Math 415: Honors Analysis I

Fall 2018

Textbook

Real Analysis, N. L. Carothers,

Cambridge University Press, 2000.

 

Course Description

Our goal is to cover the first half of the textbook (Chapters 1-11), with some exceptions and amendments. Major topics to be addressed are likely to include:

_       Construction of real numbers.

_       Topology of metric spaces, normed vector spaces. Open and closed sets, sequences and limits.

_       Bounded linear transformations. The derivative as a linear transformation.

_       Properties of compact metric spaces. The Heine-Borel Theorem. The Bolzano-Weierstrass Theorem.

_       Continuous functions, connectedness and completeness. Contraction Mapping Principle and applications. Inverse and Implicit function theorems.

_       Baire category theorem.

_       Sequences and series of functions. Uniform convergence. The Arzela-Ascoli theorem. The Weierstrass approximation theorem.

Instructor

Hang Xu

Email: hxu@math.jhu.edu

Office Location: Krieger Hall Rm 220

Office Hours: W 3:00-5:00

Course Assistant

Cheng Zhang

Email: czhang67@math.jhu.edu   

Office Hours: Th 9-11 am at Krieger Hall 213

Lecture

MW 1:30-2:45 at Maryland 309

Section

F 1:30-2:20 at Hodson 311

 

Grading

is based on 350 points distributed as follows:

11 homework problem sets, worth 10 points each, due in the section on the following Friday. The lowest score will be dropped.

Midterm on October 10 in lecture, worth 100 points.

Final 9 am-12 pm on December 13, worth 150 points.

 

Course Policies

No late homework is accepted. Staple your problem sets! Study groups are encouraged, but homework has to be written down independently.

No makeup exam in this course. If you have to miss an exam for a documented, legitimate reason, please inform me as early in the semester as possible.

You are responsible for lecture notes, any course material handed out, and attendance in class. The lectures will be conducted as if you have already read the material and attempted some homework problems. In this manner, you can focus mainly on those parts of the lectures that cover the areas of your reading you found difficulty to understand.

If you have any math questions, please feel free to ask me anytime. You can find me after lectures, in office hours, or you can reach me by email.

 

Help Room

Krieger Hall 213. The hours are 9am – 9pm on Monday through Thursday, and 9am – 5pm on Friday. This free service is a very valuable way to get one-on-one help on the current material of a class from other students outside the course. It is staffed by graduate students and advanced undergraduates. Outside of me and the Grader for the course, definitely take your questions to the Help Room. This course is simply an analysis course directed toward particular maps and differential equations. Most graduate students should be able to "see" through the many problems stated in this course. And your attempts to help guide them will be of huge benefit to you also. 

 

Students with disabilities

Students with documented disabilities or other special needs who require accommodation must obtain an accommodation letter from Student Disability Services, 385 Garland, (410)516-4720, studentdisabilityservices@jhu.edu. After that, remind the instructor of the specific needs at two weeks prior to each exam; the instructor must be provided with the official letter stating all the needs from Student Disability Services.

 

JHU ethics statement

The strength of the university depends on academic and personal integrity. In this course, you must be honest and truthful. Ethical violations include cheating on exams, plagiarism, reuse of assignments, improper use of the Internet and electronic devices, unauthorized collaboration, alteration of graded assignments, forgery and falsification, lying, facilitating academic dishonesty, and unfair competition.

Report any violations you witness to the instructor. You may consult the associate dean of student affairs and/or the chairman of the Ethics Board beforehand. See the guide on “Academic Ethics for Undergraduates” and the Ethics Board Web site (http://e-catalog.jhu.edu/undergrad-students/student-life-policies/#UAEB) for more information.

 

Tentative Schedule (will be updated as the course progresses)

Homework will be posted by every Monday and due in the section on the next Friday.

Week 1 (8/30): Chapter 1

Week 2 (9/5): Chapter 1 and Chapter 2

Homework 1: chapter 1, problems: 3, 7, 15, 17, 37, 45, 46.

Week 3 (9/10 & 9/12): Chapter 2 and Chapter 3

Homework 1 due.

Homework 2: chapter 2, problems 3, 7, 8, 16, 17, 18.

Week 4 (9/17 & 9/19): Chapter 3 and Chapter 4

Homework 2 due.

Homework 3: chapter 2, problems 22, 23, 26, 29, 30, 32, 33.

Week 5 (9/24 & 9/26): Chapter 3

Read 3.1, 3.2

Homework 3 due.

Homework 4:  chapter 3, Problems: 6, 15, 22, 23, 25, 31, 37

 

Week 6 (10/1 & 10/3): Chapter 5 and Chapter 6

Read 3.3

Homework 4 due.

Homework 5 (part a): chapter 4, Problems: 1, 3, 5, 8.

 

Week 7 (10/8 & 10/10): Chapter 6 and Midterm

Midterm on Wednesday in class

Homework 5 (part b): chapter 4, Problems: 11, 18, 33, 34, 41, 46, 48.

No homework due.

 

Week 8 (10/15 & 10/17): Chapter 7

Homework 5 due next Monday (10/22) in the lecture since this Friday (10/19) is a holiday.

Homework 6:  Chapter 5, Problems: 8, 17, 25, 42, 46 (Hint: See Chpt. 3, Problem 6), 48, 56, 57. 

 

Week 9 (10/22 & 10/24): Chapter 7 and Chapter 8

Homework 6 due.

Homework 7:  Chapter 6, Problems: 5, 12, 15, 26. Chapter 7, Problems: 5, 9, 10, 18.

 

Week 10 (10/29 & 10/31): Chapter 8 and Chapter 9

Homework 7 due.

Homework 8: Chapter 7, Problems: 35, 40, 44, 47.  Chapter 8, Problems: 2, 8, 10, 37, 40.

 

Week 11 (11/5 & 11/7): Chapter 9 and Chapter 10

Homework 8 due.

Homework 9: Chapter 8, Problems: 48, 77, 78, 81, 80, 82, 84.

 

Week 12 (11/12 & 11/14): Chapter 10 and Chapter 11

Homework 9 due.

Homework 10: Chapter 9, Problems: 5, 9, 12, 14, 28, 30, 32, 47

Week 13 (11/19 & 11/21) Thanksgiving Vacation

Week 14 (11/26 & 11/28): Chapter 11 and Chapter 12

Homework 10 due.

Homework 11: Chapter 10, Problems: 7, 12, 18, 19, 21, 23, 29;
                         Chapter 11, Problems: 1, 7, 9, 12, 14.

 

Week 15 (12/3 & 12/5): Chapter 12

Homework 11 due.

 

Final Exam