Basic Analysis I and II: Introduction to Real Analysis E59CDZA4DP

URI	`m2://res/E59CDZA4DP`
Kind	textbook
Creators	Jiří Lebl
Scope	deferred (no test decides the record, and the classifier has no model)

Expressions

e1: edition

Contents

chapter 1: Functions as Limits
section 1.1: Complex numbers
subsection 1.1.1: The complex plane
subsection 1.1.2: Complex numbers and limits
subsection 1.1.3: Complex-valued functions
subsection 1.1.4: Exercises
section 1.2: Swapping limits
subsection 1.2.1: Continuity
subsection 1.2.2: Integration
subsection 1.2.3: Differentiation
subsection 1.2.4: Exercises
section 1.3: Power series and analytic functions
subsection 1.3.1: Analytic functions
subsection 1.3.2: Convergence of power series
subsection 1.3.3: Properties of analytic functions
subsection 1.3.4: Power series as analytic functions
subsection 1.3.5: Identity theorem for analytic functions
subsection 1.3.6: Exercises
section 1.4: Complex exponential and trigonometric functions
subsection 1.4.1: The complex exponential
subsection 1.4.2: Trigonometric functions and $\pi$
subsection 1.4.3: The unit circle and polar coordinates
subsection 1.4.4: Exercises
section 1.5: Maximum principle and the fundamental theorem of algebra
subsection 1.5.1: Exercises
section 1.6: Equicontinuity and the Arzel\`a--Ascoli theorem
subsection 1.6.1: Exercises
section 1.7: The Stone--Weierstrass theorem
subsection 1.7.1: Weierstrass approximation
subsection 1.7.2: Stone--Weierstrass approximation
subsection 1.7.3: Exercises
section 1.8: Fourier series
subsection 1.8.1: Trigonometric polynomials
subsection 1.8.2: Fourier series
subsection 1.8.3: Orthonormal systems
subsection 1.8.4: The Dirichlet kernel and approximate delta functions
subsection 1.8.5: Localization
subsection 1.8.6: Parseval's theorem
subsection 1.8.7: Exercises
chapter 2: Continuous Functions
section 2.1: Limits of functions
subsection 2.1.1: Cluster points
subsection 2.1.2: Limits of functions
subsection 2.1.3: Sequential limits
subsection 2.1.4: Limits of restrictions and one-sided limits
subsection 2.1.5: Exercises
section 2.2: Continuous functions
subsection 2.2.1: Definition and basic properties
subsection 2.2.2: Composition of continuous functions
subsection 2.2.3: Discontinuous functions
subsection 2.2.4: Exercises
section 2.3: Extreme and intermediate value theorems
subsection 2.3.1: Min-max or extreme value theorem
subsection 2.3.2: Bolzano's intermediate value theorem
subsection 2.3.3: Exercises
section 2.4: Uniform continuity
subsection 2.4.1: Uniform continuity
subsection 2.4.2: Continuous extension
subsection 2.4.3: Lipschitz continuous functions
subsection 2.4.4: Exercises
section 2.5: Limits at infinity
subsection 2.5.1: Limits at infinity
subsection 2.5.2: Infinite limit
subsection 2.5.3: Compositions
subsection 2.5.4: Exercises
section 2.6: Monotone functions and continuity
subsection 2.6.1: Continuity of monotone functions
subsection 2.6.2: Continuity of inverse functions
subsection 2.6.3: Exercises
chapter 3: The Derivative
section 3.1: The derivative
subsection 3.1.1: Definition and basic properties
subsection 3.1.2: Chain rule
subsection 3.1.3: Exercises
section 3.2: Mean value theorem
subsection 3.2.1: Relative minima and maxima
subsection 3.2.2: Rolle's theorem
subsection 3.2.3: Mean value theorem
subsection 3.2.4: Applications
subsection 3.2.5: Continuity of derivatives and the intermediate value theorem
subsection 3.2.6: Exercises
section 3.3: Taylor's theorem
subsection 3.3.1: Derivatives of higher orders
subsection 3.3.2: Taylor's theorem
subsection 3.3.3: Exercises
section 3.4: Inverse function theorem
subsection 3.4.1: Inverse function theorem
subsection 3.4.2: Exercises
chapter 4: Metric Spaces
section 4.1: Metric spaces
subsection 4.1.1: Exercises
section 4.2: Open and closed sets
subsection 4.2.1: Topology
subsection 4.2.2: Connected sets
subsection 4.2.3: Closure and boundary
subsection 4.2.4: Exercises
section 4.3: Sequences and convergence
subsection 4.3.1: Sequences
subsection 4.3.2: Convergence in euclidean space
subsection 4.3.3: Convergence and topology
subsection 4.3.4: Exercises
section 4.4: Completeness and compactness
subsection 4.4.1: Cauchy sequences and completeness
subsection 4.4.2: Compactness
subsection 4.4.3: Exercises
section 4.5: Continuous functions
subsection 4.5.1: Continuity
subsection 4.5.2: Compactness and continuity
subsection 4.5.3: Continuity and topology
subsection 4.5.4: Uniform continuity
subsection 4.5.5: Cluster points and limits of functions
subsection 4.5.6: Exercises
section 4.6: Fixed point theorem and Picard's theorem again
subsection 4.6.1: Fixed point theorem
subsection 4.6.2: Picard's theorem
subsection 4.6.3: Exercises
chapter 5: Multivariable Integral
section 5.1: Riemann integral over rectangles
subsection 5.1.1: Rectangles and partitions
subsection 5.1.2: Upper and lower integrals
subsection 5.1.3: The Riemann integral
subsection 5.1.4: Integrals of continuous functions
subsection 5.1.5: Integration of functions with compact support
subsection 5.1.6: Exercises
section 5.2: Iterated integrals and the Fubini theorem
subsection 5.2.1: Exercises
section 5.3: Outer measure and null sets
subsection 5.3.1: Outer measure and null sets
subsection 5.3.2: Examples and basic properties
subsection 5.3.3: Images of null sets under differentiable functions
subsection 5.3.4: Exercises
section 5.4: The set of Riemann integrable functions
subsection 5.4.1: Oscillation and continuity
subsection 5.4.2: The set of Riemann integrable functions
subsection 5.4.3: Exercises
section 5.5: Jordan measurable sets
subsection 5.5.1: Volume and Jordan measurable sets
subsection 5.5.2: Integration over Jordan measurable sets
subsection 5.5.3: Images of Jordan measurable subsets
subsection 5.5.4: Exercises
section 5.6: Green's theorem
subsection 5.6.1: Exercises
section 5.7: Change of variables
subsection 5.7.1: Exercises
chapter 6: One-dimensional Integrals in Several Variables
section 6.1: Differentiation under the integral
subsection 6.1.1: Exercises
section 6.2: Path integrals
subsection 6.2.1: Piecewise smooth paths
subsection 6.2.2: Path integral of a one-form
subsection 6.2.3: Path integral of a function
subsection 6.2.4: Exercises
section 6.3: Path independence
subsection 6.3.1: Path independent integrals
subsection 6.3.2: Vector fields
subsection 6.3.3: Exercises
chapter 7: Real Numbers
section 7.1: Basic properties
subsection 7.1.1: Exercises
section 7.2: The set of real numbers
subsection 7.2.1: The set of real numbers
subsection 7.2.2: Archimedean property
subsection 7.2.3: Using supremum and infimum
subsection 7.2.4: Maxima and minima
subsection 7.2.5: Exercises
section 7.3: Absolute value and bounded functions
subsection 7.3.1: Exercises
section 7.4: Intervals and the size of \texorpdfstring{$\R$}{R}
subsection 7.4.1: Exercises
section 7.5: Decimal representation of the reals
subsection 7.5.1: Exercises
chapter 8: The Riemann Integral
section 8.1: The Riemann integral
subsection 8.1.1: Partitions and lower and upper integrals
subsection 8.1.2: Riemann integral
subsection 8.1.3: More notation
subsection 8.1.4: Exercises
section 8.2: Properties of the integral
subsection 8.2.1: Additivity
subsection 8.2.2: Linearity and monotonicity
subsection 8.2.3: Continuous functions
subsection 8.2.4: More on integrable functions
subsection 8.2.5: Exercises
section 8.3: Fundamental theorem of calculus
subsection 8.3.1: First form of the theorem
subsection 8.3.2: Second form of the theorem
subsection 8.3.3: Change of variables
subsection 8.3.4: Exercises
section 8.4: The logarithm and the exponential
subsection 8.4.1: The logarithm
subsection 8.4.2: The exponential
subsection 8.4.3: Exercises
section 8.5: Improper integrals
subsection 8.5.1: Integral test for series
subsection 8.5.2: Exercises
chapter 9: Sequences of Functions
section 9.1: Pointwise and uniform convergence
subsection 9.1.1: Pointwise convergence
subsection 9.1.2: Uniform convergence
subsection 9.1.3: Convergence in uniform norm
subsection 9.1.4: Exercises
section 9.2: Interchange of limits
subsection 9.2.1: Continuity of the limit
subsection 9.2.2: Integral of the limit
subsection 9.2.3: Derivative of the limit
subsection 9.2.4: Convergence of power series
subsection 9.2.5: Exercises
section 9.3: Picard's theorem
subsection 9.3.1: First order ordinary differential equation
subsection 9.3.2: The theorem
subsection 9.3.3: Examples
subsection 9.3.4: Exercises
chapter 10: Sequences and Series
section 10.1: Sequences and limits
subsection 10.1.1: Monotone sequences
subsection 10.1.2: Tail of a sequence
subsection 10.1.3: Subsequences
subsection 10.1.4: Exercises
section 10.2: Facts about limits of sequences
subsection 10.2.1: Limits and inequalities
subsection 10.2.2: Continuity of algebraic operations
subsection 10.2.3: Recursively defined sequences
subsection 10.2.4: Some convergence tests
subsection 10.2.5: Exercises
section 10.3: Limit superior, limit inferior, and Bolzano--Weierstrass
subsection 10.3.1: Upper and lower limits
subsection 10.3.2: Using limit inferior and limit superior
subsection 10.3.3: Bolzano--Weierstrass theorem
subsection 10.3.4: Infinite limits
subsection 10.3.5: Exercises
section 10.4: Cauchy sequences
subsection 10.4.1: Exercises
section 10.5: Series
subsection 10.5.1: Definition
subsection 10.5.2: Cauchy series
subsection 10.5.3: Basic properties
subsection 10.5.4: Absolute convergence
subsection 10.5.5: Comparison test and the \texorpdfstring{$p$}{p}-series
subsection 10.5.6: Ratio test
subsection 10.5.7: Exercises
section 10.6: More on series
subsection 10.6.1: Root test
subsection 10.6.2: Alternating series test
subsection 10.6.3: Rearrangements
subsection 10.6.4: Multiplication of series
subsection 10.6.5: Power series
subsection 10.6.6: Exercises
chapter 11: Several Variables and Partial Derivatives
section 11.1: Vector spaces, linear mappings, and convexity
subsection 11.1.1: Vector spaces
subsection 11.1.2: Linear combinations and dimension
subsection 11.1.3: Linear mappings
subsection 11.1.4: Convexity
subsection 11.1.5: Exercises
section 11.2: Analysis with vector spaces
subsection 11.2.1: Norms
subsection 11.2.2: Matrices
subsection 11.2.3: Determinants
subsection 11.2.4: Exercises
section 11.3: The derivative
subsection 11.3.1: The derivative
subsection 11.3.2: Partial derivatives
subsection 11.3.3: Gradients, curves, and directional derivatives
subsection 11.3.4: The Jacobian
subsection 11.3.5: Exercises
section 11.4: Continuity and the derivative
subsection 11.4.1: Bounding the derivative
subsection 11.4.2: Continuously differentiable functions
subsection 11.4.3: Exercises
section 11.5: Inverse and implicit function theorems
subsection 11.5.1: Implicit function theorem
subsection 11.5.2: Exercises
section 11.6: Higher order derivatives
subsection 11.6.1: Exercises
chapter: Introduction
section 11.7: About this book
section 11.8: About analysis
section 11.9: Basic set theory
subsection 11.9.1: Sets
subsection 11.9.2: Induction
subsection 11.9.3: Functions
subsection 11.9.4: Relations and equivalence classes
subsection 11.9.5: Cardinality
subsection 11.9.6: Exercises

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