Algebra, Calkin      310
Algebra, involutive      348
Algebra, quotient      310
Algebra, unital      304
Algebra, von Neumann      354
Algebra, von Neumann, amenable      360
Algebra, von Neumann, enveloping      358
Algebra, Wiener      423
Algebraic sum      3
Approximate identity      430
Approximate identity, bounded      430
Approximate identity, bounded, countable      431
Approximation problem      214
Approximation property      213
Balanced category      50
Balanced set      3
Ball      248
Ball, open      9
Banach inverse operator theorem      150
Banach — Alaoglu theorem      270
Banach — Steinhaus theorem      154
Banach — Stone Theorem      198
Base of a topological space      11
Basis of an object      50
Bessel inequality      74
Bicommutant      353
Biextension      3
Bilinear operator, completely bounded      122
Bilinear operator, contraction      92
Bilinear operator, jointly bounded      92
Bilinear operator, multiplicatively bounded      122
Bilinear operator, Schmidt      188
Bilinear operator, separately bounded      92
Bimorphism      33
Birestriction      3
Bound      6
C*-algebra      350
C*-algebra, operator      351
C*-identity      330 351
Calculus, Borel      384
Calculus, continuous      361
Calculus, continuous, on a compact set      363
Calculus, entire holomorphic      321
Calculus, polynomial      308
Calculus, rational      309
Canonical isometric operator      110
Cartesian product      40
Category      21
Category of Abelian groups      23
Category of groups      23
Category of linear spaces      22
Category of measure spaces      23
Category of metric spaces      23
Category of ordered sets      23
Category of sets      22
Category of topological spaces      23
Cauchy — Bunyakovskii inequality      68
CHARACTER      307
Character, group      449
Closed graph theorem      151
Closed set      10
Closure      11
Codimension      7
Coextension      3
Cohen's factorization theorem      430
Coisometry      78
Commutant      353
Commutative diagram      28
Compact      209
Compactification      197
Compactification, Alexandroff      200
Completion      168
Complex Borel measure      61
Complex-conjugate space      4
Conjugate linear isomorphism      4
Conjugate linear operator      4
Continuous mapping      16
Contraction      9
Convergence, classical      245 279
Convergence, coordinatewise      246
Convergence, generated by a prenorm      246
Convergence, weak      264
Convergence, weak*      264
Convergence, Weierstrass      245 250
Convex set      3
Convolution      312
Convolution of a function and a generalized function      429
Convolution of functions      424
Convolution of functions, on a group      454
Convolution of generalized functions      430
Coproduct of a family of objects      38
Corestriction      3
Coretraction      32
Cosimplicial $\mathcal{K}$-object      51
Covering      191
Crust      278
Dense subset      12
Derivative, generalized      289
Derivative, generalized, Sobolev      289
Diagram      28
Diameter      15
Dini condition      421
Direct complement, topological      97
Direct product, Banach      127
Direct sum of a family of linear spaces      41
Direct sum of subspaces      97
Direct sum, Banach      127
Discrete category      22
Disjoint union      2
Distance, between an element and a subset      73
Distance, from a point to a subset      9
distribution      282
Divisible abelian group      35
Domain of a morphism      21
Dual category      24
Dual equivalence of categories      53
Dual space, linearly      4
Element, adjoint      348
Element, idempotent      308
Element, inverse      305
Element, invertible      305
Element, nilpotent      308
Element, normal      348
Element, positive      365
Element, selfadjoint      348
Element, unitary      348
Elementary tensors      175
Endomorphism      22
Enflo theorem      214
Enflo — Read theorem      140
Epimorphism      33
Equation, adjoint      333
Equation, Volterra      320
Equicontinuous set of functions      208
Equivalent categories      53
Equivalent families of prenorms      253
Equivalent families of subspaces      373
Equivalent measure spaces      9
Equivalent prenorms      87
Essentially bounded function      59
Extension      3
Extreme epimorphism      37
Extreme monomorphism      36
Factor      359
Family of prenorms, $\Lambda$-weak      263
Family of prenorms, compatible      274
Family of prenorms, majorizing another family      253
Family of prenorms, strong-operator      247 259
Family of prenorms, weak      263
Family of prenorms, weak*      263
Family of prenorms, weak-measure      383
Family of prenorms, weak-operator      248
Family of spectral types      404
Family of subsets, centered      192
Family of subspaces associated with an operator      372
Final object      24
Forgetful functor      48
Fourier coefficients      73 413
Fourier integral      414
Fourier operator      417
Fourier series      73
Fourier transform      414
Fourier transform of a tempered generalized function      439
Fourier transform on $\mathbb{Z}$      415
Fourier transform on a group      451
Fourier transform on the circle      415
Fourier transform, Hilbert      442 445
Fourier transform, inverse      421
Fredholm alternative      237
Fredholm theorem      240
Free object      50
Freedom functor      49 51
Full category      22
Function, Borel      383
Function, finitary      278
Function, Gauss      279
Function, generalized      282
Function, generalized, concentrated on      292
Function, generalized, of finite order      283
Function, generalized, regular      283
Function, generalized, singular      283
Function, generalized, tempered      282
Function, generalized, with compact support      282
Function, Heaviside      290
Function, locally integrable      283
Function, rapidly decreasing      279
Function, resolvent      316
Function, simple      393
Function, square-integrable      67
Function, symmetric      338
Function, vanishing at infinity      63 193
Functional      4
Functional, conjugate bilinear      66
Functional, evaluating      263
Functional, positive      357
Functional, regular      286
Functor of continuous functions      198
Functor, Banach adjointness (Banach star)      156
Functor, contravariant      45
Functor, covariant      45
Functor, faithful      46
Functor, full      46
Functor, Gelfand      324
Functor, Hilbert adjointness      330
Functor, weak* star      275
Fundamental sequence      125
Gelfand functor      324
Gelfand spectrum      324
Gelfand theorem      323
Gelfand transform      324
Gelfand — Mazur theorem      317
Gelfand — Naimark theorem, first      352
Gelfand — Naimark theorem, second      352
GNS-representation      357
Gowers theorem      138
Greatest element      8
Grothendieck operator      183
Grothendieck theorem on $L_{1}(X,\mu)$      181
Grothendieck theorem on approximation property      214
Group, dual      450
Group, topological      449
Group, topological, abelian      449
Group, topological, locally compact      449
Hahn — Banach theorem      103
Hellinger's theorem (final form of Hilbert's spectral theorem)      404
Hermite function      73
Hermite polynomial      73
Hilbert spectral theorem in terms of operator-valued Lebesgue integral      394
Hilbert spectral theorem in terms of operator-valued Riemann — Stiltjes integral      380
Hilbert spectral theorem, geometric form      402
Hilbert — Schmidt theorem      341
Homeomorphism      17 25
Homeomorphism, uniform      25
Homomorphism      306
Homomorphism, involutive      348
Homomorphism, unital      306
Hyperplane      105
Hyperplane of support      105
Hyperplane, separating subsets      113
Ideal      309
Identity, Hilbert's      316
Image of the mapping      2
INDEX      233
Inequality, multiplicative      311
Initial object      24
Inner product      66
Integral equation      239
Integral equation, homogeneous      239
Integral, Lebesgue — Stieltjes      108
Integral, Lebesgue, with respect to a spectral measure      394
Interior point      10
Invariant for an isomorphism      26
Invariant subspace      4
Inverse morphism      25
Involution      347
Isometry      9 78
Isomorphic objects      25
Isomorphism      25
Isomorphism of algebras      306
Isomorphism, involutive      348
Isomorphism, isometric      83
Isomorphism, isometric of (pre)normed spaces      84
Isomorphism, isometric of Banach algebras      313
Isomorphism, isometric, complete      118
Isomorphism, topological      83 258
Isomorphism, topological of Banach algebras      313
Isomorphism, topological, complete      118
Isomorphism, unitary of (pre)normed spaces      84
Kernel      3
Kernel (of an integral operator)      81
Krein — Milman theorem      273
Kuratowski theorem      197
Least element      8
Left inverse morphism      32
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