Mathematical Analysis

This text carefully leads the student through the basic topics of Real Analysis. Topics include metric spaces, open and closed sets, convergent sequences, function limits and continuity, compact sets, sequences and series of functions, power series, differentiation and integration, Taylor's theorem, total variation, rectifiable arcs, and sufficient conditions of integrability.

Analysis Mathematics

Tác giả: Elias Zakon
Nhà xuất bản: University of Windsor
Nguồn: LibreTexts
Giấy phép: CC BY
Ngôn ngữ: EN
Danh mục: Khoa học cơ bản

Đọc trực tuyến Tải PDF (nguồn mở)

Mục lục· 174 mục

01: Set Theory
1.01: Sets and Operations on Sets. Quantifiers
1.1.E: Problems in Set Theory (Exercises)
1.02: Relations. Mappings
1.2.E: Problems on Relations and Mappings (Exercises)
1.03: Sequences
1.04: Some Theorems on Countable Sets
1.4.E: Problems on Countable and Uncountable Sets (Exercises)
02: Real Numbers and Fields
2.01: Axioms and Basic Definitions
2.02: Natural Numbers. Induction
2.2.E: Problems on Natural Numbers and Induction (Exercises)
2.03: Integers and Rationals
2.04: Upper and Lower Bounds. Completeness
2.4.E: Problems on Upper and Lower Bounds (Exercises)
2.05: Some Consequences of the Completeness Axiom
2.06: Powers with Arbitrary Real Exponents. Irrationals
2.6.E: Problems on Roots Powers and Irrationals (Exercises)
2.07: The Infinities. Upper and Lower Limits of Sequences
2.7.E: Problems on Upper and Lower Limits of Sequences in (E) (Exercises)
03: Vector Spaces and Metric Spaces
3.01: The Euclidean n-Space E
3.1.E: Problems on Vectors in (En) (Exercises)
3.02: Lines and Planes in E
3.2.E: Problems on Lines and Planes in (En) (Exercises)
3.03: Intervals in E
3.3.E: Problems on Intervals in E (Exercises)
3.04: Complex Numbers
3.4.E: Problems on Complex Numbers (Exercises)
3.05: Vector Spaces. The Space C. Euclidean Spaces
3.5.E: Problems on Linear Spaces (Exercises)
3.06: Normed Linear Spaces
3.6.E: Problems on Normed Linear Spaces (Exercises)
3.07: Metric Spaces
3.7.E: Problems on Metric Spaces (Exercises)
3.08: Open and Closed Sets. Neighborhoods
3.8.E: Problems on Neighborhoods Open and Closed Sets (Exercises)
3.09: Bounded Sets. Diameters
3.9.E: Problems on Boundedness and Diameters (Exercises)
3.10: Cluster Points. Convergent Sequences
3.10.E: Problems on Cluster Points and Convergence (Exercises)
3.11: Operations on Convergent Sequences
3.11.E: Problems on Limits of Sequences (Exercises)
3.12: More on Cluster Points and Closed Sets. Density
3.12.E: Problems on Cluster Points Closed Sets and Density
3.13: Cauchy Sequences. Completeness
3.13.E: Problems on Cauchy Sequences
04: Function Limits and Continuity
4.01: Basic Definitions
4.1.E: Problems on Limits and Continuity
4.02: Some General Theorems on Limits and Continuity
4.2.E: More Problems on Limits and Continuity
4.03: Operations on Limits. Rational Functions
4.3.E: Problems on Continuity of Vector-Valued Functions
4.04: Infinite Limits. Operations in E
4.4.E: Problems on Limits and Operations in (E)
4.05: Monotone Function
4.5.E: Problems on Monotone Functions
4.06: Compact Sets
4.6.E: Problems on Compact Sets
4.07: More on Compactness
4.08: Continuity on Compact Sets. Uniform Continuity
4.8.E: Problems on Uniform Continuity Continuity on Compact Sets
4.09: The Intermediate Value Property
4.9.E: Problems on the Darboux Property and Related Topics
4.10: Arcs and Curves. Connected Sets
4.10.E: Problems on Arcs Curves and Connected Sets
4.11: Product Spaces. Double and Iterated Limits
4.11.E: Problems on Double Limits and Product Spaces
4.12: Sequences and Series of Functions
4.12.E: Problems on Sequences and Series of Functions
4.13: Absolutely Convergent Series. Power Series
4.13.E: More Problems on Series of Functions
05: Differentiation and Antidifferentiation
5.01: Derivatives of Functions of One Real Variable
5.1.E: Problems on Derived Functions in One Variable
5.02: Derivatives of Extended-Real Functions
5.2.E: Problems on Derivatives of Extended-Real Functions
5.03: L'Hopital's Rule
5.3.E: Problems on (Lprime) Hopital's Rule
5.04: Complex and Vector-Valued Functions on (E1)
5.4.E: Problems on Complex and Vector-Valued Functions on (E1)
5.05: Antiderivatives (Primitives Integrals)
5.5.E: Problems on Antiderivatives
5.06: Differentials. Taylors Theorem and Taylors Series
5.6.E: Problems on Tayior's Theorem
5.07: The Total Variation (Length) of a Function f - E1 E
5.7.E: Problems on Total Variation and Graph Length
5.08: Rectifiable Arcs. Absolute Continuity
5.8.E: Problems on Absolute Continuity and Rectifiable Arcs
5.09: Convergence Theorems in Differentiation and Integration
5.9.E: Problems on Convergence in Differentiation and Integration
5.10: Sufficient Condition of Integrability. Regulated Functions
5.10.E: Problems on Regulated Functions
5.11: Integral Definitions of Some Functions
5.11.E: Problems on Exponential and Trigonometric Functions
06: Differentiation on E and Other Normed Linear Spaces
6.01: Directional and Partial Derivatives
6.1.E: Problems on Directional and Partial Derivatives
6.02: Linear Maps and Functionals. Matrices
6.2.E: Problems on Linear Maps and Matrices
6.03: Differentiable Functions
6.3.E: Problems on Differentiable Functions
6.04: The Chain Rule. The Cauchy Invariant Rule
6.4.E: Further Problems on Differentiable Functions
6.05: Repeated Differentiation. Taylors Theorem
6.5.E: Problems on Repeated Differentiation and Taylor Expansions
6.06: Determinants. Jacobians. Bijective Linear Operators
6.6.E: Problems on Bijective Linear Maps and Jacobians
6.07: Inverse and Implicit Functions. Open and Closed Maps
6.7.E: Problems on Inverse and Implicit Functions Open and Closed Maps
6.08: Baire Categories. More on Linear Maps
6.8.E: Problems on Baire Categories and Linear Maps
6.09: Local Extrema. Maxima and Minima
6.9.E: Problems on Maxima and Minima
6.10: More on Implicit Differentiation. Conditional Extrema
6.10.E: Further Problems on Maxima and Minima
07: Volume and Measure
7.01: More on Intervals in E. Semirings of Sets
7.1.E: Problems on Intervals and Semirings
7.02: (mathcalC sigma)-Sets. Countable Additivity. Permutable Series
7.2.E: Problems on (mathcalC sigma) -Sets (sigma) -Additivity and Permutable Series
7.03: More on Set Families
7.3.E: Problems on Set Families
7.04: Set Functions. Additivity. Continuity
7.4.E: Problems on Set Functions
7.05: Nonnegative Set Functions. Premeasures. Outer Measures
7.5.E: Problems on Premeasures and Related Topics
7.06: Measure Spaces. More on Outer Measures
7.6.E: Problems on Measures and Outer Measures
7.07: Topologies. Borel Sets. Borel Measures
7.7.E: Problems on Topologies Borel Sets and Regular Measures
7.08: Lebesgue Measure
7.8.E: Problems on Lebesgue Measure
7.09: LebesgueStieltjes Measures
7.9.E: Problems on Lebesgue-Stieltjes Measures
7.10: Generalized Measures. Absolute Continuity
7.10.E: Problems on Generalized Measures
7.11: Differentiation of Set Functions
7.11.E: Problems on Vitali Coverings
08: Measurable Functions and Integration
8.01: Elementary and Measurable Functions
8.1.E: Problems on Measurable and Elementary Functions in ((S mathcalM))
8.02: Measurability of Extended-Real Functions
8.2.E: Problems on Measurable Functions in ((S mathcalM m))
8.03: Measurable Functions in ((S mathcalM m))
8.3.E: Problems on Measurable Functions in ((S mathcalM m))
8.04: Integration of Elementary Functions
8.4.E: Problems on Integration of Elementary Functions
8.05: Integration of Extended-Real Functions
8.5.E: Problems on Integration of Extended-Real Functions
8.06: Integrable Functions. Convergence Theorems
8.6.E: Problems on Integrability and Convergence Theorems
8.07: Integration of Complex and Vector-Valued Functions
8.7.E: Problems on Integration of Complex and Vector-Valued Functions
8.08: Product Measures. Iterated Integrals
8.8.E: Problems on Product Measures and Fubini Theorems
8.09: Riemann Integration. Stieltjes Integrals
8.9.E: Problems on Riemann and Stieltjes Integrals
8.10: Integration in Generalized Measure Spaces
8.10.E: Problems on Generalized Integration
8.11: The RadonNikodym Theorem. Lebesgue Decomposition
8.11.E: Problems on Radon-Nikodym Derivatives and Lebesgue Decomposition
8.12: Integration and Differentiation
8.12.E: Problems on Differentiation and Related Topics
09: Calculus Using Lebesgue Theory
9.01: L-Integrals and Antiderivatives
9.1.E: Problems on L-Integrals and Antiderivatives
9.02: More on L-Integrals and Absolute Continuity
9.2.E: Problems on L-Integrals and Absolute Continuity
9.03: Improper (Cauchy) Integrals
9.3.E: Problems on Cauchy Integrals
9.04: Convergence of Parametrized Integrals and Functions
9.4.E: Problems on Uniform Convergence of Functions and C-Integrals