Instructor
Office Hours
Graders
Textbooks
Supplements
Resources
Exams
Homework
Robert Chang (rchang@reed.edu)
WF 11:00 a.m. to 12:30 p.m. at Library 390
Solis McClain (mmcclain@) and Elle Wen (welle@)
[Ru] Rudin, Principles of Mathematical Analysis, 3rd edition
[SS] Stein–Shakarchi, Real Analysis: Measure Theory, Integration, and Hilbert Spaces
Errata for and supplements to [Ru] by George Bergman
Errata for [SS]
Math 321 lecture notes by David Perkinson
Math Help Center/Drop-in Tutoring: SuMTTh 7:00–9:00 p.m. at Library 204
Individual Tutoring: Reed offers one hour per week of free, one-to-one tutoring
One in-class midterm on Wednesday, October 11
One in-person final exam on Wednesday, December 13, 9:00 a.m. to 12:00 p.m. at Bio 19
Expect weekly assignments. Work must be typeset with LaTeX and submitted on GradeScope (entry code: 6GVPEV).
Collaboration, being a nontrivial part of learning and of scholarship in general, is highly encouraged. But, in accordance with the honor principle and basic human decency, you must submit your own write-up with an acknowledgment of collaborators.
Week
1
Date
8/28
8/30
9/1
Topics and Reading
Intro; set-theoretic foundations of the number system
Review [Ru, pp. 1-8] on your own: field axioms, order axioms, ordered field axioms,
Read [Ru, pp. 4-5, 8-11]
Least upper bounds; Archimedean property
Dedekind cuts is nicely explained in [Ru, pp. 16-21].
Read [Ru, pp. 8-11]
Archimedean property; distance functions and metric spaces
Review basic set theory and cardinality [Ru, pp. 24-30] on your own.
Read [Ru, pp. 30-36]
Homework 1 (pdf, tex, solutions) due Friday, September 8
2
9/4
9/6
9/1
Labor Day, no class
Metric topology
NB: I use Bᵣ(p) in place of Rudin's Nᵣ(p) to denote a neighborhood (i.e., a ball).
Read [Ru, pp. 30-36]
Metric topology comtinued; compactness
Read [Ru, pp. 30-36]
Homework 2 (pdf, tex, solutions) due Friday, September 15
3
9/11
9/13
9/15
Compactness; compact sets and closed sets; the finite intersection property
Read [Ru, pp. 36-38]; see also notes on the finite intersection property
Nested interval property; Heine–Borel theorem
Read [Ru, pp. 38-39]
Cantor set; sequences in a metric space
Those of you shaky on ε-δ proofs should carefully review the proof of Theorem 3.3.
David Perkinson's notes (linked further up on this webpage) may also be helpful.
Read [Ru, pp. 40-42]
Homework 3 (pdf, tex, solutions) due Friday, September 22
4
9/18
9/20
9/22
Sequences; subsequences
Read [Ru, pp. 47-52]
Cauchy sequences and completeness; monotone convergence theorem
Read [Ru, pp. 52-55]
limsup and liminf (à la Rudin); some special sequences
Read [Ru, pp. 55-58]; see also notes on equivalent definitions of limsup
Regrettably, there is no time for a detailed discussion on alternative definitions of upper/lower limits. Prospective analysts should nonetheless be aware of them (Definition 2 and Definition 3 in the notes linked above). The proofs of the equivalence of all these definitions are, admittedly, technical and notationally heavy, but it draws on many important ideas we have discussed thus far: sup/inf, limits, Cauchy sequences, etc.
Homework 4 (pdf, tex, solutions) due Friday, September 29
5
9/25
9/27
9/29
Cauchy criterion; test for divergence; comparison test; Cauchy condensation test; p-series; geometric series
Read [Ru, pp. 58-63]
The number e; root test; ratio test
Read [Ru, pp. 63-69]
Summation by parts; alternating series test; absolute convergence; Riemann's theorem on rearrangements
Read [Ru, pp. 70-72, 75-78]
We are moving fast through series because most convergence tests were covered in Math 112. Review David Perkinson's Math 112 notes as necessary.
Homework 5 (pdf, tex, solutions) due Friday, October 6
6
10/2
10/4
10/6
Power series; addition and multiplication of series; the exponential function
Read [Ru, pp. 72-75, 178-180]
Continuity in metric spaces
Read [Ru, pp. 83-87]
Extreme value theorem; intermediate value theorem; the Darboux property
Read [Ru, pp. 89-95]
No homework due Friday, October 13
In-class midterm on Wednesday, October 11
OH for midterm week: M 12:30–2:30, T 12:00–2:00 p.m. at Lib 390
7
10/9
10/11
10/13
Derivatives; the mean value theorem; Taylor's theorem
Read [Ru, pp. 103-108, 110-111]; see additional comments below
In-class midterm
Darboux sums; the Riemann integral
Read [Ru, pp. 120-125]; see additional comments below
Click to show additional comments on our study of derivatives and Riemann integrals
We have to be selective because of a lack of time. The important takeaways from Chapter 5 are as follows.
Theorem 5.2: differentiability implies continuity.
Theorems 5.8, 5.11: local extrema, critical points, and the first derivative test.
Theorem 5.10: the mean value theorem.
Theorem 5.15: Taylor's theorem with the Lagrange form of the remainder.
You may enjoy L'Hôpital's rule (Theorem 5.13), which is almost never done correctly in a calculus class, at your own leisure.
We must also abandon a detailed discussion of Riemann's integration theory in favor of that of Lebesgue. Do not, however, discount the Riemann integral, whose construction will again be relevant in the development of stochastic calculus.
Note that Rudin discusses not just the vanilla version of the Riemann integral, but the fancier Riemann–Stieltjes integral, in which one integrates againt a "weight" α = α(x). It is instructive to work through Rudin's proofs yourself in the special case α(x) = x (so as to recover the theory of the standard Riemann integral) to really understand what is going on.
Highlights from Chapter 6 include the following.
Definitions 6.1, 6.3: upper/lower Darboux sums, Riemann integrability, refinements of partitions.
Theorem 6.6: an equivalent, and computationally more useful, characterization of integrability.
Theorem 6.7: justifying the "limit as mesh size goes to zero of a Riemann sum" nonsense (what does this limit mean in ε-δ?) that one encounters in a calculus class.
Theorems 6.8, 6.10: functions with finitely many points of discontinuity are integrable.
Theorem 6.19: change of variables formula.
Theorems 6.20, 6.21: the fundamental theorems of calculus.
Theorem 6.22: integration by parts, (one of) the greatest trick(s) in mathematics.
In light of Theorem 6.10, one might be tempted to investigate the necessary and sufficient condition(s) for a function to be Riemann integrable. This condition turns out to be that "the set of discontinuities has Lebesgue measure zero," which offers a compelling transition to measure theory in the second half of this semester.
Fall break
No homework due Friday, October 27
9
10/23
10/25
10/27
The Riemann integral continued; the fundamental theorems of calculus
Read [Ru, pp. 125-129, 133-134]
Uniform convergence and shortcomings of the Riemannintegral, overview of Lebesgue's measure theory
Read [Ru, pp. 143–154], [SS, pp. xv–xix]
Lebesgue outer measure
Read [SS, pp. 10-16]
Homework 6 (pdf, tex, solutions) due Friday, November 3
10
10/30
11/1
11/3
Outer measure; measure; measurable sets
Read [SS, pp. 16-19]
Measure and measurable sets continued
Read [SS, pp. 16-19]
Countable additivity; continuity of measure; approximation of measurable sets by open and closed sets (Littlewood's principle), invariance properties of the Lebesgue measure
Read [SS, pp. 19-22]
Homework 7 (pdf, tex, solutions) due Friday, November 10
11
11/6
11/8
11/10
σ-algebras of measurable sets and of Borel sets; approximation of measurable sets by Borel sets
Read [SS, pp. 23-25]
Construction of a nonmeasurable set; measurable functions
Read [SS, pp. 27-30]
Approximation of measurable functions by simple and step functions
Read [SS, pp. 30-32]
Homework 8 (pdf, tex, solutions) due Friday, November 17
12
11/13
11/15
11/17
Lusin's and Egorov's theorems (Littlewood's principles)
Read [SS, pp. 33-34]
Integration of simple fuinctions and of bounded functions with finite support (cf. the Riemann integral)
Read [SS, pp. 49-55]
Integration of measurable functions; Fatou's lemma; monotone convergence theorem
Read [SS, pp. 56-65]
Homework 9 (pdf, tex, solutions) due Wednesday, December 9
Wednesday due date is not a typo. This is a long but important homework; start early.
13
11/20
11/22
11/24
Lebesgue's dominated convergence theorem, the space L¹ of integrable functions
Read [SS, pp. 65-69]
L¹ as a Banach space (Riesz–Fischer)
Read [SS, pp. 70-74]
Thanksgiving break
Thanksgiving break
Homework 9 (pdf, tex, solutions) due Wednesday, December 9
14
11/27
11/29
12/1
Invariance properties of the integral; modes of convergence
Read [SS, pp. 70-74]
The Fubini–Tonelli theorem (proof omitted); integration in polar coordinates
Read [SS, pp. 75-86]
Abstract measure theory: Carathéodory's criterion; Hahn–Kolmogorov theoremahn-Kolmogorov theor
Read [SS, pp. 262-281]
Some additional exercises for the final exam
15
12/4
12/6
Abstract measure theory continued: product measures; Fubini–Tonelli
Read [SS, pp. 262-281]
Review