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18.S097 | January IAP 2022 | Undergraduate
Introduction to Metric Spaces

There is no official textbook. Lecture notes are provided below. You may find the following textbooks useful:

[L] = Lebl, Jiri. Basic Analysis I: Introduction to Real Analysis, Volume 1. CreateSpace Independent Publishing Platform, 2018. ISBN: 9781718862401. This book is available as a free PDF download.

[TBB] = Thomson, Brian S., Judith B. Bruckner, and Andrew M. Bruckner. Elementary Real Analysis. CreateSpace Independent Publishing Platform, 2008. ISBN: 9781434841612. This book is available as a free PDF download (PDF - 5.96 MB).

[R] = Rudin, Walter. Principles of Mathematical Analysis. McGraw Hill, 1976. ISBN: 9780070542358.

Lecture notes in one file: Introduction to Metric Spaces (PDF)

#### Lecture 1: Motivation, Intuition, and Examples

Lecture 1: Motivation, Intuition, and Examples (PDF)

Lecture 1: Motivation, Intuition, and Examples (TEX)

Outline: Motivation, definition, and intuition behind metric spaces. Redefining 18.100A Real Analysis and 18.100P Real Analysis in terms of metrics: open/closed sets, convergence, Cauchy sequences, and continuity.

References: [L, §7.1], [TBB, §§13.1–13.4], [R, §2.2]

#### Lecture 2: General Theory

Lecture 2: General Theory (PDF)

Lecture 2: General Theory (TEX)

Outline: Some general theory of metric spaces regarding convergence, open and closed sets, continuity, and their relationship to one another.

References: [L, §§7.2–7.4.1], [TBB, §§13.5–13.6, 4.3–4.4]

#### Lecture 3: Compact Sets in Rⁿ

Lecture 3: Compact Sets in Rⁿ (PDF)

Lecture 3: Compact Sets in Rⁿ (TEX)

Outline: Norms and analysis on finite sets (as motivation for compact sets). Topological compactness and sequential compactness. Today will focus purely on theorems regarding compact subsets of Rⁿ. The Heine-Borel theorem and the Bolzano-Weierstrass theorem.

References: [L, §7.4.2], [TBB, §4.5], [R, §2.3]

#### Lecture 4: Compact Metric Spaces

Lecture 4: Compact Metric Spaces (PDF)

Lecture 4: Compact Metric Spaces (TEX)

Outline: Compact sets on general metric spaces. Sequential compactness is equivalent to topological compactness on metric spaces.

References: [L, §§7.4.2–7.5], [TBB, §13.12], [R, §4.3]

#### Lecture 5: The Fixed Point Theorem

Lecture 5: The Fixed Point Theorem (PDF)

Lecture 5: The Fixed Point Theorem (TEX)

Outline: Compact is equivalent to totally bounded and complete. The Banach fixed point theorem and ODEs. The Arzela-Ascoli theorem.

References: [L, §7.6], [TBB, §§13.9–13.10, 13.11.4], [R, §9.3]

#### Lecture 6: Where We Go from Here

Lecture 6: Where We Go from Here (PDF)

Lecture 6: Where We Go from Here (TEX)

Outline: How these topics were motivated, and a preview of how these topics come up in later classes.

Course Info
Instructor

- Paige Bright

Departments

- Mathematics

As Taught In
January IAP 2022
Level
Undergraduate
Topics

- Mathematics
  Mathematical Analysis

Learning Resource Types
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