Course Meeting Time and Location Monday, Wednesday and Friday 11:00  11:55 am 310 Linde
Course Instructor Contact Information and Office Hours lazebnik@caltech.edu 204 Kellogg Office hour: Wednesday 5pm, Linde 204
TA Contact Information and Office Hours
Liyang Yang lyyang@caltech.edu Office hour: Tuesday 89pm, Linde 259
Gefei Dang gdang@caltech.edu
Course Schedule and Textbook
Date  Topic  10/01/18  Several flawed applications of techniques from calculus (see Section 1.2 from Tao's Analysis I)  1. interchanging secondorder partial derivatives for a function which is not twice continuously differentiable, 2. flawed calculation of the length of the hypotenuse of a triangle using polygonal approximations, 3. interchanging the order of integration in a double integral, 4. interchanging a limit and an integral, 5. interchanging infinite sums, 6. flawed application of L'Hopital's rule.  10/03/18  Construction of the natural numbers from settheoretic axioms (informal), constructing the integers from the natural numbers, constructing the rationals from the integers. Nonexistence of a rational number whose square is 2, no least upper bound property for the rationals. Defining the real numbers as the set of cauchy rational sequences (modulo an equivalence relation).  10/05/18  Defining the real numbers using Dedekind cuts, the least upper bound property for the real numbers. Definition of a metric space, first example: Euclidean nspace with the Euclidean (L^2) metric.  10/08/18  Examples of metric spaces: (1) R^n with the Euclidean (L^2) metric, (2) R^n with the L^1 (taxicab) metric, (3) R^n with the L^\infty metric, (4) L^p function spaces (informal), (5) discrete metric, (6) Riemann sphere with the spherical metric. Inequalities between L^1, L^2, L^\infty metrics on R^n.  10/10/18  Convergence in metric spaces, examples, definitions of: ball, interior/exterior/boundary point, adherent point, closure, open/closed sets, examples.  10/12/18  Subspaces of metric spaces, subsequences, cauchy sequences, completeness. Proof that the real numbers (with the L^2 metric) are complete.  10/15/18  Definition of compactness for metric spaces (using existence of subsequential limits). Compact metric spaces are complete. Compact metric spaces are closed and bounded, whereas the converse is not true in general. HeineBorel Theorem (i.e. converse is true in R^n with the L^2 metric). Statement of equivalent definition of compactness using open covers.  10/17/18  Equivalent definition of compactness using open covers  proof of the equivalence with the definition using subsequential limits. Proof that the intersection of a sequence of nested, nonempty compact sets is nonempty.  10/19/18  Epsilondelta definition of a continuous function between metric spaces, two equivalent definitions using sequences and preimages of open sets. Continuous images of compact sets are compact, the extreme value theorem. Definition of uniform continuity, e.g. exponential function is continuous but not uniformly continuous on the real line, but continuous functions on compact metric spaces are always uniformly continuous.  10/22/18  Definition of connected subsets of metric spaces, characterizing connected subsets of the real line, continuous images of connected sets are connected, deducing the intermediate value theorem.  10/24/18  Definition of pointwise convergence of a sequence of functions. One can not, in general, interchange (pointwise) limits with integrals, derivatives, etc.... Definition of uniform convergence of a sequence of functions. A (uniform) limit of a sequence of continuous functions is continuous.  10/26/18  Interchanging a limit "x > x_0" in a metric space with a uniform limit "n > \infty$ for a uniformly convergent sequence f_n. Introducing the metric of uniform convergence on a space of bounded functions.  10/29/18  The subspace of continuous bounded functions C(X,Y) is complete inside the space of bounded functions B(X,Y) (with the L^\infty metric) provided that the metric space Y is complete. A criterion (uniform convergence) for interchanging a Riemann integral with a limit.  10/31/18  Interchanging infinite sums with integrals for uniformly convergent series, finding power series expansion for log(1r) as an application. (f_n) > f uniformly with all (f_n) differentiable does not in general imply differentiability of f. With an extra assumption of differentiability of f it still does not follow that (f_n)' > f' pointwise.  11/2/18  A criterion for being able to interchange a limit with a derivative. Brief discussion of Weierstrass approximation theorem and a general strategy for the proof (convolving the given continuous function with a polynomial approximation to the identity).  11/5/18  Defining approximations to the identity, proving the existence of polynomial approximations to the identity, definition of convolution, intuition for convolving a function with an approximation to the identity.  11/7/18  Proof that convolving a compactly supported continuous function f (on R) with an approximation to the identity actually approximates f. Proof that convolving with a polynomial yields a polynomial. Deducing the Weierstrass approximation theorem.  11/9/18  Review of the Riemann integral and the fundamental theorems of calculus  11/12/18  Definition of the RiemannStieltjes integral, some motivating examples (the classical Riemann integral, integrating against an approximation to the identity, integrating against increasing step functions), basic properties.  11/14/18  Criterion for RiemannStieltjes integrability, proving that continuous functions are RiemannStieltjes integrable, and bounded functions f with finitely many discontinuities (where the function "\alpha" we are integrating against is continuous at the points of discontinuity for f) is RiemannStieltjes integrable.  11/16/18  Evaluating the RiemannStieltjes integral when (1) "\alpha" is C^1, (2) "\alpha" is an increasing step function  11/19/18  Brief motivation for Fourier series. Introducing the space of continuous 1periodic functions on the real line. Definitions of inner product space (e.g. L^2), normed vector space (e.g. L^{\infty}).  11/21/18  Space of continuous 1periodic functions on the real line with L^2 inner product, CauchySchwarz inequality for inner product spaces, defining character with frequency n, trigonometric polynomials. Orthonormality of sequence of characters of frequency n.  11/26/18  Fourier transform of periodic functions, notions of periodic convolutions and periodic approximations to the identity. Statement of Weierstrass approximation theorem for approximating continuous periodic functions by trig. polynomials.  11/28/18  Proving existence of trig. polynomials which are approximations to the identity, deducing Weierstrass approximation theorem with trig. polynomials.  11/30/18  Fourier and Plancherel theorems  12/3/18  Explicit example of a continuous periodic function whose Fourier series diverges at a single point.  12/5/18  Some discussion of Fourier series for periodic, discontinuous functions; the start of a proof for a pointwise convergence theorem for Fourier series of differentiable functions.   
Course Policies You are encouraged to collaborate with one another on homework. It is not, however, permitted to consult solutions manuals or online forums for help with homework. Late work is only accepted with written permission from the Dean. There will be a small bonus (5 points per assignment) for work typeset using LaTeX. The textbook for the course is Rudin's Principles of Mathematical Analysis.
Assignments (solutions posted below) Date Posted  Assignment  Due Date  10/05/18  HW1  10/12/18 (noon)  10/12/18  HW2  10/19/18 (noon)  10/19/18  HW3  10/26/18 (noon)  10/26/18  HW4  11/2/18 (noon)  11/2/18  Take the midterm. There are two parts to the midterm: the first part is a closed book exam of four problems, to be completed in one sitting of no more than four hours (I think four hours is more than you will need). No class notes, textbooks, computers/calculators, or collaboration for the first part. The second part consists of two problems for which you can use your notes or your textbook, but no internet and no collaboration. The second part is not timed  you just need to finish writing the solutions before Friday 11/9 at noon. Here is a link to the first part (do not open this until you are ready to take the exam), and here is a link to the second part (you can open this whenever you would like  this part is not timed.)  11/9/18 (noon)  11/7/18  HW5  11/16/18 (noon)  11/14/18  HW6  11/21/18 (4pm)  11/28/18  HW7  12/5/18 (Wednesday, noon)  12/5/18  Take the final. The format of the final is the same as that of the midterm. The first part is a closed book exam, consisting of four problems, to be completed in a single sitting of no more than five hours. No class notes, textbooks, computers/calculators, or collaboration for the first part. The second part consists of two problems for which you can use your notes or your textbook (Rudin), but no internet and no collaboration. The second part is not timed  you just need to finish writing the solutions by 12/12 at noon. Please hand in both parts (preferably attached together somehow) by 12/12 at noon. Here is a link to the first part (do not open this until you are ready to take the exam), and here is a link to the second part (you can open this whenever you would like  this part is not timed.)  12/12/18 (Wednesday, noon) 
Midterm and Final Exam
Collaboration Table  Homework  Exams 

You may consult:    Course textbook (including answers in the back)  YES  NO  Other books  YES  NO  Solution manuals  NO  NO  Internet  YES  NO  Your notes (taken in class)  YES  NO  Class notes of others  YES  NO  Your hand copies of class notes of others  YES  NO  Photocopies of class notes of others  YES  NO  Electronic copies of class notes of others  YES  NO  Course handouts  YES  NO  Your returned homework / exams  YES  NO  Solutions to homework / exams (posted on webpage)  YES  NO  Homework / exams of previous years  NO  NO  Solutions to homework / exams of previous years  NO  NO  Emails from TAs  YES  NO  You may: 

 Discuss problems with others  YES  NO  Look at communal materials while writing up solutions  YES  NO  Look at individual written work of others  YES  NO  Post about problems online  NO  NO  For computational aids, you may use: 

 Calculators  YES*  NO  Computers  YES*  NO 
* You may use a computer or calculator while doing the homework, but may not refer to this as justification for your work. For example, "by Mathematica" is not an acceptable justification for deriving one equation from another. Also, since computers and calculators will not be allowed on the exams, it's best not to get too dependent on them. 
Updating...
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