551TsikkouFall2018

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Course Information

Description: Real Variables 1

Instructor: Charis Tsikkou. Email

Class Schedule: Mondays, Wednesdays, Fridays 10:30 - 11:20 AM in Armstrong Hall 407

Office Hours: Mondays, Wednesdays 01:30 - 02:30 PM, Fridays 09:30 - 10:30 AM, and by appointment, in Armstrong Hall 405

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

Textbook

Textbook:

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)

Evaluation

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

  • Homework will be assigned once every two weeks, and due two 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 50-minutes in-class exams, on September 24 and November 5.
  • 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

  • Thursday, December 13, 2018, 02:00 - 04:00 PM in Armstrong Hall 407.
  • Final is cumulative.

Course Schedule (subject to changes)

Date Topic Resources HW
Wednesday, Aug 15 Preliminaries on Sets, Mappings, and Relations
Friday, Aug 17 Preliminaries on Sets, Mappings, and Relations
Monday, Aug 20 Preliminaries on Sets, Mappings, and Relations
Wednesday, Aug 22 The Field, Positivity, and Completeness Axioms; The Natural and Rational Numbers 1.1, 1.2 HW 1 posted, solutions
Friday, Aug 24 Countable and Uncountable Sets 1.3
Monday, Aug 27 Countable and Uncountable Sets 1.3
Wednesday, Aug 29 Open Sets, Closed Sets, and Borel Sets of Real Numbers 1.4
Friday, Aug 31 Open Sets, Closed Sets, and Borel Sets of Real Numbers 1.4
Monday, Sep 3 No Class (Labor Day Recess)
Wednesday, Sep 5 Open Sets, Closed Sets, and Borel Sets of Real Numbers 1.4
Friday, Sep 7 Open Sets, Closed Sets, and Borel Sets of Real Numbers 1.4
Monday, Sep 10 Open Sets, Closed Sets, and Borel Sets of Real Numbers 1.4
Wednesday, Sep 12 Sequences of Real Numbers 1.5
Friday, Sep 14 Sequences of Real Numbers 1.5
Monday, Sep 17 Sequences of Real Numbers 1.5
Wednesday, Sep 19 Continuous Real-Valued Functions of a Real Variable 1.6
Friday, Sep 21 Continuous Real-Valued Functions of a Real Variable 1.6
Monday, Sep 24 Exam I Exam I, solutions
Wednesday, Sep 26 Measures and Measurable Sets 17.1 HW 2 posted, solutions
Friday, Sep 28 Measures and Measurable Sets 17.1
Monday, Oct 1 Lebesgue Outer Measure 2.1, 2.2
Wednesday, Oct 3 The σ -Algebra of Lebesgue Measurable Sets 2.3
Friday, Oct 5 The σ -Algebra of Lebesgue Measurable Sets 2.3
Monday, Oct 8 The σ -Algebra of Lebesgue Measurable Sets 2.3
Wednesday, Oct 10 The σ -Algebra of Lebesgue Measurable Sets 2.3
Friday, Oct 12 No Class (Fall Break)
Monday, Oct 15 Outer and Inner Approximation of Lebesgue Measurable Sets 2.4
Wednesday, Oct 17 Nonmeasurable Sets; Linear Transformations 2.6
Friday, Oct 19 Linear Transformations
Monday, Oct 22 Countable Additivity, Continuity and the Borel-Cantelli Lemma 2.5
Wednesday, Oct 24 Sums, Products, and Compositions; Sequential Pointwise Limits and Simple Approximation 3.1, 3.2 HW 3 posted, solutions
Friday, Oct 26 Sums, Products, and Compositions; Sequential Pointwise Limits and Simple Approximation 3.1, 3.2
Monday, Oct 29 Littlewood's Three Principles, Egoroff's Theorem, and Lusin's Theorem 3.3
Wednesday, Oct 31 The Riemann Integral 4.1
Friday, Nov 2 The Lebesgue Integral of a Measurable Nonnegative Function 4.3
Monday, Nov 5 Exam II Exam IIa, solutions, Exam IIb, solutions
Wednesday, Nov 7 The Lebesgue Integral of a Measurable Nonnegative Function 4.3
Friday, Nov 9 The General Lebesgue Integral 4.4
Monday, Nov 12 The General Lebesgue Integral 4.4
Wednesday, Nov 14 The General Lebesgue Integral 4.4 HW 4 posted, solutions
Friday, Nov 16 The General Lebesgue Integral 4.4
Monday, Nov 19 No Class (Fall Recess)
Wednesday, Nov 21 No Class (Fall Recess)
Friday, Nov 23 No Class (Fall Recess)
Monday, Nov 26 The Lebesgue Integral of a Bounded Measurable Function over a Set of Finite Measure 4.2
Wednesday, Nov 28 The Lebesgue Integral of a Bounded Measurable Function over a Set of Finite Measure 4.2
Friday, Nov 30 The Lebesgue Integral of a Bounded Measurable Function over a Set of Finite Measure 4.2
Monday, Dec 3 The Lebesgue Integral of a Measurable Nonnegative Function 4.3
Wednesday, Dec 5 No Class (State Holiday)
Friday, Dec 7 Countable Additivity and Continuity of Integration 4.5
Thursday, Dec 13 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]