**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 **

**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 ÒAcademic Ethics for UndergraduatesÓ 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.

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

Homework 1 due.

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

Homework 2 due.

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

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.

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.

Homework 6:

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

Homework 6 due.

Homework 7:

Homework 7 due.

Homework 8:

Homework 8 due.

Homework 9:

Homework 9 due.

Homework 10:

Homework 10 due.

Homework 11:

Homework 11 due.