Jacob Bernstein

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Math 405: Introduction to Real Analysis


Course Description

This is an introduction to real analysis. Topics covered in the course will include, The Logic of Mathematical Proofs, Construction and Topology of the Real Line, Continuous Functions, Differential Calculus, Integral Calculus, Sequences and Series of Functions. There will be 10 problem sets (20% of final grade), two in class midterm exams (20% each) and one final exam (40%).

Lectures are Monday and Wednesday 1:30-2:45 in Hodson 311. Section meets Friday 1:30-2:20 in Hodson 311.

Problem sets will be due in section (or in blackboard) on Fridays (see below for dates). The assignments will be posted to blackboard. No late homework will be accepted. The lowest grade will be dropped.

Lecturer: Jacob Bernstein. jberns15@jhu.edu
Lecturer Office hours: Thrusday 10-11am in Krieger 408 or by appointment.

TA: Lili He. lhe31@jhu.edu
TA Office hours: Wednesdays 3-4pm in Krieger 201.

References

The course text is
  • R. Strichartz, “The Way of Analysis," Rev. Ed. (Available on Amazon for ~$40)
We will primarily use the following supplementary text (dual licensed under Creative Commons Attribution-Noncommercial-Share Alike 4.0 License and Creative Commons Attribution-Share Alike 4.0 License) is

Exams

There will be three exams. Two in class midterms and a final.

The dates of the exames are
First Midterm: Wednesday, October 6th.
Second Midterm: Wednesday, November 17th.
Final Exam: Tuesday, December 14 (6pm-9pm)

(Tentative) Schedule

Week 1 (8/30 & 9/1): Logic of Mathematical Proofs

Read JL: 0.3; See also Sections 1-4 of M. Taylor's notes. RS: 1.1, 1.2, 1.3, 1.4.
If you need a more in depth introduction to basic logic and proofs see this book.
No homework due.

Week 2 (9/6): Construction of the Real Numbers (no class Monday)

Read JL: 1.1 and 1.2. See RS: 2.1, 2.2;
Problem Set 1 due.

Week 3 (9/13 & 9/15): Construction of the Real Numbers (cont.); Topology of the Real Number Line

Read JL: 1.3, 1.4 and 1.5. See RS: 2.3
Problem Set 2 due.

Week 4 (9/20 & 9/22): Topology of the Real Number Line (cont.)

Read JL: 2.1, 2.2. See RS: 3.1, 3.2, 3.3
Problem Set 3 due.

Week 5 (9/37 & 9/29): Topology of the Real Line (cont.); Series

Read JL: 2.3, 2.4 and 2.5
Problem Set 4 due.

Week 6 (10/4 & 10/6): First Midterm; Series

Read JL: 2.5, 2.6. See RS: 4.1 4.2
No homework due.
Practice Midterms:
The exam will not cover properties of continuous functions.

Week 7 (10/11 & 10/13: Continuous Functions

Read: JL: 3.1, 3.2, 3.3, 3.4. See RS: 5.1, 5.2
Problem Set 5 due.

Week 8 (10/18 & 10/20): Continuous Functions (cont.); Differential Calculus

Read: JL: 4.1. See RS: 5.3, 5.4, 6.1
Problem Set 6 due.

Week 9 (10/25 & 10/27): Differential Calculus (cont.)

Read JL: 4.2, 4.3. See RS: 6.1
Problem Set 7 due.

Week 10 (11/1 & 11/3): Integral Calculus

Read JL: 5.1, 5.2. See RS: 6.2
Problem Set 8 due.

Week 11 (11/8 & 11/10): Integral Calculus (cont.)

Read JL: 5.3, 5.4, 5.5.
Problem Set 9 due.

Week 12 (11/18 & 11/20): Second Midterm;

Read JL: 6.1 6.2. See RS: 7.2, 7.3
Practice Midterms:
No homework due.

Week 13: Thanksgiving Break

No homework due.

Week 14 (11/29 & 12/1): Sequences and Series of Functions

Read JL: 6.1 6.2. See RS: 7.3, 7.4
Problem Set 10 due.

Week 15 (12/6): Picard iteration and the existence theory for ODEs.

Read JL: 6.3.

Final: Tuesday, December 14

Practice Finals:

Students with disabilities

Students with documented disabilities or other special needs who require accommodation must register with Student Disability Services. After that, remind the instructor of the specific needs at least 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 students and/or the chairman of the Ethics Board beforehand. Read the "Statement on Ethics" at the Ethics Board website for more information.

If a student is found responsible through the Office of Student Conduct for academic dishonesty on a graded item in this course, the student will receive a score of zero for that assignment, and the final grade for the course will be further reduced by one letter grade.


Fall 2021 -- Department of Mathematics, Johns Hopkins University.