Course Meeting Time and Location Lecture 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 259 Linde Hall
Office Hours: Thursday 8pm, Linde 259
Course Schedule and Textbook
Course Schedule and Textbook
Date  Topic  10/02/19  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.  10/04/19  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).  10/07/19  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, L^p metrics, L^\infty metric.  10/09/19  Examples of metric spaces: R^n with the L^1 (taxicab) metric, discrete metric, Riemann sphere with the spherical metric. Inequalities between L^1, L^2 metrics on R^n. Discussion of C([0,1]) and application to solving an integral equation. Definition of ball in metric space.  10/11/19  Convergence in metric spaces, examples, definitions of: ball, interior/exterior/boundary point, adherent point, closure, open/closed sets, examples. Subspaces of metric spaces, subsequences, cauchy sequences, completeness. Sketch of a proof that the real numbers (with the L^2 metric) are complete.  10/14/19  Further discussion of completeness (analogy between L^p in R^n and function spaces). 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.  10/16/19  HeineBorel Theorem. Equivalence of two definitions of compactness: (1) existence of subsequential limits, (2) existence of finite subcovers.  10/18/19  Proof that the intersection of a sequence of nested, nonempty compact sets is nonempty. 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.  10/21/19  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. Definition of connected subsets of metric spaces  10/23/19  characterizing connected subsets of the real line, continuous images of connected sets are connected, deducing the intermediate value theorem. 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.  10/25/19  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.  10/28/19  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.  10/30/19  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.  11/1/19  A criterion for being able to interchange a limit with a derivative. Weierstrass example of a continuous, nowheredifferentiable function.  11/4/19  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.  11/6/19  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.  11/8/19  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.  11/11/15  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.  11/13/15  Evaluating the RiemannStieltjes integral when (1) "\alpha" is C^1, (2) "\alpha" is an increasing step function  11/15/15  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}). Cauchy Schwarz inequality.         
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. Homework assignments will constitute 50% of the final grade, the midterm 20%, and the final 30%.
Assignments Date Posted  Assignment  Due Date  10/4/19  HW1  10/11/19 (noon)  10/11/19  HW2  10/18/19 (noon)  10/18/19  HW3  10/25/19 (noon)  10/25/19  HW4  11/1/19 (noon)  11/1/19  Complete the midterm examination. The midterm is a 4 hour exam (to be completed in one sitting). No notes, internet, or textbooks allowed. Please hand in the exam by noon on Friday, 11/8. This is the exam (do not open this link until you are ready to take the exam): Midterm.  11/8/19 (noon)  11/8/19  HW5  11/15/19 (noon)  11/15/19  HW6  11/22/19 (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...
kirilllazebnik, Oct 16, 2019, 6:07 PM
Ċ Angus Gruen, Oct 20, 2019, 4:47 PM
Ċ Angus Gruen, Nov 3, 2019, 10:28 PM
