# 18.100b: Real Analysis, Spring 2016

Lecturer: Joshua Zahl
jzahl@mit.edu
Class: TR 1-2:30: 4-163
Office hours: T 11-12, R 10-11: 2-169

TA: Kostya Tolmachov
tolmak@mit.edu
Office hours: M 11-12, W 1-2: 2-333A

## Textbook

The course textbook will be Mathematical Analysis, Second Edition by Tom Apostol.

## Course Policies

The grade breakdown will be 50% Homework, 15% midterm, 35% final. There will be an in-class midterm on April 7, 2016. There will also be a final (between May 16-20; date to be determined).

### Homework

Homework can be turned in during class, dropped in my mailbox (2-169) or emailed to me by 2:30pm on the posted due date. The homework consists of three parts (A,B,C). Please turn in each part as a separate, stapled packet, and write your name, the homework number, and the part clearly on the top of each packet.

There will be 11 homework assignments. Your lowest homework score will be dropped and the "homework" portion of your grade will be an average of the 10 remaining scores. If you fail to turn in a homework assignment by the posted due date (regardless of reason), this assignment will receive a score of 0 and will be counted as your lowest homework score (and thus dropped). If you miss multiple homeworks due to a documented health or other issue, please contact me and we can discuss how to proceed.

### Collaboration Policy

You are encouraged to collaborate on homework assignments. However, each student must write up his or her own solutions in his or her own words. In addition to the course textbook, you are encouraged to consult other textbooks and online sources such a wikipedia, mathworld, etc. However, consulting online solution manuals (to Apostol or other textbooks) is not permitted.

## Course calendar

 February 11, 2016 HW 1 Due Solutions February 18, 2016 HW 2 Due Solutions February 25, 2016 HW 3 Due Solutions March 3, 2016 HW 4 Due Solutions March 10, 2016 HW 5 Due Solutions March 17, 2016 HW 6 Due Solutions March 31, 2016 HW 7 Due Solutions April 7, 2016 Midterm Solutions April 14, 2016 HW 8 Due Solutions April 21, 2016 HW 9 Due Solutions April 28, 2016 HW 10 Due Solutions May 5, 2016 HW 11 Due Solutions May 18, 1:30pm Final exam

## Final exam review material

Here is some material that might be helpful in reviewing for the final exam.

This review sheet is for 18.100C. The final exam for 18.100C is not the same as for 18.100B, but the material is similar. In particular, you should be able to do all of the "practice questions," (for problem 3: "uniform closure" = "closure in the metric space C([0,1])" ). You can ignore the first half of the sheet; this lists sections from Rudin which were covered in 18.100C.

The following excercises from Apostol are relevant for the course. I don't expect you to do all of them, but it is a good idea to look over them and try to figure out (at least briefly) how you would go about solving each one. Then write down detailed solutions for just a few.

2.4-2.9, 2.12, 2.15-2.22
3.1, 3.4-3.11, 3.13, 3.14, 3.16-3.24, 3.26-3.32, 3.36-3.52
4.5-4.11, 4.13-4.33, 4.36-4.45, 4.47-4.57, 4.60-4.72
5.1, 5.4, 5.10, 5.12, 5.15-5.21, 5.23, 5.25, 5.26, 5.27
6.2-6.7, 6.11-6.13
7.1, 7.2, 7.12-7.7.15, 7.19, 7.20, 7.22, 7.24, 7.25, 7.28, 7.29, 7.32
9.1-9.3, 9.5-9.8, 9.16-9.17, 9.22

## Course syllabus

Feb 2, 2016: definition of Z, Q, axiomatic definition of R; field axioms, order axioms, intervals (a,b), [a,b] (a,b] and [a,b), geometric interpretation of an interval, upper bounds, least upper bound, least upper bound axiom for R, why the least upper bound axiom fails for Q, construction of the reals using Dedekind cuts.

Feb 4, 2016: Cartesian products and functions, injec/surj/bijec. N and W have same cardinality, N and N^2 have same cardinality, any subset of a countable set is countable, N is not the same cardinality as R.

Feb 9, 2016 function composition, definition of |A|, when is |A| = |B|, |A|\leq |B|, |A| < |B|, Cantor-Bernstein-Schoeder theorem

Feb 11, 2016: Definition of R^n, vector addition, scalar multiplication, Cauchy-Schwarz, triangle inequality, open balls in R^n, interior point, defn of open set, examples of open sets, union of open sets is open, finite intersection of open sets is open, infinite intersection of open sets need not be open, every open subset of R is a countable union of disjoint open intervals.

Feb 18, 2016: defn of closed set; countable unions, arbitrary intersections, accumulation pt, nested interval theorem in R, open covers and compactness, multi-dim intervals are compact, -closed subsets of compact sets are compact, Heine Borel: closed + bdd -> compact. Compact -> Bdd.

Feb 23, 2016: metric space, open & closed sets in metric space; arbitrary union / countable intersection of open sets is open, relative metric, compactness -> closed & bounded. Converse is false, defn of a topological space, not every top. space comes from a metric, two different metrics can induce the same topology or different topologies, sequences & limits, convergent & divergent sequences (depends on the ambient metric space!), sequence is convergent iff every sub-sequence is convergent, Cauchy sequence, defn of a complete metric space.

Feb 25, 2016: M compact iff every sequence has a convergent subsequence, compact metric spaces are complete, R^n is a complete metric space.

March 1, 2016: Continuous functions: \eps - \delta defn, continuous iff function commutes with limits, composition of continuous functions, inverse image of open set is open, image of compact set is compact, uniform continuity, continuous functions on compact sets are uniformly continuous

March 3, 2016: contraction mapping theorem, Picard-Lindelof existence theorem for ODE [rough sketch & main ideas], Bolzano thm / intermediate value theorem, connected components of a metric space, two-valued functions, intermediate value theorem on a connected metric space

March 8, 2016: definition of \lim_{x->c}f(x) for a function f: (a,b)->R, one sided limits, sums, products, quotients of limits, definition of the derivative of f(x) at c, one-sided derivatives, infinite derivatives, sum, product, quotient rule, chain rule

March 10, 2016: Local extrema, Rolle's theorem, mean value theorem, intermediate value theorem for derivatives

March 15, 2016: higher order derivatives, generalized mean value theorem, Taylor's theorem with remainder, derivatives of vector-valued functions, partial derivatives of multivariate functions

March 17, 2016: functions of bounded variation, total variation, functions of bounded variation as the difference of two increasing functions

March 29, 2016: an increasing function is discontinuous at a countable set of points, Cantor-Lebesgue function, Cantor set

March 31, 2016: the Riemann-Stieltjes integral

April 5, 2016: Midterm review

April 12, 2016: Riemann's condition for the existence of the Riemann-Stieltjes integral (with alpha increasing), integrators of bounded variation

April 14, 2016: Lebesgue's criterion for the existence of the Riemann integral, Oscillation of a function at a point / on an interval

April 21, 2016: Mean value theorem for Riemann-Stieltjes integrals, first and second fundamental theorems of calculus

April 26, 2016: Sequences of functions, pointwise convergence, uniform convergence, uniformly bounded sequences, equicontinuity, Arzela Ascoli theorem

April 28, 2016: Arzela Ascoli (cont'd), Peano existence theorem for ODE

May 3, 2016: Stone-Weierstrass theorem

May 6, 2016: Stone-Weierstrass on a compact metric space, trigometric polynomials on [0,1] satisfy hypothesis of Stone-Weierstrass, intro to Fourier series on the circle