Analysis Course vol.1
Authors
Description
The author presents the language of sets and functions through precise conceptualization and a logical systematization of ideas. His goal is to study sets of real numbers and real functions of one variable. The concepts presented are illustrated by examples and accompanied by numerous exercises of varying difficulty. Proper use of this analysis course requires prior knowledge of calculus.
In addition to presenting general concepts and basic facts about sets and functions, the book introduces real numbers, the foundations of their theory, sequences and series; the topology of the real line; limits of functions; derivatives; the Riemann integral; and sequences and series of functions. The author adopted an informal and descriptive style in the first chapters and an axiomatic point of view in the others.
Target audience
Higher education
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 Set
2. Operations between sets
3 Functions
4. Composition of functions
5 Families
Exercises
2 Finite, Countable, and Uncountable Sets
1. Natural Numbers
2. Well-ordering and the Second Principle of Induction
3 Finite and infinite sets
4 Countable Sets
5 Non-countable 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 Accumulation Points
4 compact sets
Exercises
6 Limits of Functions
1. Definition and properties of the limit
2 Examples of limits
3 Lateral boundaries
4 Limits at Infinity
5. Adherence values of a function; lim sup and lim inf
Exercises
7 Continuous Functions
1. The concept of a continuous function
2 Discontinuities
3 Continuous functions on intervals
4 Continuous functions in compact assemblies
5 Uniform continuity
Exercises
8 Derivatives
1. Definition and properties of the derivative at a point
2 Differentiable functions on an interval
3. Taylor's Formula
Taylor's 4 series, analytical functions
Exercises
9 Riemann Integral
1 Upper integral and lower integral
2 Integrable Functions
3. The Fundamental Theorem of Calculus
4 Classic 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 Analytical Functions
5 Equicontinuity
Exercises
Bibliography
Index of Notes
Index of Authors
Index
