MATH 308 Real Analysis
Overview
MATH 308 Real Analysis is a required course for students in the Mathematics track of the Applied Mathematics and Computational Sciences track, and an and elective for students in other majors.
Course Description
Calculus was developed, starting with Leibniz and Newton, on the foundation of intuitive, and not always very precise or well-defined, concepts. Even basic ideas such as limits and continuity took centuries to be formalized. Major steps in this direction were the introduction of ε-δ definitions by Weierstrass, that did away with the intuitive but imprecise notions of “approaching” a number, or “moving” along the real axis, and the work by Cantor on the properties of real numbers.
Real Analysis can appear at first sight as a more rigorous version of Calculus, where every construction found in Calculus is placed on a firm mathematical basis and hidden assumptions about the nature of the real numbers are revealed. However, Real Analysis is much more than that. The concepts that are introduced to rigorously describe real numbers, subsets of the real line, properties of functions and the convergence of sequences of functions, have powerful generalizations to higher dimensions and to more abstract spaces, that is, in contexts where intuition can (and often does) fail us. Such concepts form the foundation of a large part of modern Mathematics. Especially notions such as compactness,completeness, Cauchy sequences, uniform convergence, etc. are found everywhere in Mathematics. Meeting such concepts for the first time in the familiar context of functions of one real variable helps to build intuition for them before revisiting them in more abstract settings.
Moreover, this course is also an excellent setting for learning how to read and write proofs since we will be able to see several proof techniques that are used throughout Mathematics. Unlike other courses, where either no or few proofs are given, one of the aims in Real Analysis is to prove (almost) everything. Real Analysis is not about developing new computational skills but about developing a mathematical mindset and rigorous thinking that will be applicable for the rest of your careers as applied mathematicians, (data) scientists, engineers, financial analysts, etc.
Syllabus
Literature
Understanding Analysis. Stephen Abbott. Springer, 2nd edition (2015).