### Classical Analysis

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 second-order 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 set-theoretic axioms (informal), constructing the integers from the natural numbers, constructing the rationals from the integers. Non-existence 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 n-space 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 Heine-Borel 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, non-empty compact sets is non-empty. Epsilon-delta 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(1-r) 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, nowhere-differentiable 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 Riemann-Stieltjes 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 Riemann-Stieltjes integrability, proving that continuous functions are Riemann-Stieltjes 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 Riemann-Stieltjes integrable. 11/13/15 Evaluating the Riemann-Stieltjes 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 1-periodic 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
HomeworkExams
You may consult:
Course textbook (including answers in the back)YESNO
Other booksYESNO
Solution manualsNONO
InternetYESNO
Class notes of othersYESNO
Your hand copies of class notes of othersYESNO
Photocopies of class notes of othersYESNO
Electronic copies of class notes of othersYESNO
Course handoutsYESNO
Solutions to homework / exams (posted on webpage)YESNO
Homework / exams of previous yearsNONO
Solutions to homework / exams of previous yearsNONO
Emails from TAsYESNO
You may:

Discuss problems with othersYESNO
Look at communal materials while writing up solutionsYESNO
Look at individual written work of othersYESNO
For computational aids, you may use:

CalculatorsYES*NO
ComputersYES*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.

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HW2_108a.tex
(6k)
kirilllazebnik,
Oct 16, 2019, 6:07 PM
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Angus Gruen,
Oct 20, 2019, 4:47 PM
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Angus Gruen,
Nov 3, 2019, 10:28 PM