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

- 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**

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:
- Writing a Research Paper in Mathematics, by Ashley Reiter
- How to Write Mathematics, by Paul Halmos
- Notes on logic, proof techniques, and mathematical writing by Prof. Jenny Wilson.

- The Hume Center provides additional support for WIM courses.

- WIM final drafts are due by 11:59 PM on Sunday, December 9th. No extensions.

**Office Hours**

- 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)

**Lectures**

**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**R**^{n}**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**R**^{n}, the Bolzano–Weierstrass theorem in**R**^{n}, 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.

**Homework**

- Problem sets are due (online) on Fridays at 10 AM (starting Week 2).
- Homework 1 (due Friday, October 5th at 10 AM) — solutions.
- Homework 2 (due Friday, October 12th at 10 AM) — solutions.
- Homework 3 (due Monday, October 22nd at 10 AM — late submissions will not be accepted) — solutions.
- Homework 4 (due Monday, November 5th at 10 AM) — late submissions will not be accepted) — solutions.
- Homework 5 (due Friday, November 9th at 10 AM) — solutions.
- Homework 6 (due Saturday, November 17th at 10 AM) — solutions.
- Homework 7 (due Friday, December 7th at 10 AM) — solutions.
- All problem sets will be submitted online using Gradescope.
- If you do not have access to the course on Gradescope, the entry code is MRZJ3P.
- Gradescope submission guide, written by Dr. Kimport, Stanford Math Dept.
- Submitting homework to Gradescope, a helpful guide written by Gradescope themselves.
- They also made this video!
- Problem sets 4–7 must be submitted in the form of a PDF produced using LaTeX. (see next section)

**Examinations**

- 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!

**LaTeX**

**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.

- My First LaTeX Document: Here's the .pdf, and here's the .tex file that produced it.
**Additional resources:**- LaTeX wikibook
- LaTeX cheat sheet
- Writing Scientific documents using LaTeX (.pdf, .tex)
- LaTeX comprehensive symbols list
- DeTeXify (draw a symbol in a box, magically find out what its LaTeX command is!)
- Sharelatex's tutorial videos page
- There are tons of others. Let me know if you encounter anything that was particularly helpful, and I'll add it!

**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!!*