18.S097, Introduction to Metric Spaces, IAP 2022

Instructor: Paige Dote
Class hours:TR 1–2:30 in 2-131
Office hours: F 2:30–4 virtual
Syllabus:Course Syllabus
Dedication: I would like to thank Professor Larry Guth for being the advisor for this IAP class, and Professor Minicozzi for helping me navigate the logistics of creating a class like this. I also thank the numerous other students who read over my notes and PSETs, giving invaluable suggestions about problems to include and discussing pedagogical concerns of mine.
Grading:Based on weekly problem sets. You may use TeX or you may write your solutions by hand; in the latter case please make sure they are easy to read! You may consult anyone and anything but you need to absorb and write out the solutions yourself. Anything close to copying is not permitted.
Description: This course will provide a basic introduction to metric spaces. The goal of the class is to teach the material that 18.100B covers that 18.100A doesn't, such as metrics, compact sets, and theorems that follow from these objects.
Prerequisites: 18.100A/B/P/Q. The intended audience for this class is students who have taken 18.100A or 18.100P, but you are more than welcome to attend lectures/take the class on listener if you have taken 18.100B or 18.100Q. I will be assuming basic knowledge of convergence, Cauchy sequences, open sets, and continuity. That being said, please ask me if you find any of these topics confusing still.
Materials:There is no official textbook, I will provide lecture notes. But there are some books/notes you may find useful:

[L] Lebl: Lebl's Real Analysis Vol. 1 available for download.
[TBB] Thomson, Bruckner, and Bruckner: Elementary Real Analysis textbook available for download.
[R] and of course, Rudin's Principles of Mathematical Analysis.

Schedule (in reverse time order)

N/A N/A Lecture compilation and PSET compilation.
Lecture Compilation
PSET Compilation
Thu Jan 20 Picard's Theorem. Where we go from here.
Lecture notes §6
Problemset 4, due Wed Jan 26
Tue Jan 18 Equivalent definitions of compact metric spaces. The Banach Fixed Point Theorem. ODEs.
Lecture notes §5
References: [L, §7.6], [TBB, §§13.9–13.10, 13.11.4], [R, §9.3]
Thu Jan 13 Heine-Borel and Bolzano-Weierstrass. Compact subsets of metric spaces.
Lecture notes §4
Problemset 3, due Sun Jan 23
References: [L, §§7.4.2–7.5], [TBB, §13.12], [R, §4.3]
Tue Jan 11 Compact subsets of Rn, motivated by norms and analysis on finite sets.
Lecture notes §3
References: [L, §7.4.2], [TBB, §4.5], [R, §2.3]
Thu Jan 6 General theory of metric spaces. Developing the importance of open sets in relation to convergent sequences and continuous functions.
Lecture notes §2
Problemset 2, due Sun Jan 16
References: [L, §§7.2–7.4.1], [TBB, §§13.5–13.6,4.3–4.4]
Tue Jan 4 Definition of metric spaces, with many examples. Redefining terms from 18.100A/P in terms of metrics.
Lecture notes §1
Problemset 1, due Sun Jan 9
References: [L, §7.1], [TBB, §§13.1–13.4], [R, §2.2]

Last updated: Mar 20, 2022