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Basic Analysis I & II:
Introduction to Real Analysis, Volumes I & II
Jiří Lebl
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Front Matter
0
Introduction
0.1
About this book
0.1.1
About Volume I
0.1.2
About Volume II
0.2
About analysis
0.3
Basic set theory
0.3.1
Sets
0.3.2
Induction
0.3.3
Functions
0.3.4
Relations and equivalence classes
0.3.5
Cardinality
0.3.6
Exercises
1
Real Numbers
1.1
Basic properties
1.1.1
Exercises
1.2
The set of real numbers
1.2.1
The set of real numbers
1.2.2
Archimedean property
1.2.3
Using supremum and infimum
1.2.4
Maxima and minima
1.2.5
Exercises
1.3
Absolute value and bounded functions
1.3.1
Exercises
1.4
Intervals and the size of
R
1.4.1
Exercises
1.5
Decimal representation of the reals
1.5.1
Exercises
2
Sequences and Series
2.1
Sequences and limits
2.1.1
Monotone sequences
2.1.2
Tail of a sequence
2.1.3
Subsequences
2.1.4
Exercises
2.2
Facts about limits of sequences
2.2.1
Limits and inequalities
2.2.2
Continuity of algebraic operations
2.2.3
Recursively defined sequences
2.2.4
Some convergence tests
2.2.5
Exercises
2.3
Limit superior, limit inferior, and Bolzano–Weierstrass
2.3.1
Upper and lower limits
2.3.2
Using limit inferior and limit superior
2.3.3
Bolzano–Weierstrass theorem
2.3.4
Infinite limits
2.3.5
Exercises
2.4
Cauchy sequences
2.4.1
Exercises
2.5
Series
2.5.1
Definition
2.5.2
Cauchy series
2.5.3
Basic properties
2.5.4
Absolute convergence
2.5.5
Comparison test and the
p
-series
2.5.6
Ratio test
2.5.7
Exercises
2.6
More on series
2.6.1
Root test
2.6.2
Alternating series test
2.6.3
Rearrangements
2.6.4
Multiplication of series
2.6.5
Power series
2.6.6
Exercises
3
Continuous Functions
3.1
Limits of functions
3.1.1
Cluster points
3.1.2
Limits of functions
3.1.3
Sequential limits
3.1.4
Limits of restrictions and one-sided limits
3.1.5
Exercises
3.2
Continuous functions
3.2.1
Definition and basic properties
3.2.2
Composition of continuous functions
3.2.3
Discontinuous functions
3.2.4
Exercises
3.3
Extreme and intermediate value theorems
3.3.1
Min-max or extreme value theorem
3.3.2
Bolzano’s intermediate value theorem
3.3.3
Exercises
3.4
Uniform continuity
3.4.1
Uniform continuity
3.4.2
Continuous extension
3.4.3
Lipschitz continuous functions
3.4.4
Exercises
3.5
Limits at infinity
3.5.1
Limits at infinity
3.5.2
Infinite limit
3.5.3
Compositions
3.5.4
Exercises
3.6
Monotone functions and continuity
3.6.1
Continuity of monotone functions
3.6.2
Continuity of inverse functions
3.6.3
Exercises
4
The Derivative
4.1
The derivative
4.1.1
Definition and basic properties
4.1.2
Chain rule
4.1.3
Exercises
4.2
Mean value theorem
4.2.1
Relative minima and maxima
4.2.2
Rolle’s theorem
4.2.3
Mean value theorem
4.2.4
Applications
4.2.5
Continuity of derivatives and the intermediate value theorem
4.2.6
Exercises
4.3
Taylor’s theorem
4.3.1
Derivatives of higher orders
4.3.2
Taylor’s theorem
4.3.3
Exercises
4.4
Inverse function theorem
4.4.1
Inverse function theorem
4.4.2
Exercises
5
The Riemann Integral
5.1
The Riemann integral
5.1.1
Partitions and lower and upper integrals
5.1.2
Riemann integral
5.1.3
More notation
5.1.4
Exercises
5.2
Properties of the integral
5.2.1
Additivity
5.2.2
Linearity and monotonicity
5.2.3
Continuous functions
5.2.4
More on integrable functions
5.2.5
Exercises
5.3
Fundamental theorem of calculus
5.3.1
First form of the theorem
5.3.2
Second form of the theorem
5.3.3
Change of variables
5.3.4
Exercises
5.4
The logarithm and the exponential
5.4.1
The logarithm
5.4.2
The exponential
5.4.3
Exercises
5.5
Improper integrals
5.5.1
Integral test for series
5.5.2
Exercises
6
Sequences of Functions
6.1
Pointwise and uniform convergence
6.1.1
Pointwise convergence
6.1.2
Uniform convergence
6.1.3
Convergence in uniform norm
6.1.4
Exercises
6.2
Interchange of limits
6.2.1
Continuity of the limit
6.2.2
Integral of the limit
6.2.3
Derivative of the limit
6.2.4
Convergence of power series
6.2.5
Exercises
6.3
Picard’s theorem
6.3.1
First order ordinary differential equation
6.3.2
The theorem
6.3.3
Examples
6.3.4
Exercises
7
Metric Spaces
7.1
Metric spaces
7.1.1
Exercises
7.2
Open and closed sets
7.2.1
Topology
7.2.2
Connected sets
7.2.3
Closure and boundary
7.2.4
Exercises
7.3
Sequences and convergence
7.3.1
Sequences
7.3.2
Convergence in euclidean space
7.3.3
Convergence and topology
7.3.4
Exercises
7.4
Completeness and compactness
7.4.1
Cauchy sequences and completeness
7.4.2
Compactness
7.4.3
Exercises
7.5
Continuous functions
7.5.1
Continuity
7.5.2
Compactness and continuity
7.5.3
Continuity and topology
7.5.4
Uniform continuity
7.5.5
Cluster points and limits of functions
7.5.6
Exercises
7.6
Fixed point theorem and Picard’s theorem again
7.6.1
Fixed point theorem
7.6.2
Picard’s theorem
7.6.3
Exercises
8
Several Variables and Partial Derivatives
8.1
Vector spaces, linear mappings, and convexity
8.1.1
Vector spaces
8.1.2
Linear combinations and dimension
8.1.3
Linear mappings
8.1.4
Convexity
8.1.5
Exercises
8.2
Analysis with vector spaces
8.2.1
Norms
8.2.2
Matrices
8.2.3
Determinants
8.2.4
Exercises
8.3
The derivative
8.3.1
The derivative
8.3.2
Partial derivatives
8.3.3
Gradients, curves, and directional derivatives
8.3.4
The Jacobian
8.3.5
Exercises
8.4
Continuity and the derivative
8.4.1
Bounding the derivative
8.4.2
Continuously differentiable functions
8.4.3
Exercises
8.5
Inverse and implicit function theorems
8.5.1
Implicit function theorem
8.5.2
Exercises
8.6
Higher order derivatives
8.6.1
Exercises
9
One-dimensional Integrals in Several Variables
9.1
Differentiation under the integral
9.1.1
Exercises
9.2
Path integrals
9.2.1
Piecewise smooth paths
9.2.2
Path integral of a one-form
9.2.3
Path integral of a function
9.2.4
Exercises
9.3
Path independence
9.3.1
Path independent integrals
9.3.2
Vector fields
9.3.3
Exercises
10
Multivariable Integral
10.1
Riemann integral over rectangles
10.1.1
Rectangles and partitions
10.1.2
Upper and lower integrals
10.1.3
The Riemann integral
10.1.4
Integrals of continuous functions
10.1.5
Integration of functions with compact support
10.1.6
Exercises
10.2
Iterated integrals and Fubini theorem
10.2.1
Exercises
10.3
Outer measure and null sets
10.3.1
Outer measure and null sets
10.3.2
Examples and basic properties
10.3.3
Images of null sets under differentiable functions
10.3.4
Exercises
10.4
The set of Riemann integrable functions
10.4.1
Oscillation and continuity
10.4.2
The set of Riemann integrable functions
10.4.3
Exercises
10.5
Jordan measurable sets
10.5.1
Volume and Jordan measurable sets
10.5.2
Integration over Jordan measurable sets
10.5.3
Images of Jordan measurable subsets
10.5.4
Exercises
10.6
Green’s theorem
10.6.1
Exercises
10.7
Change of variables
10.7.1
Exercises
11
Functions as Limits
11.1
Complex numbers
11.1.1
The complex plane
11.1.2
Complex numbers and limits
11.1.3
Complex-valued functions
11.1.4
Exercises
11.2
Swapping limits
11.2.1
Continuity
11.2.2
Integration
11.2.3
Differentiation
11.2.4
Exercises
11.3
Power series and analytic functions
11.3.1
Analytic functions
11.3.2
Convergence of power series
11.3.3
Properties of analytic functions
11.3.4
Power series as analytic functions
11.3.5
Identity theorem for analytic functions
11.3.6
Exercises
11.4
Complex exponential and trigonometric functions
11.4.1
The complex exponential
11.4.2
Trigonometric functions and
π
11.4.3
The unit circle and polar coordinates
11.4.4
Exercises
11.5
Maximum principle and the fundamental theorem of algebra
11.5.1
Exercises
11.6
Equicontinuity and the Arzelà–Ascoli theorem
11.6.1
Exercises
11.7
The Stone–Weierstrass theorem
11.7.1
Weierstrass approximation
11.7.2
Stone–Weierstrass approximation
11.7.3
Exercises
11.8
Fourier series
11.8.1
Trigonometric polynomials
11.8.2
Fourier series
11.8.3
Orthonormal systems
11.8.4
The Dirichlet kernel and approximate delta functions
11.8.5
Localization
11.8.6
Parseval’s theorem
11.8.7
Exercises
Back Matter
Further Reading
Index
🔗
Chapter
0
Introduction
0.1
About this book
0.2
About analysis
0.3
Basic set theory
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