Ash R.B. — Real Variables with Basic Metric Space Topology :: Электронная библиотека попечительского совета мехмата МГУ

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Ash R.B. — Real Variables with Basic Metric Space Topology

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Название: Real Variables with Basic Metric Space Topology

Автор: Ash R.B.

Аннотация:

This is a text for a first course in real variables. The subject matter is fundamental for more advanced mathematical work, specifically in the areas of complex variables, measure theory, differential equations, functional analysis, and probability. In addition, many students of engineering, physics, and economics find that they need to know real analysis in order to cope with the professional literature in their fields. Standard mathematical writing, with its emphasis on formalism and abstraction, tends to create barriers to learning and focus on minor technical details at the expense of intuition. On the other hand, a certain amount of abstraction is unavoidable if one is to give a sound and coherent presentation. This book attempts to strike a balance that will reach the widest audience possible without sacrificing precision.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2007

Количество страниц: 213

Добавлена в каталог: 28.03.2010

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Предметный указатель

"And" connective      15
"Implies" connective      15—16
"Not" connective      15
"Or" connective      15
$F_{\sigma}$ set      148
$G_{\delta}$ set      148
Abel's theorem      55
Algebra of functions      137
Applications: of compactness      33—37
Applications: of the Mean Value Theorem      86—91
Applications: of the Weierstrass M-test      125—129
Arzela — Ascoli theorem      132—133
Bake Category Theorem      144—149
Bake Category Theorem, proof of      145—146
Ball      11
Bolzano — Weierstrass theorem      34—35
Bolzano — Weierstrass Theorem, corollary      35—36
Boundary point      23 42
Bounded function      65 125
Bounded sequence      34
Bounded sets      29—35 161—162
Bounded variation, functions of      106—111
Calculus, Fundamental Theorem of      104—105
Cantor diagonal process      8
Cantor set      76—80
Cantor's Nested, Set Property      29 164—165
Category 1 and category 2      145
Cauchy product of power series      139
Cauchy sequence      24 40—41 124 164
Cauchy — Schwarz inequality      52
Cauchy — Schwarz inequality, for integrals      115
Cauchy — Schwarz inequality, for sums      52
Chain rule      81—82
Change of variable formula      112
Closed ball      12
Closed sets      12—13
Closed sets, continuity and      61—62
Closed sets, relation between open sets and      13—14
Closed sets, unions and intersection of      25—28
Closure      22
Cluster point      21
Co-finite set      23
Codomain      57
Compact set      30
Compactness      28—37 161—164
Compactness, and closed, bounded sets      30
Compactness, and continuity      64—70
Compactness, applications of      33—37
Compactness, Bolzano — Weierstrass Theorem      34—35
Compactness, criteria for      162
Compactness, definition      30
Compactness, Heine — Borel Theorem      31—32 35 130 164
Compactness, nested set property      29
Complement      2 3 6
Complete metric space      41
Completeness, and least upper bounds      37—42
Composition of functions      59
Connectedness      150—152
Continuity      57—80
Continuity, and compactness      64—70
Continuity, definition      57
Continuity, discontinuities, types of      70—76
Continuity, epsilon-delta approach      57—58
Continuity, global vs. local concept of      59 67
Continuity, semicontinuous functions      152—158
Continuity, uniform      66—70
Continuous functions      57ff
Contradiction, proof by      16—17
Contrapositive      17
Convergence      see also “Pointwise convergence; Uniform convergence” 10—15
Convergence pointwise      117—120
Convergence, of power series      52—55
Convergence, radius of      54
Convergent subsequence      33—34
Countable sets      7—10
Countably infinite sets      7—10
De Morgan laws      5 6 27
Decreasing function      74 85
Decreasing sequence      41
Deleted open ball      21
Dense set      68—69 145—146
Derivative      81ff
Diagonal process      8 46 131 163
Differentiable function      81ff
Differentiation      81—92
Differentiation, definition      81—83
Differentiation, Generalized Mean Value Theorem      85—86
Differentiation, Mean Value Theorem      83—85
Differentiation, Mean Value Theorem, applications of      86—91
Dini's theorem      124—125
Direct image      62—63
Disconnected sets      150—151
Discontinuities, infinite      71
Discontinuities, jump      71
Discontinuities, nonsimple      73
Discontinuities, of the first kind      71—72
Discontinuities, of the second kind      73—74
Discontinuities, point      71
Discontinuities, removable      71
Discontinuities, simple      71
Discontinuities, types of      70—76
Disjoint sets      5—6 151
Distance      10—15
Distance function      10
Distance, from a point to a set      79
Distributive law for sets      2
Domain      57
Empty set      5
Epsilon-delta characterization      57—58 71 73 76
Equicontinuity, and Arzela — Ascoli Theorem      130—133
Equicontinuous family of functions      131
Euclidean metric      10 52
Euclidean p-space      11
Euclidean plane      2
Eventually (ev)      49
Everywhere continuous, nowhere differentiable function      126—129
Existential quantifier      18
Extended real numbers      35
Extension: of a bounced continuous function      143
Extension: of a uniformly continuous function      69
Extension: Tietze Extension Theorem      143
Finite sets      7
Function      33
Function, bounded      65 125
Function, continuous      57ff
Function, decreasing      74 85
Function, differentiable      81ff
Function, everywhere continuous and nowhere differentiable      126—129
Function, increasing      74 85
Function, left-continuous      111
Function, lower semicontinuous      152
Function, monotone      74
Function, of bounded variation      106—111
Function, right-continuous      110
Function, semicontinuous      152—158
Function, uniformly continuous      66—70
Function, upper semicontinuous      152
Function, variation of a      106
Fundamental theorem of calculus      104—105
Fundamental Theorem of Calculus, intuitive view of      105
Generalized Mean Value Theorem      85—86
Global concept of continuity      59 67
Greatest lower bound      38
Heine — Borel theorem      31—32 35 130 164
Homeomorphism      144 158
Horizontal line test for uniform convergence      119—120
Image      62—63
Improper integrals      114
Increasing function      74 85
Increasing sequence      41

Induction      19
Inductive procedure      19
Infimum (inf)      38
Infinite discontinuity      71
Infinite sets      7—9
Infinitely often (i.o.)      49
Infs, properties of      39—40
integral      see “Riemann — Stieltjes integral”
Integral test      115
Integration by parts      111—112
Interchange of operations      117—119 122—124
Interior of a set      145
Intermediate Value Theorem      75—76
Intermediate value theorem, for derivatives      87—88
Intersection      2 3 6
Invalid interchange of operations, examples of      117—119
Inverse image      59
Irrational numbers      147
Isolated point      21
Jump discontinuity      71
Jump function      102—103
l'Hospital's rule      81 88—90
Largest subsequential limit      46
Least upper bounds      38
Least upper bounds, and completeness      37—42
Left-continuous function      111
lim inf      46
lim sup      46
Limit concept, generalization of      45—48
Limit operations, and uniform convergence      122—125
Limit point      21
Limit, definition of      11
Limit, lower      46
Limit, upper      46
Line integrals, and Riemann — Stieltjes integral      104
Local concept of continuity      59 67
Local maximum      83
Local minimum      83
Logic      15—21
Logic, mathematical induction      18—19
Logic, negations      19—21
Logic, proof, types of      16—17
Logic, quantifiers      17—18
Logic, truth tables      15—16
Lower bound      38
Lower limit      46
Lower limit, properties of      49
Lower semicontinuous (LSC) function      152—158
Lower sum      94
Mapping      57
Mathematical induction      18—19
Maximum      65 83
Mean value theorem      81 83—85
Mean Value Theorem, applications of      86—91
Mean Value Theorem, for integrals      112—113
Mean value theorem, generalized      85—86
Metric      10
Metric space      10—13
Metric space, compactness criteria in      162—164
Minimum      65 83
Monotone function      74
Monotone sequence      41 124—125
Mutually exclusive sets      5
Negations      19—21
Neighborhood      155
Nested, Set Property      29 164—165
Nonsimple discontinuity      73
Nowhere dense sets      78 144—145 148
Open ball      11
Open sets      12
Open sets, continuity and      61—62
Open sets, relation between closed sets and      13—14
Open sets, unions and intersections of      25—28
Open subsets of R      42
Order of summation, reversal of      137—139
Partition      93
Partition, refinement of      95
Partition, size of      93
Path-connectedness      150—151
Perfect set      78
Piecewise continuous function      99
Point discontinuity      71
Pointwise bounded sequence of functions      132
Pointwise convergence      117—120
Pointwise convergence, vertical line test for      119—120
Power series, convergence of      52—55
Predicate      17—18
Preimage      59—61
Probability, and Riemann — Stieltjes integral      104
Proof: by cases      22
Proof: by contradiction      17
Proof: types of      17
Proof: via contrapositive      17
Proper subset      4
Proposition      15
Quantifiers      17—18 19
Radius of convergence      54
Ratio Test      53
Rational numbers      7 147
Real numbers      1 165
Real numbers, extended      35
Real numbers, upper/lower limits of sequences of      45—56
refinement      95
Relative topology      142
Removable discontinuity      71
Riemann integral      93ff
Riemann — Stieltjes integral      93—116
Riemann — Stieltjes integral, change of variable formula      112
Riemann — Stieltjes integral, definitions      93—96
Riemann — Stieltjes integral, evaluation formula for      103
Riemann — Stieltjes integral, existence of      96
Riemann — Stieltjes integral, improper      114
Riemann — Stieltjes integral, integration by parts      111—112
Riemann — Stieltjes integral, line integrals and      104
Riemann — Stieltjes integral, Mean Value Theorem for      112—113
Riemann — Stieltjes integral, probability and      104
Riemann — Stieltjes integral, properties of      98—106
Riemann — Stieltjes integral, upper bounds on      113
Riemann — Stieltjes sum      94
Right-continuous function      110
Rolle's theorem      83—86
Root test      53—55
Semicontinuous functions      152—158
Separated sets      151
SEQUENCE      33—34 45—52
Sequence, and limit concept      45
Sequence, bounded      34
Sequence, Cauchy      24 40—41 124 164
Sequence, convergence of      10—15
Sequence, monotone      41 124—125
Sequence, unbounded      36
Sequences of real numbers, upper and lower limits of      45—56
Set-theoretic difference      6
Sets      1ff
Sets, $F_{\sigma}$       148
Sets, $G_{\delta}$       148
Sets, and category      2 145
Sets, bounded      29—35 161—162
Sets, Cantor      76—80
Sets, closed      12—14 25—28
Sets, closure of      21—22
Sets, co-finite      23
Sets, compact      30
Sets, complement of      2 3 6
Sets, connected      150—152
Sets, countable      7—10
Sets, dense      68 69 145—146
Sets, disjoint      5—6 151
Sets, distributive law for      2
Sets, empty      5

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