Nicolaescu L.I. Introduction to Real Analysis

University of Notre Dame, 2020. — 696 p.What follows started as notes for the Freshman Honors Calculus course at the University of Notre Dame. The word “calculus” is a misnomer since this course was intended to be an introduction to real analysis or, if you like, “calculus with proofs”. For most students this class is the first encounter with mathematical rigor and it can be a bit disconcerting. In my view the best way to overcome this is to confront rigor head on and adopt it as standard operating procedure early on. A proof is an argument that uses the basic rules of Aristotelian logic and relies on facts everyone agrees to be true. The course is based on these basic rules of the mathematical discourse. It starts from a meagre collection of obvious facts (postulates) and ends up constructing the main contours of the impressive edifice called real analysis. No prior knowledge of calculus is assumed, but being comfortable performing algebraic manipulations is something that will make this journey more rewarding.
The first 9 chapters correspond to subjects covered in the Freshman course. I rarely was able to complete the brief Chapter 10 on complex numbers. Chapters 11 and above deal with several variables calculus topics, corresponding to the sophomore Honors Calculus offered at the University of Notre Dame.The basics of mathematical reasoning.
The Real Number System.
Special classes of real numbers.
Limits of sequences.
Limits of functions.
Continuity.
Differential calculus.
Applications of differential calculus.
Integral calculus.
Complex numbers and some of their applications.
The geometry and topology of Euclidean spaces.
Continuity.
Multi-variable differential calculus.
Applications of multi-variable differential calculus.
Multidimensional Riemann integration.
Integration over submanifolds.True PDF