Math 171 — Stanford University, Fall 2018 — Schaeffer
Fundamental Concepts of Analysis
Course Information and Policies
Math 171 is a 10-week course on analysis. We will cover
The course policies are here.
- Axiomatic and constructive approaches to the real numbers;
- The topology of metric spaces;
- Completeness, compactness, and continuity;
- Theories of differentiation and integration;
- Sequences of functions;
- Basic functional analysis.
The textbook for this course is Foundations of Mathematical Analysis by Johnsonbaugh and Pfaffenberger.
Writing in the Major
- Math 171 is a WIM class. All enrolled students must complete a 4–6 page term paper on a topic of their choice (regardless of actual major).
- More information on how your paper should be structured is in this document. Information on how your project will be graded is here.
- A list of pre-approved WIM topics can be found here.
- Project topics are due, via this webform, officially by 11/07 (but practically by 11/09). This is part of your grade on HW5.
- Writing resources for mathematics:
- The Hume Center provides additional support for WIM courses.
- WIM final drafts are due by 11:59 PM on Sunday, December 9th. No extensions.
- Schaeffer: 4:30–6:00 PM on Mondays (in 381-G)
- Kuperberg: 3:30–5:30 PM on Wednesdays (in 380-L), 1:00–4:00 PM on Thursdays (in 380-W)
- Lecture 1 (09/25): Sets, Cartesian products, functions; the Peano axioms for N [Corresponding sections in J&P: 1, 2, 7]
- Lecture 2 (09/27): Addition, multiplication, and order in N; relations, equivalence relations; constructions of Z and Q [J&P: 6]
- Lecture 3 (09/29): Synthetic and constructive approaches to R [J&P: 3–5]
- Lecture 4 (10/01): Consequences of the axioms in R (Archimedean property, density of Q) [J&P: 3–5]
- Lecture 5 (10/03): Sequences in R, the Bolzano–Weierstrass theorem, the Cauchy condition [J&P: 10, 11, 18, 19]
- Lecture 6 (10/05): Some basic limits and inequalities, existence of roots in R, convergence of base-r representations
- My lecture notes from the first two weeks.
(Note: These are provided as a record of what we may have covered in class. Lectures could be slightly different. These are not formal course materials.)
- Lecture 7 (10/08): Metric spaces, the Cauchy–Schwarz inequality, the Euclidean metric on Rn
- Lecture 8 (10/10): Topological spaces, the topology of metric spaces, compactness, compact sets are closed (in metric spaces)
- Lecture 9 (10/12): Completeness of Rn, the Bolzano–Weierstrass theorem in Rn, the Heine–Borel theorem
- My lecture notes from the third week. — The completed proof that the Euclidean metric is a metric is on Piazza.
- Lecture 10 (10/15): Sequential compactness and the Baire category theorem.
- Lecture 11 (10/17): "Small" sets vs. "big" sets in R.
- Lecture 12 (10/19): Continuous functions, continuity and compactness.
- Lecture 13 (10/22): Continuity and connectedness.
- TeXed lecture notes for the whole course so far, a work in a constant state of progress (currently up to lecture 16—integration lectures in different notes below).
- Lecture 14 (10/24): Uniform continuity.
- Lecture 15 (10/26): Limits of real functions and discontinuities of the first and second kinds.
- Lecture 16 (10/29): Differentiation, Rolle's theorem, mean value theorem(s).
- Lecture 17 (10/31): Riemann–Darboux integration of step functions.
- Lecture 18 (11/02): Properties of the lower and upper Riemann–Darboux integrals.
- Notes on integration, lectures 17–18.
- Lecture 19 (11/05): Sufficient conditions for Riemann–Darboux integrability, the fundamental theorem of calculus.
- Lecture 20 (11/07): FTC continued; sequences of functions, pointwise vs. uniform convergence.
- Lecture 21 (11/09): Uniformly convergent sequences of continuous/integrable functions are continuous/integrable.
- Lecture 22 (11/12): Series of functions.
- Lecture 23 (11/14): Series of functions.
- Lecture 24 (11/26): Stone–Weierstrass theorem.
- Lecture 25 (11/28): Stone–Weierstrass theorem.
- Lecture 26 (11/30): The Takagi function; measure spaces.
- Lecture 27 (12/03): Measure spaces.
- Lecture 28 (12/05): Measure spaces.
- Lecture 29 (12/07): Lebesgue's convergence theorems and consequences.
- Problem sets are due (online) on Fridays at 10 AM (starting Week 2).
- All problem sets will be submitted online using Gradescope.
- Problem sets 4–7 must be submitted in the form of a PDF produced using LaTeX. (see next section)
- The midterm exam will be take-home. It will be released Wednesday October 24th after class. You will have 3 hours from when you download the exam to complete it, plus an extra 30 minutes to scan/upload your completed exam to Gradescope. It will be due at the end of Friday (that is, 11:59:59 PM).
- To read the parameters of the midterm exam before downloading it, click here.
- Copy of the midterm and solutions.
- The final exam will be take-home. It will be released Saturday December 9th. You will have 8 hours from when you download the exam to complete it (though it will be designed for 4.5 hours), plus an extra 30 minutes to scan/upload your completed exam to Gradescope. It will be due at 4:30 PM on Thursday, December 13th.
- To read the parameters of the final exam before downloading it, click here.
- Final exam solutions and remarks!
- Getting LaTeX on your computer: You can either install LaTeX on your computer, or you can use an online editor. That's up to you!
(Personally, I do most of my LaTeX on Overleaf (formerly ShareLaTeX), nowadays, but I do have MacTeX installed on my home laptop and use TeXshop for my offline typesetting.)
- LaTeX installation is somewhat complicated, but one of the recommended "Distributions" here will probably work for you.
- Here's a list of online editors. I only have experience with Overleaf/ShareLaTeX, which is free and works reasonably well.
- Producing your first document: Like most programming languages, probably the easiest way to get started in LaTeX is to start from a file written by an experienced user and then customize it.
- My First LaTeX Document: Here's the .pdf, and here's the .tex file that produced it.
To see what the above looks like on Sharelatex, you have to either create a blank project and paste the source into main.tex, or upload the .tex file as a separate file in the project.
- The file above should help you get started and you should feel free to use it as a template to create your homework files.
- Overleaf also has some nice homework templates.
- Additional resources:
Lectures, Readings, and Handouts
- Some detailed notes on logic, proof techniques, and mathematical writing (written by Prof. Jenny Wilson)
Please read these if you haven't had a lot of experience with proofs!!