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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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Limit of a net 19 Limit of a sequence 18 Limit point 10 Limit point, strict 10 Lindenstrauss — Tzafriri theorem 149 Linear basis 7 Linear complement 3 Linear dimension 7 Linear isomorphism 25 Linear span 3 Linearly independent system of vectors 7 Mapping, bijective 2 Mapping, continuous 9 Mapping, continuous at a point 9 Mapping, injective 2 Mapping, isometric 9 Mapping, open 17 Mapping, surjective 2 Mapping, topologically injective 17 Mapping, topologically surjective 17 Mapping, uniformly continuous 9 Matrix of an operator 90 Maximal element 6 Mean convergence 60 Mean-square convergence 60 Measurable mapping 8 Measurable mapping, proper 8 Measurable set 8 Measurable set with a countable basis 8 Measure space 8 Measure, absolutely continuous 399 Measure, Haar 451 Measure, left-invariant 451 Measure, personal, of a vector 397 Measure, regular 451 Measure, spectral 390 Measure, spectral, of an operator 391 Measure, unimodular 451 Measures, equivalent 399 Measures, mutually singular 404 Measures, orthogonal 404 Metric subspace 9 Metric, discrete 13 Metric, inherited 9 Milyutin theorem 199 Minkowski functional 64 Model 27 Monomorphism 33 Mor-functor, covariant 47 Multiplication 304 Multiplicative inequality for operator prenorms 77 Multiplicative property of index 235 Multiplicity 343 Natural equivalence 52 Natural equivalence of contravariant functors 52 Natural transformation of covariant functors 52 Neighborhood 10 Net 19 Nikol'skii theorem 241 Norm 55 Norm of the nth floor 115 Norm, admissible 164 Norm, Hilbert 70 Norm, nuclear 183 226 Norm, operator 77 Norm, quantum 115 Norm, quantum, generated by 117 Norm, quantum, standard 116 Norm, Schmidt 188 221 Normed space 55 Open mapping principle 150 Open set 10 248 Operator 3 Operator equation 239 Operator equation, homogeneous 239 Operator of bilateral shift 79 Operator of composition 257 Operator of indefinite integration 80 Operator of translation by... 82 Operator, adjoint, weak* 266 Operator, associated with a bilinear operator 173 Operator, associated with the resolution of the identity 378 Operator, bounded 76 Operator, completely bounded 118 Operator, completely contractive 118 Operator, completely isometric 118 Operator, continuous 256 Operator, contraction 78 Operator, cyclic 398 Operator, diagonal 79 Operator, differentiation 82 256 Operator, finite-dimensional 4 Operator, Fredholm 233 Operator, generated by 94 Operator, hermitian 336 Operator, Hilbert — Schmidt 221 Operator, homogeneous 405 Operator, idempotent 98 Operator, identity 78 Operator, integral 81 Operator, n-dimensional 4 Operator, normal 337 Operator, nuclear 183 226 Operator, one-dimensional 4 Operator, positive 346 Operator, Schmidt 221 Operator, selfadjoint 336 Operator, shift 79 Operator, unitary 84 Operator, Volterra 82 Operators, equivalent, isometrically 89 Operators, equivalent, topologically 89 Operators, equivalent, unitarily 90 Operators, equivalent, weakly isometrically 88 Operators, equivalent, weakly topologically 88 Operators, equivalent, weakly unitarily 90 Operators, partially isometric 336 Order 5 Order in the algebra 368 Order of a generalized function 283 Order, discrete 6 Order, linear 6 Ordered set 6 Ordered set, directed 19 Ordered set, linearly 6 Orthogonal vectors 71 Orthogonalization process 73 Orthonormal system of vectors 71 Paley — Wiener theorem 448 Parallelogram identity 70 Part of an operator, negative 370 Part of an operator, positive 370 Path 17 28 Plancherel theorem 445 Point at infinity 200 Point of increase 374 Point, discontinuity 374 Point, extreme 273 Point, interior 248 Point, linearly interior 113 Point, regular 297 305 Point, singular 297 305 Point, singular, essentially 299 Polar identity 66 Polar identity for a prenorm 70 Polynomial of an element of an algebra 308 Pontryagin's duality principle 452 Pre-inner product 66 Premetric 14 Prenorm 55 Prenorm of a bilinear operator 92 Prenorm, accompanying 247 Prenorm, admissible 278 Prenorm, generated by a (pre-)inner product 70 Prenorm, Hilbert 70 Prenorm, majorizing 87 Prenorm, operator 76 Prenorm, projective 178 Prenorm, standard 278 Prenormed space 55 Presheaf 52 Primitive generalized 291 Principle of nested closed subsets 126 Principle, extension-by-continuity 131 Product of a family of objects 38 Projection 98 Proper subset 3 Pythagorean equality 71 Quadratic form 66 336 Quantization 116 Quantization, maximal 117 Quantization, minimal 117 Quotient (pre)norm 57 Quotient space topological 12 Rademacher functions 72 Radius spectral 319 Range of a morphism 21 Rarefied set 126 Reflection, orthogonal 335 Representation 307 Representation, involutive 348 Representation, universal 358 Representing object 54 Resolution of the identity 375 Restriction 3 Retraction 32 Riemann — Stiltjes, integral 376 Riemann — Stiltjes, integral sum 376 Riesz theorem 109 144 Riesz theorem on compactness of unit ball 207 Riesz — Fischer theorem 134 Right inverse morphism 32 s-numbers 218 Sakai theorem 355 Schauder basis 75 Schmidt series 218 Schmidt theorem 215 Sequence in a category 160 Sequence in a category, dual 160 Sequence in a category, exact 160 Sequence in a category, short 161 Sequence, 287 427 Sequence, square-integrable 67 Sequence, two-sided 413 Sequence, two-sided, rapidly decreasing 437 Set, fat 126 Set, meager 126 Shift of a set 451 Similar endomorphisms 29 Simplicial 51 Singleton 14 Space, 278 Space, 279 Space, 279 Space, accompanying 247 Space, Banach 126 Space, countably normed 247 Space, dual 77 Space, Frechet 255 Space, Hilbert 126 Space, locally convex 250 Space, metric 9 Space, metric, totally bounded 201 Space, normed, quantized 115 Space, normed, underlying 116 Space, polynormed 246 Space, polynormed, normal 275 Space, polynormed, strongest 247 Space, polynormed, weakest 257 Space, pre-Hilbert 67 Space, predual 354 Space, premetric 15 Space, premetric, bounded 15 Space, prenormable 254 Space, quantized, maximal 117 Space, quantized, minimal 117 Space, quantum 115 Space, reflexive 111 Space, Schwartz 279 Space, second dual 110 Space, topological vector 250 Space, Varopoulos 183 Spectral mapping law for an entire calculus 322 Spectral mapping law for continuous calculus 364 Spectral picture 402 Spectral picture, ordered 404 Spectral radius formula 319 Spectral type 399 Spectral types, independent 404 Spectrum 297 Spectrum of an operator 305 Spectrum, continuous 298 Spectrum, essential 299 Spectrum, Gelfand 324 Spectrum, point 298 Spectrum, residual 299 Square root (of an operator) 366 Standard simplicial category 24 State 358 Step-function 377 Strictly positive functional 357 Subalgebra 309 Subbase of a topological space 11 Subcategory 22 Subcovering 191 Subnet 192 Subset in an ordered set bounded 6 Subset, bounded 254 Subset, compact 191 Subset, relatively compact 199 Subset, selfadjoint 349 Subset, totally bounded 201 Subspace in the algebraic dual space sufficient 262 Subspace, cyclic 397 Subspace, polynormed 247 Subspace, quantum 116 Subspace, topological 11 Subspace, topologically complemented 97 Sum of linear spaces 3 Sum, Hilbert of Hilbert spaces 400 Sum, Hilbert of operators 401 Support 292 Support of a Borel measure 404 Tensor product, algebraic 173 Tensor product, Banach 177 Tensor product, Banach of operators 182 Tensor product, Banach, functor of 183 Tensor product, Hilbert 187 Tensor product, Hilbert, functor of 190 Tensor product, projective 177 Tensor product, projective of prenorms 178 Tensor trace 184 Theorem, on Borel functional calculus 385 Theorem, on continuous functional calculus 361 Theorem, on existence of algebraic tensor product 175 Theorem, on existence of completion 170 Theorem, on existence of projective tensor product 179 Theorem, on injectivity of Fourier transform 452 Theorem, on the inversion of Fourier transform 436 Theorem, on the stability of index 325
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