Version 5.6. (Newer edition 6 available ISBN: 979-8851944635) A first course in rigorous mathematical analysis. Covers the real number system, sequences and series, continuous functions, the derivative, the Riemann integral, sequences of functions, and metric spaces. Originally developed to teach Math 444 at University of Illinois at Urbana-Champaign and later enhanced for Math 521 at University of Wisconsin-Madison and Math 4143 at Oklahoma State University. The first volume is either a stand-alone one-semester course or the first semester of a year-long course together with the second volume. It can be used anywhere from a semester early introduction to analysis for undergraduates (especially chapters 1-5) to a year-long course for advanced undergraduates and masters-level students. See http://www.jirka.org/ra/Table of Contents (of this volume I):
Introduction
1. Real Numbers
2. Sequences and Series
3. Continuous Functions
4. The Derivative
5. The Riemann Integral
6. Sequences of Functions
7. Metric Spaces
This first volume contains what used to be the entire book "Basic Analysis" before edition 5, that is chapters 1-7. Second volume contains chapters on multidimensional differential and integral calculus and further topics on approximation of functions.
Author(s): Jirí Lebl
Series: Basic Analysis: Introduction to Real Analysis 01
Edition: 6.0
Publisher: CreateSpace Independent Publishing Platform
Year: 2023
Language: English
Tags: 18.100A; maths; mathematics; math; calculus; Massachusetts Institute of Technology; MIT
Title Page
Introduction
About this book
About analysis
Basic set theory
Real Numbers
Basic properties
The set of real numbers
Absolute value and bounded functions
Intervals and the size of R
Decimal representation of the reals
Sequences and Series
Sequences and limits
Facts about limits of sequences
Limit superior, limit inferior, and Bolzano–Weierstrass
Cauchy sequences
Series
More on series
Continuous Functions
Limits of functions
Continuous functions
Extreme and intermediate value theorems
Uniform continuity
Limits at infinity
Monotone functions and continuity
The Derivative
The derivative
Mean value theorem
Taylor's theorem
Inverse function theorem
The Riemann Integral
The Riemann integral
Properties of the integral
Fundamental theorem of calculus
The logarithm and the exponential
Improper integrals
Sequences of Functions
Pointwise and uniform convergence
Interchange of limits
Picard's theorem
Metric Spaces
Metric spaces
Open and closed sets
Sequences and convergence
Completeness and compactness
Continuous functions
Fixed point theorem and Picard's theorem again
Further Reading
Index
List of Notation
