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Helemskii A.Ya. — Lectures and Exercises on Functional Analysis, Vol. 233
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Название: Lectures and Exercises on Functional Analysis, Vol. 233Автор: 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: Рубрика: Математика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 2006Количество страниц: 468Добавлена в каталог: 09.04.2008Операции: Положить на полку |
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63 95 285 3 201 206 *-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 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, 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 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 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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