Course Description

Real analysis is a standard subject covered in every Mathematics program. It consists of a rigorous treatment of several topics from single and multivariable calculus (limits, continuity, differentiability, integration, etc), and lays the theoretical foundations for the theory of differential equations, functional analysis, differential geometry, an many other subjects. At Queen's, the second year course MATH/MTHE 281 (Introduction to Real Analysis) sets the stage for MATH/MTHE 328.

We will start by reviewing the construction of the field of real numbers and the consequences of its completeness (Bolzano-Weierstrass, convergence of Cauchy sequences). We will spend some time discussing several topological notions (interior points, boundary points, open sets and their structure, cluster points, isolated points, nowhere dense sets, compact sets, Heine-Borel, connectedness) and their relation to continuity for real-valued functions of a single real variable. We will then discuss uniform continuity, Heine-Borel, and its consequences. We will construct the famous middle-third Cantor set and explore its properties. We will then proceed to cover G-delta and F-sigma sets, their relation to the set of continuity of a real-valued function, Osgood's theorem on countable intersections of open dense sets, and Volterra's theorem on the intersection of dense sets of continuity. We will move on to the notion of differentiability, the inverse function theorem, and Darboux's theorem. We will have a small detour to the study of uniform convergence (Ascoli-Arzelà), before moving on to the set-up of multivariable calculus. We will discuss the inner products and norms in R^n, the topology induced by them, and Cauchy-Schwarz. We will discuss limits and continuity for vector-valued functions of several variables in terms of open sets. We will review partial derivatives, conditions for equality of mixed partials, before moving on to the notion of differentiability and its relation to continuity and to partial derivatives. One of the most important topics of single- and multi-variable calculus, the Chain Rule, will be discussed next. We will cover the derivative of inverse function, and the mean value theorem for real-valued functions of several variables. A careful discussion of the general formulation of Dini's implicit function theorem and the inverse function theorem will follow. In the process, we will see the Banach fixed point theorem and some of its applications (including the important Picard-Lindelöf theorem on the existence and uniqueness of solutions to differential equations). We will conclude with Gronwall's inequality and the analysis of the dependence of solutions to differential equations upon their initial conditions.

Potential other topics include: constrained optimization and Lagrange multipliers, introduction to Lebesgue integration, Hausdorff measure and fractals, introduction to convex analysis, functions of bounded variation and rectifiable curves.

Prerequisites

The only prerequisite for this course is MATH/MTHE 281, whose content will be used extensively throughout the course.

Lectures

There will be three in-person lectures per week. Each lecture will be 50 minutes long. Here's the schedule:

• Tuesday 12:30pm - 1:20pm in Stirling Hall B

• Thursday 11:30am - 12:20pm in Stirling Hall B

• Friday 1:30pm - 2:20pm in Stirling Hall B

Teaching Team

Instructor: Francesco Cellarosi (fc19@queensu.ca).

Teaching Assistant: TBD

Textbooks

While preparing this course, I have consulted several textbooks at various levels. The course will be based on a combination of materials from many different sources, and I will share my notes with the students.

Here are two excellent references which we will use a lot:

• Real Mathematical Analysis by Charles C. Pugh. Second Edition. Springer, 2015. ISBN 978-3-319-17770-0 (eBook ISBN 978-3-319-17771-7).

• A Passage to Modern Analysis by William J. Terrell. American Mathematical Society, 2019. ISBN 978-1-4704-5135-6.