Analysis Course vol.1
Authors
Description
The author presents the language of sets and functions through a precise conceptualization and logical systematization of ideas. His aim is to study the sets of real numbers and real functions of one variable. The concepts given are illustrated by examples and accompanied by many exercises of varying difficulty. The best use of this analysis course requires previous knowledge of calculus.
In addition to the general concepts and basic facts about sets and functions, the author presents the real numbers, the foundations of their theory, sequences and series; the topology of the line; the limits of functions; derivatives; the Riemann integral; and sequences and series of functions. The author has adopted an informal and descriptive style in the first chapters and an axiomatic point of view in the rest.
Target audience
Higher education
Name: Analysis Course vol.1
Author(s): e Elon Lages Lima
Pages: 320
Publication: IMPA, 2019
ISBN: 978-85-244-0468-9
Edition: 15
1 Sets and Functions
1 Sets
2 Operations between sets
3 Functions
4 Composition of functions
5 Families
Exercises
2 Finite, Enumerable and Non-Enumerable Sets
1 Natural Numbers
2 Good Order and the Second Principle of Induction
3 Finite and Infinite Sets
4 Enumerable Sets
5 Non-Enumerable Sets
Exercises
3 Real numbers
1 Bodies
2 Ordered bodies
3 Real numbers
Exercises
4 Sequences and Series of Real Numbers
1 Sequences
2 Limit of a sequence
3 Arithmetic properties of limits
4 Subsequences
5 Cauchy sequences
6 Infinite limits
7 Numerical series
Exercises
5 Topology of the Line
1 Open sets
2 Closed sets
3 Points of accumulation
4 Compact sets
Exercises
6 Limits of Functions
1 Definition and properties of the limit
2 Examples of limits
3 Lateral limits
4 Limits at infinity
5 Adherence values of a function; lim sup and lim inf
Exercises
7 Continuous functions
1 The notion of a continuous function
2 Discontinuities
3 Continuous functions on intervals
4 Continuous functions on compact sets
5 Uniform continuity
Exercises
8 Derivatives
1 Definition and properties of the derivative at a point
2 Derivable functions on an interval
3 Taylor’s formula
4 Taylor’s series, analytic functions
Exercises
9 Riemann Integral
1 Upper integral and lower integral
2 Integrable functions
3 The Fundamental Theorem of Calculus
4 Classical formulas of Integral Calculus
5 The integral as a limit of sums
6 Characterization of integrable functions
7 Logarithms and exponentials
Exercises
10 Sequences and Series of Functions
1 Simple convergence and uniform convergence
2 Properties of uniform convergence
3 Power series
4 Analytic functions
5 Equicontinuity
Exercises
Bibliography
Index of Notations
Index of Authors
Index of References
