Welcome to the class Introduction to Real Analysis. Real analysis concerns the study of real numbers and functions of a real variable. This includes the study of sequences and series, convergence, continuity of functions, differentiation of functions, integration of functions, and related-areas. Real analysis also serves as the foundation for understanding rigorously a variety of mathematical fields, including calculus, partial differential equations, probability theory, and optimization. This class will cover topics central to the study of functions of a real variable and central to the study of related fields of mathematical analysis.

Selection of Topics

• Sequences and Series of Functions.
• Different Types of Convergence.
• Continuity.
• Differentiation of Functions.
• Integration of Functions.
• Lebesgue Measure.
• Lebesgue Integral.
• L^p Spaces.
• Hilbert Spaces.
• Fourier Series.

For more information see the syllabus Δ. Please be sure to read the prerequisites and grading policies for the class.

Prerequisites:

Knowledge of how to write rigorous mathematical proofs and a familiarity with epsilon-delta-based proofs.

Grading:

The grade for the class will be based on the homework assignments (see policy below), midterm exam, and final project as follows:

Homework Assignments 30%
Midterm Exam 30%
Final Exam 40%

Homework Policy:

Assignments will be made weekly and posted on the class website. Prompt submission of the homework assignments is required. While no late homework submissions will be accepted, one missed assignment will be allowed without penalty. While it is permissible for you to discuss materials with classmates, the submitted homework must be your own work.

Exams:

A midterm exam will be given in the class on Tuesday, March 1st.

• Midterm Solutions [PDF] Δ

Final exam will be take-home due by 4:30pm, Monday, March 21st in my mailbox on the 6th floor of South Hall.

• Final Exam (Take-Home) [PDF] Δ

Supplemental Class Notes and Materials:

• Syllabus [PDF] Δ

Class Annoucements:

• Guest lecturer on Tuesday, February 15th.
• No class lecture on Thursday, February 17th. For the make-up lecture time, be sure to vote using your full name : [link to class poll].
• Make-up class lecture will be held on Tuesday, February 22nd from 4:30pm - 5:30pm. The location will be South Hall 4519.
• Midterm Solutions [PDF] Δ
• Take-home final exam is due by 4:30pm, Monday, March 21st in my mailbox on the 6th floor of South Hall.
• Final Exam (Take-Home) [PDF] Δ
• I will hold my normal office hours TR 9:30am - 11:00am to answer any questions concerning the take-home final exam.
• I've corrected a few minor typos in the exam. I also clarified some of the hints in Problem 3 and 4. Please see the new PDF file above for the take-home exam.

Midterm Outline

• Continuity
  • Definition in terms of epsilon-delta criteria.
  • Definition in terms of limit.
  • Compact sets mapped by continuous functions.
  • Maximum and minimum of continuous function on compact set.
• Differentiation of Functions
  • Definition in terms of epsilon-delta criteria.
  • Definition in terms of limit.
  • Hypotheses ensuring Sum, Product, Division, Chain Rules.
  • Mean Value Theorem
    • General form in terms of f(x), g(x).
    • Special form in terms of f(x), g(x) = x.
    • Consequences for monotone functions.
    • Local maxima and minima.
• Integration of Functions.
  • Riemann-Steijtles Integration.
  • Basic properties (Theorems 6.7, 6.12, 6.13).
  • Relationship between operations of integration and differentiation.
  • Integration by parts.
• Sequences and Series of Functions
  • Definition of pointwise convergence
    • in terms of epsilon-delta criteria
    • in terms of limit.
  • Definition of uniform convergence
    • in terms of epsilon-delta criteria
    • in terms of limit.
  • Consequences of uniform convergence
    • Sequences of continuous functions
    • Criteria so integration and limit are interchangeable.
    • Criteria so differentiation is consistent with limit of f_n -> f.
• Be prepared to apply above theorems and techniques discussed in lecture to justify results for specific problems.
• Be prepared to answer true/false questions by providing justification when true or provide counter-example when false.

TA and Office Hours:

The TA for the course is Jon Karl Sigurdsson. His office hour is on Fridays in SH 6431 P from 11:30 to 12:30. His Mathlab hours are on Thursdays in SH 1607 form 5:00 to 7:00.

Homework Assignments:

Turn all homework into Sigurdsson mailbox in South Hall on the 6th Floor by 4:30pm on the due date. Graded homeworks will be returned in class. Numbered exercises are from Principles of Mathematical Analysis, Walter Rudin, third edition.

HW1: (Due Tuesday, January, 11th) Chapter 4: 1, 2, 3, 4, 7, 10, 14, 18.
HW2: (Due Thursday, January, 20th) Chapter 5: 1, 2, 9, 11, 12, 13, 18, 22, 27.
HW3: (Due Tuesday, February 1st) Chapter 6: 1, 4, 7, 8, 10, 11, 16, 17.
HW4: (Due Thursday, February 10th) Chapter 6: 2, 3, 5, 12, 13, 18, 19.
HW5: (Due Tuesday, February 15th) Chapter 7: 1, 2, 4, 6, 8, 12.
HW6: (Due Thursday, February 24th) Chapter 7: 3, 5, 7, 9, 11, 14, 16.
HW7: (Due Friday, March 18th) Chapter 7: 20, 21, 22, 23.

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