Berberian S.K. — Fundamentals of Real Analysis :: Электронная библиотека попечительского совета мехмата МГУ

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Berberian S.K. — Fundamentals of Real Analysis

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Название: Fundamentals of Real Analysis

Автор: Berberian S.K.

Аннотация:

Integration theory and general topology form the core of this textbook for a first-year graduate course in real analysis. After the foundational material in the first chapter (construction of the reals, cardinal and ordinal numbers, Zom's Lemma, and transfinite induction), measure, integral, and topology are introduced and developed as recurrent themes of increasing depth.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 1st edition

Год издания: 1998

Количество страниц: 479

Добавлена в каталог: 04.04.2008

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Предметный указатель

Norm      314
Normed space      314
Normed space, finite-dimensional      468
Null functions on a measure space      314
Null set      156
One-one correspondence      10
One-one correspondence, function      10
Onto function      10
Open ball in a metric space      127
Open covering      274
Open set in a metric space      128
Open set in a topological space      131
Operator      9
Order ideal      63
Order ideal, isomorphism      23
Order ideal, monomorphism      23
Order ideal, morphism      23
Order ideal, relation      20
Ordinal numbers      60
Ordinal numbers, countable      67
Ordinal numbers, denumerable      67
Ordinal numbers, finite      67
Ordinal numbers, infinite      67
Ordinal numbers, natural ordering of      62
Ordinal numbers, sum of      68
Ordinal numbers, uncountable      67
Ordinality      59
Ordinality of a cardinal number      69
Outer measure      95 365
Outer measure, Lebesgue      87
Outer regularity of Lebesgue measure      105
p-Norm      332
Partial ordering      20
Partially ordered set      21
Partition of a set      18
Peano’s existence theorem      416
Picard’s existence theorem      410
Planar Lebesgue measure      379
Point mass      106
Pointwise Cauchy sequence of functions      286
Pointwise Cauchy sequence of functions, convergent sequence of functions      153 286
Pointwise Cauchy sequence of functions, limit of a sequence of functions      153 286
Pointwise Cauchy sequence of functions, totally bounded set of functions      399
Polar decomposition of a signed measure      444
Positive integers      2
Positive linear form      162 168
Positive part of a function      152
Power series      84 288 297 353
Power set      12
Pre-ordered set      21
Pre-ordering relation      21
Primitive, a.e.      248
Principle of Mathematical Induction      23
Principle of transfinite induction      42
Product measure space      378
Product of sets      13
Product of sets, $\sigma$ -algebras      371
Product of sets, $\sigma$ -finite measures      378
Product of sets, order relations      22 24
Product of sets, real or complex measures      379
Product of sets, topological spaces      352
Projection mapping      14
Proper subset      4
Proposition      2
Pseudometric      118 123
Pseudometric space      118 123
pth-power integrable function      332
Purely negative subset      428
Purely positive subset      428
Quasicompact space      274
Quasicompact subset      274
Quotient mapping      19
Quotient of a pre-order relation      22
Quotient set      19
Radon — Nikodym theorem      195 441
Range of a relation      8
Rare subset      303
Rational numbers      2
Real measure      319
Real numbers      2
Real numbers, extended      73
Real part of a complex-valued function      349
Real variable, function of      33
Real-valued function      33
Reflexive relation      17 20
Regularization of a function, lower semicontinuous      268
Regularization of a function, upper semicontinuous      268
Relation      6
Relative topology induced on a subset      138 146 276
Restriction of a function      11
Reverse of a relation      8
Reverse of an order relation      21
Riemann integrability, Lebesgue’s criterion      268
Riesz representation theorem      327 341
Riesz — Fischer theorem      341

Right limit of a function      144
Right-derivative of a function      145
Right-differentiable function      145
Ring of subsets      432
Rising Sun Lemma      244
Russell’s paradox      5
Schroder — Bernstein theorem      47
Second category, subset of the      304
Sections of a function on a product space      383
Sections of a subset of a product space      372
Self-adjoint linear mapping      468
Semicontinuous approximations of a Lebesgue-integrable function      239
Semicontinuous regularization of a function, lower      268
Semicontinuous regularization of a function, upper      268
Seminorm      314
Seminormed space      460
Seminormed space, complete      461
Separable metric space      279
Separated space      146 275
SEQUENCE      11
Sesquilinear form      344
Signed measure      424
Signed measure, finite      190
Similar pre-ordered sets      23
Simple function      154
Simple ordering      21
Simply ordered set      21
Singleton      8
Singular function      264
Singular function, Lebesgue’s      210
Sphere in a metric space      129
Stone — Weierstrass theorem      359 361
Subcovering      274
Subset      3
Subtractible functions      423
Subtractible functions, measures      423
Subtractive set function      320
Sup-metric      119 290
Sup-norm      119
Superset      3
Supremum      73
Surjection      10
Surjective function      10
Symmetric difference of sets      107
Symmetric relation      17
Term-by-term differentiation      296
Theorem      3
Theorem on nested intervals      32
Topological space      131
topology      131
Topology of uniform convergence      399
Total ordering      21
Total variation of a finite signed measure      198
Total variation of a function      202
Total variation of a signed measure      431
Totally bounded metric space      278
Totally bounded set of functions      399
Transfinite induction, principle of      42
Transformation      9
Transitive relation      17 20
Trichotomy, law of      21
Trichotomy, law of, for cardinal numbers      51
Trichotomy, law of, for ordinal numbers      63
Trigonometric polynomials      363
Trivial measure space      106
Trivial measure space, relation      18
Trivial measure space, topology on a set      131
Tukey’s lemma      43
u.s.c      231 232
Ultimately      80
Uncountable set      38
Uniform boundedness principle      305
Uniform convergence      286
Uniform convergence, topology of      399
Uniform limit      286
Uniformly Cauchy      286
Uniformly continuous function      300
Union of subsets      4 13
Unique extension theorem      370
Unitary space      122
Upper bound      22
Upper derivate of a function      236
Upper envelope of a family of functions      230
Upper semicontinuous function      231 232
Upper semicontinuous regularization of a function      268
Upward directed partially ordered set      77
Weierstrass approximation theorem      361
Weierstrass M-test      288
Weierstrass — Bolzano property      277
Weierstrass — Bolzano theorem      276
Well-ordered set      41 59
Well-ordering of a set      41
Well-ordering theorem      42
Zermelo’s theorem      42
Zorn’s Lemma      42

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