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Lang S. — Real and Functional Analysis (Graduate Texts in Mathematics Series #142)
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Название: Real and Functional Analysis (Graduate Texts in Mathematics Series #142)Автор: Lang S. Аннотация: This book is meant as a text for a first-year graduate course in analysis. In a sense, the subject matter covers the same topics as elementary calculus - linear algebra, differentiation, integration - but treated in a manner suitable for people who will be using it in further mathematical investigations. The book begins with point-set topology, essential for all analysis. The second part deals with the two basic spaces of analysis, Banach and Hilbert spaces. The book then turns to the subject of integration and measure. After a general introduction, it covers duality and representation theorems, some applications (such as Dirac sequences and Fourier transforms), integration and measures on locally compact spaces, the Riemann-Stjeltes integral, distributions, and integration on locally compact groups. Part four deals with differential calculus (with values in a Banach space). The next part deals with functional analysis. It includes several major spectral theorems of analysis, showing how one can extend to infinite dimensions certain results from finite-dimensional linear algebra; a discussion of compact and Fredholm operators; and spectral theorems for Hermitian operators. The final part, on global analysis, provides an introduction to differentiable manifolds. The text includes worked examples and numerous exercises, which should be viewed as an integral part of the book. The organization of the book avoids long chains of logical interdependence, so that chapters are as independent as possible. This allows a course using the book to omit material from some chapters without compromising the exposition of material from later chapters.
Язык: Рубрика: Математика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Издание: 3rd EditionГод издания: 1993Количество страниц: 580Добавлена в каталог: 04.04.2008Операции: Положить на полку |
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361 361 253 80 219 129 19 128 181 108 218 220 437 97 182 209 199 123 155 112 123 137 148 256 Absolutely continuous 191 199 Adherent 22 Adjoint 106 392 438 469 Adjoint of differential operator 303 Alaoglu's theorem 71 Algebra 51 72 113 Algebra automorphism 61 Algebra of functions 51 Algebra of subsets 113 Almost all 122 Almost compact support 560 Almost everywhere 122 Alternating 507 Alternating, product 507 509 Anosov theorem 381 Antifunctional 391 Antilinear 95 391 Approximate 129 Approximation, 147 Approximation, Dirac 228 Approximation, Stone — Weierstrass 52 61 62 Arcwise connected 29 Ascoli's theorem 57 Atlas 523 Automorphism 67 Averaging theorem 145 221 Baire's theorem 387 Banach algebra 73 Banach isomorphism 67 Banach space 51 Barner's theorem 291 Base 23 Bessel function 245 Bessel inequality 102 Bijective 3 Bilinear 67 91 Block 498 Bonnet mean value 286 Borel set or measurable 114 Bound of linear map 65 Boundary of manifold 541 Boundary, point 22 Bounded 18 Bounded, functional 253 Bounded, linear map 65 252 Bounded, measure 199 Bounded, variation 279 284 Bourbaki's theorem 13 Bruhat-Tits 50 C*-algebra 410 Calculus of variation 358 Cantor set 49 Caratheodory, criterion 179 Caratheodory, measure 179 Carried (measure) 179 192 Cesaro summation 230 Change of variables formula 505 CHARACTER 327 407 Characteristic function 3 Chart 523 Circumcenter 50 Closed ball 20 Closed graph theorem 395 Closed map 48 Closed operator 469 Closed set 21 Closure 22 Codimension 390 Commute 81 Compact 31 Compact, groups 459 Compact, hermitian operator 442 Compact, operator 415 Compact, support 167 252 Complement 6 Complementary subspace 389 complete 21 51 Complete, metric space 21 Completion 77 Completion of a measure 173 Concentrated (measure) 192 Connected component 29 Connected set 27 continuous 24 Converges, strongly 435 Converges, weakly 107 435 Convex 84 Convolution 73 176 223 234 239 Convolution of measures 275 327 Countable 10 Countably additive 120 196 Counting measure 120 Covering 21 Decomposable form 507 508 Decomposition 195 Dense 22 Denumerable 7 Derivation of a distribution 446 Derivative 334 Differentiable 333 Differential 534 Differential, equation 135 Differential, form 508 547 Differential, form on a manifold 503 Differentiating under integral sign 175 225 355 Differentiation of sequence 356 Dini's theorems 60 178 254 Dirac distribution 298 Dirac family 228 485 Dirac measure 120 Dirac sequence 227 Direct image of a measurable space 114 Direct image of a measure 174 Direct sum 389 Discrete subgroup 312 Discrete support 305 Discrete topology 18 Distance 19 45 distribution 296 Dominated Convergence Theorem 141 184 210 Dual exponent 209 Dual space 68 Duality, 185 190 220 Duality, 211 Egoroff's theorem 172 Eigenfunction 108 Eigenspace 442 Eigenvalue 426 Eigenvector 426 Endomorphism 66 73 enumerate 7 Equicontinuous 57 Equivalent maps (for a measure) 139 Equivalent norms 20 38 Essential image 221 Essential sup 185 218 Essentially self-adjoint 472 Exterior derivative 410 Extreme point 86 Family 4 Fatou's lemma 141 Finite intersection property 31 Flow 366 377 Fourier coefficients 98 176 Fourier inversion 241 289 Fourier series 230 Fourier transform 238 288 Fredholm operator 417 Frontier 562 Fubini's theorem 162 Function 4 Functionals 68 104 Fundamental lemma of integration 111 129 Gelfand transform 407 409 Gelfand — Mazur theorem 402 Gelfand — Naimark theorem 411 Generated (-algebra) 114 Global flow 377 545 Gradient 383 Haar functional and measure 313 324 Hahn decomposition 203 Hahn — Banach theorem 70 Hahn's theorem 153 Half space 85 539 Harmonic functions 231 Hausdorff measure 180 Hausdorff space 32 Heat, equation 232 234 248 Heat, operator 232 235 248 449 485 Hermite polynomials 276 Hermitian form 95 Hermitian operator 107 438 Hilbert basis 98 Hilbert nullstellensatz 57 Hilbert space 99 Hilbert — Schmidt operators 435 461 Hilbertian operator 439 Hoelder condition 44 Hoelder inequality 210 Homeomorphism 25 Homogeneous space 323 Hyperbolic 381 Hyperplane 539 Ideal 55 Ideal topology 21 Image 3 Implicit Mapping Theorem 364 532 INDEX 420 Induced topology 23 Inductively ordered 12 Initial condition 366 371 Injective 3 Integrable 132 Integral curve 365 544 Integral in one variable 331 Integral of step maps 126 Integral operators 213 432 478 Integral, curve 544 Integral, equation 432 Integral, general theory 129 Integral, Lebesgue 166 Integral, mean value theorem 286 Integration by parts 282 Interior 22 497 Invariant subspace 81 442 450 Inverse image 6 Inverse image of differential form 513 Inverse mapping theorem 361 Invertible 66 74 Irreducible representation 459 Isometry 67 Isomorphism 66 361 527 Jacobian 503 Karamata theorem 277 Kernel 87 Kolmogoroff inequality 215 Krein — Milman theorem 88 Laguerre polynomials 276 Landau approximation 229 Laplace operator 476 Lattice points 249 Least upper bound 12 Lebesgue integral 166 Lebesgue measure 167 Lebesgue theorem 75 lim inf 140 Linear differential equations 384 Linear extension theorem 75 Lipschitz condition 366 497 Local coordinates 524 Local flow 366 Local isomorphism 361 527 Local order 302 Local representation of vector field 544 Localization of measure 270 Locally closed 527 Locally compact 39 Locally compact groups 313 Locally finite 271 536 Locally integrable 170 Locally invertible 361 Locally zero (distribution) 299 Lorch's theorem 468 Lusin's theorem 266 Manifold 524 Manifold with boundary 540 Mapping 3 Marriage problem 49 Maximal element 11 Mazur's theorem 88 Mean value theorem 341 Measurable map 114 Measurable set 113 155 Measurable space 113 Measure 120 Measure 0 497 Measure, associated with a differential form 554 Measure, associated with a functional 263 Measured space 120 Mehler family 233 Metric space 19 45 Metrizable 45 midpoint 50 Modular function 327 Monotone Convergence Theorem 139 Monotone families 174 Morphism 527 Morse — Palais lemma 455 Multilinear 68 Negative definite 450 Negligible 563 Neighborhood 24 Newton's method 380 Non-degenerate 188 Non-degenerate critical point 455 Non-measurable set 177 Non-singular 455 Norm 18 normal 33 Normed algebra 73 Normed vector space 18 Null space 96 One point compactification 40 Open ball 18
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