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Course Information

Description: Real Variables 2

Instructor: Charis Tsikkou. Email

Class Schedule:Tuesdays, Thursdays 01:00 - 02:15 PM in Armstrong Hall 121

Office Hours: Tuesdays, Thursdays 12:00 - 01:00 PM and by appointment in Armstrong Hall 405

Info Sheet containing more or less the stuff on this webpage.



Halsey Royden, Patrick Fitzpatrick, Real Analysis (Classic Version), 4th edition, Updated Printing

Errata for “Real Analysis”, 4th edition by Halsey Royden and Patrick Fitzpatrick

Alternative Textbooks:

Walter Rudin, Principles of Mathematical Analysis (3rd edition)

Elias M. Stein & Rami Shakarchi, Real Analysis: Measure Theory, Integration, and Hilbert Spaces (1st edition)

Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications (2nd edition)


Grading Scheme

  • 35% Final Exam
  • 25% Exam I
  • 25% Exam II
  • 15% Homework Assignments
  • Letter grades will be assigned according to the scheme
A 90-100% | B 80-90% | C 70-80% | D 60-70% | F 0-60%


  • Homework will be assigned once every four weeks, and due four weeks later (please see the course schedule below for exact dates).
  • Homework will always be collected at the beginning of the class.
  • Late turn-ins will not be accepted.
  • The accuracy of your work will be checked on selected problems.

Exams I and II

  • There will be two 75-minutes in-class exams, on February 19 and April 9.
  • These exams will test material covered after the previous exam (they are not cumulative).
  • Calculators will not be allowed.
  • No make-up exam will be given.

Final Exam

  • Friday, May 3, 2019, 11:00 AM - 01:00 PM in Armstrong Hall 121.
  • Final is cumulative.

Course Schedule (subject to changes)

Date Topic Resources HW
Tuesday, Jan 8 Review
Thursday, Jan 10 Review
Tuesday, Jan 15 Review
Thursday, Jan 17 Countable Additivity and Continuity of Integration; Uniform Integrability: The Vitali Convergence Theorem 4.5, 4.6 HW 1 posted, solutions
Tuesday, Jan 22 Uniform Integrability: The Vitali Convergence Theorem 4.6
Thursday, Jan 24 Uniform Integrability and Tightness: A General Vitali Convergence Theorem 5.1
Tuesday, Jan 29 Convergence in Measure 5.2
Thursday, Jan 31 No Class (Adverse Weather)
Tuesday, Feb 5 Characterizations of Riemann and Lebesgue Integrability 5.3
Thursday, Feb 7 Continuity of Monotone Functions 6.1
Tuesday, Feb 12 Differentiability of Monotone Functions: Lebesgue's Theorem 6.2
Thursday, Feb 14 Differentiability of Monotone Functions: Lebesgue's Theorem 6.2 HW 2 posted, solutions
Tuesday, Feb 19 Exam I Exam I, solutions
Thursday, Feb 21 Differentiability of Monotone Functions: Lebesgue's Theorem 6.2
Tuesday, Feb 26 Differentiability of Monotone Functions: Lebesgue's Theorem 6.2
Thursday, Feb 28 Differentiability of Monotone Functions: Lebesgue's Theorem 6.2
Tuesday, Mar 5 Functions of Bounded Variation: Jordan's Theorem 6.3
Thursday, Mar 7 Absolutely Continuous Functions 6.4
Tuesday, Mar 12 No Class (Spring Recess)
Thursday, Mar 14 No Class (Spring Recess)
Tuesday, Mar 19 Absolutely Continuous Functions 6.4
Thursday, Mar 21 Absolutely Continuous Functions 6.4 HW 3 posted, solutions
Tuesday, Mar 26 Normed Linear Spaces 7.1
Thursday, Mar 28 The Inequalities of Young, Hölder, and Minkowski 7.2
Tuesday, Apr 2 Lp is Complete: The Riesz-Fischer Theorem 7.3
Thursday, Apr 4 Lp is Complete: The Riesz-Fischer Theorem 7.3
Tuesday, Apr 9 Exam II Exam II, solutions
Thursday, Apr 11 Approximation and Separability 7.4 HW 4 posted, solutions
Tuesday, Apr 16 Approximation and Separability; Convex Functions 7.4, 6.6
Thursday, Apr 18 Convex Functions 6.6
Tuesday, Apr 23 Product Measures: The Theorems of Fubini and Tonelli 20.1
Thursday, Apr 25 Product Measures: The Theorems of Fubini and Tonelli 20.1
Friday, May 3 Final Exam

Doing Well in This Class

Although the prerequisites for the class are minimal, the material is dense and not trivial, especially if you have not seen mathematical proofs before. As is often the case in math courses, we will constantly build upon previous stuff; therefore, not leaving gaps in your understanding of the material is crucial for succeeding. This will require a sustained effort on your part, and in addition to attending lectures, you are encouraged to take advantage of instructor's office hours. Of course, this is not a substitute for also working on your own; it is essential to think about the material, read the suggested texts, and solve homework problems by yourself. This last bit is a prerequisite to being able to solve problems under the pressure of an exam.

Academic Integrity Statement

The integrity of the classes offered by any academic institution solidifies the foundation of its mission and cannot be sacrificed to expediency, ignorance, or blatant fraud. Therefore, instructors will enforce rigorous standards of academic integrity in all aspects and assignments of their courses. For the detailed policy of West Virginia University regarding the definitions of acts considered to fall under academic dishonesty and possible ensuing sanctions, please see the West Virginia University Academic Standards Policy (http://catalog.wvu.edu/undergraduate/coursecreditstermsclassification). Should you have any questions about possibly improper research citations or references, or any other activity that may be interpreted as an attempt at academic dishonesty, please see your instructor before the assignment is due to discuss the matter.

Inclusivity Statement

The West Virginia University community is committed to creating and fostering a positive learning and working environment based on open communication, mutual respect, and inclusion.

If you are a person with a disability and anticipate needing any type of accommodation in order to participate in your classes, please advise your instructors and make appropriate arrangements with the Office of Accessibility Services. (https://accessibilityservices.wvu.edu/)

More information is available at the Division of Diversity, Equity, and Inclusion (https://diversity.wvu.edu/) as well. [adopted 2-11-2013]