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 Krieger 300. Section meets Friday 1:30-2:20 in Krieger 300.

Problem sets will be due in class on Wednesdays (see below for dates). No late homework will be accepted. The lowest grade will be dropped.

Lecturer office hours: Tuesday, 3-5pm or by appointment in Krieger 408.

TA office hours: Thursday, 4-5pm in Krieger 211.

The syllabus is here.

References

The course text is
  • R. Strichartz, “The Way of Analysis," Rev. Ed. (Available on Amazon for ~$40)

Exams

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

The dates of the exames are
First Midterm: Monday, February 23.
Second Midterm: Monday, April 6.
Final Exam: Wednesday, May 13, 9am-12pm.

(Tentative) Schedule

Week 1 (1/26 & 1/28) : Logic of Mathematical Proofs

Read 1.1, 1.2, 1.3, 1.4, 1.5; See M. Taylor's notes.
No homework due.

Week 2 (2/2 & 2/4): Construction of the Real Numbers

Read 2.1, 2.2;
Problem Set 1 due. Solutions to selected problems.

Week 3 (2/9 & 2/11): Construction of the Real Numbers (cont.); Topology of the Real Number Line

Read 2.3
Problem Set 2 due. Solutions to selected problems.

Week 4 (2/16 & 2/18): Topology of the Real Number Line (cont.)

Read 3.1,3.2, 3.3
Problem Set 3 due. Solutions to selected problems.

Week 5 (2/23 & 2/25): First Midterm; Continuous Functions

Read 4.1
No homework due.
Practice Midterm (Solutions). More Practice. (Solutions). (Note the exam will not cover properties of continuous functions).
First Midterm Solutions

Week 6 (3/2 & 3/4): Continuous Functions (cont.)

Read 4.2
Problem Set 4 due.

Week 7 (3/9 & 3/11): Differential Calculus

Read 5.1, 5.2
Problem Set 5 due. Solutions to selected problems.

Week 8: Spring Break

No homework due.

Week 9 (3/23 & 3/25): Differential Calculus (cont.); Integral Calculus

Read 5.3, 5.4, 6.1
Problem Set 6 due.
Solutions to selected problems.

Week 10 (3/30 & 4/1): Integral Calculus (cont.);

Read 6.1
Problem Set 7 due.
Solutions to selected problems.

Week 11 (4/6 & 4/8): Second Midterm; Integral Calculus (cont.);

Read 6.2
No homework due.
Practice Midterm (Solutions). More Practice (Solutions). Even more Practice (Solutions).
Second Midterm Solutions

Week 12 (4/13 & 4/15): Sequences and Series of Functions

Read 7.2, 7.3
Problem Set 8 due.
Solutions to selected problems.

Week 13 (4/20 & 4/22): Sequences and Series of Functions (cont.)

Read 7.3, 7.4
Problem Set 9 due.

Week 14 (4/27 & 4/29): Picard iteration and the existence theory for ODEs.

Read: Handout
Problem Set 10 due.

Final (5/13)

I will have office hours by appointment.
Practice Final (with solutions). More practice. Solutions.
Solutions to Final.

Spring 2015 -- Department of Mathematics, Johns Hopkins University.