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

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Ќазвание: Real Variables with Basic Metric Space Topology

јвтор: Ash R.B.


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: en

–убрика: ћатематика/

—татус предметного указател€: √отов указатель с номерами страниц

ed2k: ed2k stats

√од издани€: 2007

 оличество страниц: 213

ƒобавлена в каталог: 28.03.2010

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ѕредметный указатель
"And" connective      15
"Implies" connective      15Ч16
"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      33Ч37
Applications: of the Mean Value Theorem      86Ч91
Applications: of the Weierstrass M-test      125Ч129
Arzela Ч Ascoli theorem      132Ч133
Bake Category Theorem      144Ч149
Bake Category Theorem, proof of      145Ч146
Ball      11
Bolzano Ч Weierstrass theorem      34Ч35
Bolzano Ч Weierstrass Theorem, corollary      35Ч36
Boundary point      23 42
Bounded function      65 125
Bounded sequence      34
Bounded sets      29Ч35 161Ч162
Bounded variation, functions of      106Ч111
Calculus, Fundamental Theorem of      104Ч105
Cantor diagonal process      8
Cantor set      76Ч80
Cantor's Nested, Set Property      29 164Ч165
Category 1 and category 2      145
Cauchy product of power series      139
Cauchy sequence      24 40Ч41 124 164
Cauchy Ч Schwarz inequality      52
Cauchy Ч Schwarz inequality, for integrals      115
Cauchy Ч Schwarz inequality, for sums      52
Chain rule      81Ч82
Change of variable formula      112
Closed ball      12
Closed sets      12Ч13
Closed sets, continuity and      61Ч62
Closed sets, relation between open sets and      13Ч14
Closed sets, unions and intersection of      25Ч28
Closure      22
Cluster point      21
Co-finite set      23
Codomain      57
Compact set      30
Compactness      28Ч37 161Ч164
Compactness, and closed, bounded sets      30
Compactness, and continuity      64Ч70
Compactness, applications of      33Ч37
Compactness, Bolzano Ч Weierstrass Theorem      34Ч35
Compactness, criteria for      162
Compactness, definition      30
Compactness, Heine Ч Borel Theorem      31Ч32 35 130 164
Compactness, nested set property      29
Complement      2 3 6
Complete metric space      41
Completeness, and least upper bounds      37Ч42
Composition of functions      59
Connectedness      150Ч152
Continuity      57Ч80
Continuity, and compactness      64Ч70
Continuity, definition      57
Continuity, discontinuities, types of      70Ч76
Continuity, epsilon-delta approach      57Ч58
Continuity, global vs. local concept of      59 67
Continuity, semicontinuous functions      152Ч158
Continuity, uniform      66Ч70
Continuous functions      57ff
Contradiction, proof by      16Ч17
Contrapositive      17
Convergence      see also УPointwise convergence; Uniform convergenceФ 10Ч15
Convergence pointwise      117Ч120
Convergence, of power series      52Ч55
Convergence, radius of      54
Convergent subsequence      33Ч34
Countable sets      7Ч10
Countably infinite sets      7Ч10
De Morgan laws      5 6 27
Decreasing function      74 85
Decreasing sequence      41
Deleted open ball      21
Dense set      68Ч69 145Ч146
Derivative      81ff
Diagonal process      8 46 131 163
Differentiable function      81ff
Differentiation      81Ч92
Differentiation, definition      81Ч83
Differentiation, Generalized Mean Value Theorem      85Ч86
Differentiation, Mean Value Theorem      83Ч85
Differentiation, Mean Value Theorem, applications of      86Ч91
Dini's theorem      124Ч125
Direct image      62Ч63
Disconnected sets      150Ч151
Discontinuities, infinite      71
Discontinuities, jump      71
Discontinuities, nonsimple      73
Discontinuities, of the first kind      71Ч72
Discontinuities, of the second kind      73Ч74
Discontinuities, point      71
Discontinuities, removable      71
Discontinuities, simple      71
Discontinuities, types of      70Ч76
Disjoint sets      5Ч6 151
Distance      10Ч15
Distance function      10
Distance, from a point to a set      79
Distributive law for sets      2
Domain      57
Empty set      5
Epsilon-delta characterization      57Ч58 71 73 76
Equicontinuity, and Arzela Ч Ascoli Theorem      130Ч133
Equicontinuous family of functions      131
Euclidean metric      10 52
Euclidean p-space      11
Euclidean plane      2
Eventually (ev)      49
Everywhere continuous, nowhere differentiable function      126Ч129
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      126Ч129
Function, increasing      74 85
Function, left-continuous      111
Function, lower semicontinuous      152
Function, monotone      74
Function, of bounded variation      106Ч111
Function, right-continuous      110
Function, semicontinuous      152Ч158
Function, uniformly continuous      66Ч70
Function, upper semicontinuous      152
Function, variation of a      106
Fundamental theorem of calculus      104Ч105
Fundamental Theorem of Calculus, intuitive view of      105
Generalized Mean Value Theorem      85Ч86
Global concept of continuity      59 67
Greatest lower bound      38
Heine Ч Borel theorem      31Ч32 35 130 164
Homeomorphism      144 158
Horizontal line test for uniform convergence      119Ч120
Image      62Ч63
Improper integrals      114
Increasing function      74 85
Increasing sequence      41
Induction      19
Inductive procedure      19
Infimum (inf)      38
Infinite discontinuity      71
Infinite sets      7Ч9
Infinitely often (i.o.)      49
Infs, properties of      39Ч40
integral      see УRiemann Ч Stieltjes integralФ
Integral test      115
Integration by parts      111Ч112
Interchange of operations      117Ч119 122Ч124
Interior of a set      145
Intermediate Value Theorem      75Ч76
Intermediate value theorem, for derivatives      87Ч88
Intersection      2 3 6
Invalid interchange of operations, examples of      117Ч119
Inverse image      59
Irrational numbers      147
Isolated point      21
Jump discontinuity      71
Jump function      102Ч103
l'Hospital's rule      81 88Ч90
Largest subsequential limit      46
Least upper bounds      38
Least upper bounds, and completeness      37Ч42
Left-continuous function      111
lim inf      46
lim sup      46
Limit concept, generalization of      45Ч48
Limit operations, and uniform convergence      122Ч125
Limit point      21
Limit, definition of      11
Limit, lower      46
Limit, upper      46
Line integrals, and Riemann Ч Stieltjes integral      104
Local concept of continuity      59 67
Local maximum      83
Local minimum      83
Logic      15Ч21
Logic, mathematical induction      18Ч19
Logic, negations      19Ч21
Logic, proof, types of      16Ч17
Logic, quantifiers      17Ч18
Logic, truth tables      15Ч16
Lower bound      38
Lower limit      46
Lower limit, properties of      49
Lower semicontinuous (LSC) function      152Ч158
Lower sum      94
Mapping      57
Mathematical induction      18Ч19
Maximum      65 83
Mean value theorem      81 83Ч85
Mean Value Theorem, applications of      86Ч91
Mean Value Theorem, for integrals      112Ч113
Mean value theorem, generalized      85Ч86
Metric      10
Metric space      10Ч13
Metric space, compactness criteria in      162Ч164
Minimum      65 83
Monotone function      74
Monotone sequence      41 124Ч125
Mutually exclusive sets      5
Negations      19Ч21
Neighborhood      155
Nested, Set Property      29 164Ч165
Nonsimple discontinuity      73
Nowhere dense sets      78 144Ч145 148
Open ball      11
Open sets      12
Open sets, continuity and      61Ч62
Open sets, relation between closed sets and      13Ч14
Open sets, unions and intersections of      25Ч28
Open subsets of R      42
Order of summation, reversal of      137Ч139
Partition      93
Partition, refinement of      95
Partition, size of      93
Path-connectedness      150Ч151
Perfect set      78
Piecewise continuous function      99
Point discontinuity      71
Pointwise bounded sequence of functions      132
Pointwise convergence      117Ч120
Pointwise convergence, vertical line test for      119Ч120
Power series, convergence of      52Ч55
Predicate      17Ч18
Preimage      59Ч61
Probability, and Riemann Ч 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      17Ч18 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      45Ч56
refinement      95
Relative topology      142
Removable discontinuity      71
Riemann integral      93ff
Riemann Ч Stieltjes integral      93Ч116
Riemann Ч Stieltjes integral, change of variable formula      112
Riemann Ч Stieltjes integral, definitions      93Ч96
Riemann Ч Stieltjes integral, evaluation formula for      103
Riemann Ч Stieltjes integral, existence of      96
Riemann Ч Stieltjes integral, improper      114
Riemann Ч Stieltjes integral, integration by parts      111Ч112
Riemann Ч Stieltjes integral, line integrals and      104
Riemann Ч Stieltjes integral, Mean Value Theorem for      112Ч113
Riemann Ч Stieltjes integral, probability and      104
Riemann Ч Stieltjes integral, properties of      98Ч106
Riemann Ч Stieltjes integral, upper bounds on      113
Riemann Ч Stieltjes sum      94
Right-continuous function      110
Rolle's theorem      83Ч86
Root test      53Ч55
Semicontinuous functions      152Ч158
Separated sets      151
SEQUENCE      33Ч34 45Ч52
Sequence, and limit concept      45
Sequence, bounded      34
Sequence, Cauchy      24 40Ч41 124 164
Sequence, convergence of      10Ч15
Sequence, monotone      41 124Ч125
Sequence, unbounded      36
Sequences of real numbers, upper and lower limits of      45Ч56
Set-theoretic difference      6
Sets      1ff
Sets, $F_{\sigma}$      148
Sets, $G_{\delta}$      148
Sets, and category      2 145
Sets, bounded      29Ч35 161Ч162
Sets, Cantor      76Ч80
Sets, closed      12Ч14 25Ч28
Sets, closure of      21Ч22
Sets, co-finite      23
Sets, compact      30
Sets, complement of      2 3 6
Sets, connected      150Ч152
Sets, countable      7Ч10
Sets, dense      68 69 145Ч146
Sets, disjoint      5Ч6 151
Sets, distributive law for      2
Sets, empty      5
1 2
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