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Lang S. — Real and Functional Analysis (Graduate Texts in Mathematics Series #142)
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: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 3rd Edition

Год издания: 1993

Количество страниц: 580

Добавлена в каталог: 04.04.2008

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Предметный указатель
$C^{p}$-invertible      361
$C^{p}$-isomorphism      361
$C_{c}$-functional      253
$H^{p}$ spaces      80
$H_{s}$ spaces      219
$l^{1}$      129
$L^{1}$ seminorm      19 128
$L^{2}$      181
$L^{2}$ bound      108 218 220 437
$L^{2}$ norm      97 182
$L^{p}$      209
$\mu$-continuous      199
$\mu$-measurable      123 155
$\sigma$-algebra      112
$\sigma$-finite      123 137 148
$\sigma$-regular measure      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, $L^{1}$      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, $L^{1}$      185 190 220
Duality, $L^{p}$      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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