Analysis of Metric Spaces

The world according to Rudin

What you’ll learn
Effectively locate and use the information needed to prove theorems and establish mathematical results.
Effectively write mathematical solutions in a clear and concise manner.
Demonstrate an intuitive understanding of set theory and metric spaces
Understand the classic book Principles of Mathematical Analysis by Rudin

Requirements
Mathematical maturity. In other words, be interested in mathematics.

Description
The methods of calculus are limited to Euclidean spaces. In this course, we show how the incredibly powerful tools of calculus, beginning with the limit concept, can be generalized to so-called metric spaces. Almost every space used in advanced analysis is in fact a metric space, and limits in metric spaces are a universal language for advanced analysis. The basic techniques of calculus were invented for the real line R. What should we do when we want to handle something more general than R? The fundamental notions of calculus begin with the idea that one point of R can be close to another point of R, and this is called “Approximation” or “taking a limit.” Metrics are a way to transfer this key notion of being “close to” to a more general setting. The “points” of a metric space can be complicated objects in their own right. For example, they may themselves be functions on some other space. Ideas like this are ubiquitous in advanced mathematics today. One tries to throw away complicated details of the space being considered, and this makes it easier to see which theorem or technique can be applied next. In this way, the mathematician tries to avoid getting overwhelmed by the details, or to say it differently, we try to see the overall forest rather than the trees.

Overview
Section 1: Introduction

Lecture 1 Introduction and Chapter 2 of Rudin

Lecture 2 Basic Set Theory

Lecture 3 Sequences

Lecture 4 Real and Rational Numbers

Lecture 5 The real numbers are uncountable

Lecture 6 Russel’s paradox: not _everything_ can be a set

Lecture 7 More on Paradox… and the start of Metric Spaces

Lecture 8 Neighbourhoods in Metric space

Lecture 9 Interior Points, Open, Closed, and Clopen

Lecture 10 Dense Sets and Perfect sets

Lecture 11 Closed, bounded, compact

Lecture 12 Open is not the opposite of closed

Lecture 13 Not limits, but Limit Points. Closures.

Lecture 14 Compactness: a very clever generalization of closed intervals of the number line

Lecture 15 More on compactness

Lecture 16 Review

Lecture 17 Properties of compact sets and of perfect sets

Lecture 18 The real line: Sequences and Series

Lecture 19 The Cantor set: Is it connected?

Lecture 20 Sequences in Metric Spaces

Lecture 21 More on sequences in Metric Spaces

Lecture 22 Another clever generalization: completeness through Cauchy sequences

Lecture 23 Comparing the two kinds of completeness

Lecture 24 Continuity: a property of maps of metric spaces

Have you tried to read the classic book Introduction to Mathematical Analysis by Rudin and been stimied? Take this course!