Real Analysis I

Course Overview

This is a 400-level theoretical mathematical course in one of the two cornerstones of modern mathematics: Analysis. For your reference, the Calculus I and Calculus II courses are 100-level while Calculus III is a 200-level course; linear Algebra is a 200-level course; and Introduction to Mathematical Rigor is a 300 level course. Therefore, keep in mind that more effort an maturity is expected from the students in this 400 level course. The objective of this course is to formalize all the concepts learned in the 3 Calculus courses series. Proofs are the foundation of this course and mastering proof techniques is a fundamental objective of this course.

Brief Course Description: The course will be divided in seven parts:

Real Numbers: ℝ: In this part we will learn the axiomatic description of the real numbers. That is, ℝ will be defined as any set of numbers that satisfies certain number of axioms coupled with other properties. Then, we will prove that such set is unique. Furthermore, we will construct a set that contains all of those properties proving that such set exists.

Sequences: In this part, we will introduce the concept of sequences and the notion of convergence, which is arguably one of the most important notions in mathematics.

Topology of the Real Numbers and Metric Spaces: This section is a very gentle introduction to topology (restricted to the nice properties of the real numbers). This section is probably where the real flavour of analysis is tasted. A lot of definitions and ingenious proofs will be discussed.

Continuous Functions: In this section and the following we will revisit many of the concepts learned in Calculus I regarding continuity but will be formalized and proved. Continuity will be defined formally rather than intuitively and many of the properties learned in MTH 141 will be proven and expanded.

Differentiable Functions: In this section we will revisit many of the concepts learned in Calculus I and Calculus III regarding differentiability but will be formalized and proved. What does differentiability mean will be defined formally rather than intuitively and many of the properties learned in MTH 141 and MTH 243 will be proven and expanded.

Functions of Bounded Variation and Rectifiable Curves: This section deals with the notion of variation of a function, which refers to a measure of how “wiggly” a function is. All of the concepts given here are probably new to most students.

(Time Allowing) - Infinite Series Part I: This section is a formalization of the material taught in Calculus II or MTH 142. The concepts of series, their properties and the notion of convergence will be given and formalized.

Prerequisites

• MTH 215, MTH 243 and MTH 307.

Textbooks

• Invitation to Real Analysis by Cesar E. Silva. American Mathematical Society (AMS).

Grading

• Midterm Exam 1: 20%
• Midterm Exam 2: 20%
• Homework: 20%
• Short Quizzes: 15%
• Cumulative Final Exam: 25%