Math 171: Fundamental Concepts of Analysis, Spring 2016

Monday, Wednesday, Friday 9:30-10:20am in 380-380F

Teaching Staff

Instructor Sheel Ganatra Course Assistant Alex Zamorzaev
Office 380-382E Office 380-380M
e-mail ganatra (at) math (dot) stanford (dot) edu e-mail alzaor (at) stanford (dot) edu
Office Hours This week: Wednesday 3-5pm, Thursday 2-3pm, Friday 2:30-3:30pm Office Hours This week: Thursday 4-7pm, Friday 4-7pm.

Course Description and Prerequisites

Math 171 is Stanford's honors analysis class and will have a strong emphasis on rigor and proofs. The class will take an abstract approach, especially around metric spaces and related concepts. Math 171 is required for honors majors, and satisfies the WIM (Writing In the Major) requirement.

For some students, Math 115 may be a suitable alternative to 171. Both Math 115 and Math 171 cover similar material, but 171 will be more fast-paced and have a more abstract/proof-based flavor. If you are unsure which of these two classes will be more appropriate for you, please come and talk to me as soon as possible (well before the drop deadline, which is April 15).

Textbook and topics

Foundations of Mathematical Analysis by Johnsonbaugh and Pfaffenberger.

We will cover approximately chapters I-X of the book. Thematically, the content we will cover falls into three areas:

A more detailed lecture plan (updated after each lecture) is below.

Course Grade

The course grade will be based on the following:

Homework Assignments

Homeworks will be posted here on an ongoing basis (roughly a week before they are due) and will be due at 4pm on the date listed. You can hand write your solutions, but you are encouraged to consider typing your solutions with LaTeX (now is a good time to start learning LaTeX if you haven't already, as you will be required to typeset the WIM assignment). Please submit your homework either directly to our Course Assistant Alex if he is in his office (380-380M) or slide the homework under his door if he is away. If you have typed your solutions, you are welcome to simply e-mail your homework to to Alex.

Note: we are looking not just for valid proofs, but also a readable, well explained ones (and indeed, you will be partly graded on readability). This means you should try to use complete sentences, insert explanations, and err on the side of writing out "for all" and "there exist", etc. symbols if there is any chance of confusion.

Late homeworks will not be accepted. In order to accomodate exceptional situations such as serious illness, your lowest homework score will be dropped at the end of the quarter. You are encouraged to discuss problems with each other, but you must work on your own when you write down solutions. The Honor Code applies to this and all other written aspects of the course.

As homeworks are completed, solutions (in PDF and LaTeX) will be uploaded here. The LaTeX files may be useful as templates for your own LaTeX work (whether on homework or the writing assignment).

Due date Assignment
Fri, Apr 8 Homework 1. Solutions: PDF. TeX
Fri, Apr 15 Homework 2. Solutions: PDF. TeX
Fri, Apr 22 Homework 3. Solutions: PDF. TeX
Fri, Apr 29 Homework 4. Solutions: PDF. TeX
Fri, May 6 Homework 5. Solutions: PDF. TeX
Fri, May 13 Homework 6. Solutions: PDF. TeX
Fri, May 20 Homework 7 (half weight, in light of WIM assignment). Solutions: PDF. TeX
Fri, May 27 Homework 8 (not half weight, but also shorter than usual) Solutions: PDF. TeX

Writing Assignment

The writing assignment is now posted here. The first draft will be due Friday, May 20 at 4pm to Alex and the final draft will be due Tuesday, May 31 at 4pm to Alex ( edit: The deadline has been extended to June 1 at 4 pm, should you require an additional day.) Your paper should be about 4-7 pages in length. It is required that you typeset your assignment; we strongly recommend you use LaTeX. We have made available copies of the TeX solutions to homework assignemnts above, in case they are helpful references as you learn LaTeX. A couple of the homework assignments will be shorter than usual, to give you time to work on the writing assignment.

Clear writing is an important part of mathematical communication, and is an important part of our course. The broad idea of this assignment is to write a clear exposition of a specific mathematical topic detailed in the assignment beyond what we have covered in class, which is accessible to someone at a similar stage in a similar class.

Professor Keith Conrad at the University of Connecticut has written a helpful guide to common errors in mathematical writing, available here.

Midterm and Final Exam

Midterm exam

The Midterm Exam was held on Wednesday April 27 from 8:30 am - 10:20 am in 380-380F. Here is a copy of the exam. Here is a set of solutions.

Also, here are some old exams from previous versions of the course which we used for practice (bear in mind that the exact topics, time of exam, and format all differ from year to year):

Final Exam

The Final Exam will be held on Saturday, June 4th from 8:30 am - 11:30 am in classroom 380-380X.

The final exam is a closed book, closed notes exam. The topics range through all of the topics we have covered in the class. Please see the Lecture Plan below for a review of all of the important topics and book sections (plus Professor Simon's review notes, and notes on integration) covered.

A helpful way to prepare for the final exam is to make sure you can solve all of the homework problems, and/or related problems in the book. It is also important to know and be able to articulate statements of all of the major definitions and Theorems/results covered in class to date (Indeed, a part of an exam question might even be to state an important result, before working with it. Or at the very least, when you are explaining your argument, you may need to cite such a result). You will not be asked to recall from memory proofs of important theorems, but you will certainly be asked to use these results, or reason through parts of their proof --- so some understanding of how various results are proved is definitely important.

Also, here is an old exam and some practice problems. See also the old midterm exams above (and note that some of them had problems which did not appear on our midterm, but might appear on the final)

Lecture Plan

Lecture topics by day will be posted on an ongoing basis below. Future topics are tentative and will be adjusted as necessary.

Day Lecture topics Book chapters Remarks
Mar. 28 Welcome and overview of class. A rapid review of real numbers and their axioms (algebraic, ordering, and the supremum/least upper bound property). Uniqueness of supremums (least upper bounds). Sequences, limits, convergence and uniqueness. Monotone bounded sequences. 3-7, 10, 16. For lectures Mar. 28 - April 11: See also Prof. Simon's review notes.
Mar. 30 More limits. Monotone bounded sequences are convergent. The algebra of limits and the squeeze theorem. Subsequences and the Bolzano-Weierstrass theorem. lim sup and lim inf of a bounded sequence. 10-13, 16, 18, 20, 21.
Apr. 1 More lim sup and lim inf: alternate characterizations, and a sequence converges if and only if lim sup and lim inf agree. Unbounded sequences. Countable and uncountable sets. 20, 21, 9.
Apr. 4 More countable and uncountable sets. Criteria for countability and examples of countable sets. Cantor's diagonalization argument and examples of uncountable sets. The continuum hypothesis (briefly). 9.
Apr. 6 Series, and their sum. Series with non-negative entries. Geometric series. Some tests for convergence and non-convergence. Absolute convergence. 22-26.
Apr. 8 More on absolute and conditional convergence and associated tests. Power series and their radius of convergence. 27-28
Apr. 11 Cauchy sequences in R. Functions (limits and continuity). An introduction to metric spaces. 19, 30-33, 35.
Apr. 13 More metric spaces. A series of examples: R^n with its Euclidean metric, the discrete metric, the metric spaces l^1, l^2,... and l^\infty, and convergence properties of their elements. l^k metrics on R^n. Convergence in metric spaces. 36-37.
Apr. 15 More convergence in metric spaces. Continuous functions between metric spaces, and examples. A characterization of continuity in terms of convergent sequences. 37, 40.
Apr. 18 Continuity in terms of convergent sequences continued. Compositions of continuous functions. Limit points of subsets and closed sets. Open balls and open sets. 40, 38, 39.
Apr. 20 Guest lecture by Prof. Alex Wright: More open sets and closed sets. V is open if and only if its complement is closed (and vice versa). Unions and intersections of closed and open sets. Characterizing continuous functions in terms of their effect on open (respectively closed) sets. 38-40.
Apr. 22 Guest lecture by Prof. Alex Wright: More continuity in terms of open and closed sets. Equivalent metrics (i.e., metrics that have the same open sets, or the same topology) have the same continuous functions. An example of two metrics on R^n with the same continuous functions. Z has the discrete topology, and hence all functions out of it are continuous. 38-40.
Apr. 25 The example of Z continued. The relative metric for a subset X in M, and open and closed sets for subsets endowed with relative metrics. A brief introduction to compactness. 41.
Apr. 27 Midterm exam from 8:30-10:20am (no class).
Apr. 29 An introduction to compactness: continuous functions from [a,b] to R are always bounded (and always attain their minima and maxima) because of the Heine-Borel theorem: any collection of open intervals covering [a,b] always has a finite subcollection which still covers [a,b]. The definition of compactness for a general metric space. 34, 42.
May 2 The definition of compactness for a general metric space (every open cover has a finite subcover). Examples and non-examples; the Heine-Borel theorem implies that [a,b] is compact. Properties of compact sets: any continuous real valued function on a compact set is bounded and attains its maximum and minimum. Sequential compactness (any sequence in a metric space has a convergent subsequence) is equivalent to compactness; which is often known as the Bolzano-Weierstrass characterization of compactness. One application: The image of any compat metric space under a continuous function is compact. 42, 43, 44.
May 4 A metric space is compact if and only if it is sequentially compact, continued. Compact subsets are closed and bounded. Subsets of R^n are compact if and only if they are closed and bounded. 42, 43, 44.
May 6 A closed subset of a compact metric space is compact. Subsets of R^n are compact if and only if they are closed and bounded, continued. Continuous injections out of compact sets have continuous inverses. Continuous functions out of compact metric spaces are uniformly continuous. Cauchy sequences and complete metric spaces. Q and (0,1) are examples of non-complete metric spaces. 42, 43, 44, 46.
May 9 Complete metric spaces continued. Cauchy sequences are bounded, and if they have convergent subsequences they themselves are convergent (with the same limit). Compact metric spaces are complete. R^n and l^1 are complete. Explanation of WIM assignment. 46
May 11 Connected metric spaces. Intervals in R are connected, and the image of a connected set under a continuous function is connected. The intermediate value theorem. An introduction to integration; intervals in R^n and their partitions. 45 For lectures May 11-current: See Prof. Simon's notes on integration.
May 13 Partitions of intervals continued. Step functions and how to integrate them. The Riemann integral of a function defined on an interval in R^n. Prof. Simon's notes.
May 16 The Riemann integral of a function defined on an interval in R^n. A criterion for checking integrability and application: continuous functions are Riemann integrable. An example of a non-Riemann integrable function. The set of Riemann integrable functions is a vector space, and the integral is a linear map on this vector space. Prof. Simon's notes.
May 18 Sets of (Lebesgue) measure zero. Countable unions of measure zero sets are measure zero. Lebesgue's theorem on the Riemann integral. Prof. Simon's notes.
May 20 The Lebesgue integral for non-negative functions in L_+(R). Well-definedness of the Lebesgue integral, which follows from Lemma 3.4 in Prof. Simon's notes. Prof. Simon's notes.
May 23 Class cancelled.
May 25 The Lebesgue integral of a function in L^1(R) and some properties: linearity, maxima, minima and absolute values stay in L^1(R), etc. Prof. Simon's notes.
May 27 Guest lecture by Prof. Angela Hicks: Riemann integrable functions are Lebesgue integrable, and (when both defined) the Lebesgue and Riemann integral coincide. Hence, the Lebesgue integral completely subsumes the Riemann integral. L^1(R) as a (semi-)normed vector space, and hence a pseudometric space. The associated metric space is complete, and hence a Banach space. Prof. Simon's notes.
May 27 Review.