Math 5201- Introduction to Real Analysis- Autumn 2016
Instructor: Krystal Taylor; taylor.2952; MW 706; office hours: MW 11:20 am- 12:20-pm or by apt.
Recitation leader: Joseph Migler; migler.1; MW 700; office hours: TR 11:20 am- 12:20-pm or by apt.
Lecture: MWF 10:20 am – 11:15 am, Bolz Hall – Room: 432
Recitation: TR (9:10 am – 10:05 am, Derby Hall – Room: 060) & (10:20 am – 11:15 am, University Hall – Room: 024 )
Grading: quizzes (15%), home work (25 %), midterm (25%), final (35%)
Text: W. Rudin, Principles of Mathematical Analysis, 3rd Ed., McGraw-Hill
Class notes: We are going to follow the book, but expect some deviations.
Review sessions: I am going to run periodic review sessions to take extra questions and to help you prepare for both the midterm and final exams.
Expectations: This is a graduate course and I expect you to approach it like professionals. This includes the following:
i) I expect the attendance to be at or near %100.
ii) I expect you to know all the material from the previous lectures throughout the semester. This is necessary to follow the lectures and to do well on quizzes.
iii) I expect you to read the material we are going to cover ahead of time.
iv) I expect all the electronic devices to be off during class and attention focused squarely on the blackboard.
v) I expect you to ask questions in class and suggest alternative approaches to proofs.
Intermediate Value Theorem (see Theorem 3.15 on this sheet for a simple proof of the I.V.T.)
Set 1: 2, 4, 7, 8, 10, 13 (optional: 15) (due: at the beginning of lecture, Wednesday, Aug. 31) solutions
Set 7 (due October 26): (Rudin Ch. 4) #6, 10, 14, 18, 23; (solutions)
Set 9 (there are a total of 6 problems due Nov. 9): (Rudin Ch. 6) #1, 2, 3, 8, 16. (Bergman) 6.3:0 solutions
Set 10 (there are a total of 5 problems due Nov. 16): (Rudin Ch. 6) #10 abc, 12; (Bergman) 6.4:6, 6.3:1; (problem given in recitation Nov. 10) solutions
Set 11 (there are a total of 4 problems due Nov. 22): (Rudin Ch. 7) 9, 11; (Bergman) 7. 1: 1; (Riemann-integration) Exercise 11
Set 12 (there are a total of 6 problems due Dec. 7): (Rudin Ch. 7) 13, 18, 20;(Bergman) 7.1:2; (problems) X172; exercise: Let A and B be dense sets on the real line. Is the intersection of A and B dense?
quiz 1 key
quiz-6-(solutions discussed in class)
quiz-8 (solution in text, solution to bonus discussed in recitation Nov. 10)
Homework policy: Homework assignments will appear on each Wednesday of the semester and will due on the following Wednesday at the beginning of lecture (some adjustments around University holidays may announced in class). Late homework creates difficulty for the grader and puts the student behind. For the benefit of all involved, late homework will not be accepted unless extenuating circumstances are present.
Midterm date: Wednesday, October 12, ( in class) MIDTERM REVIEW
Final Exam Date: Tuesday, December 13, 10-11:45 am
Review for the final (follow W. Rudin, Principles of Mathematical Analysis, 3rd Ed., McGraw-Hill):
- know the definition of the supremum (infimum) of a set, know when this value exists, and be ready to apply
- know elementary properties of the derivative of a function.
- know the definition and properties of the closure of a set (see quiz 2 along with related proofs in Rudin)
- prove Theorem 2.37 and deduce that every bounded sequence has a convergent subsequence
- know the statement of the Heine-Borel Theorem (Theorem 2.41) and be ready to apply
- prove the Intermediate Value Theorem (see Theorem 3.15 on this sheet for a simple proof of the I.V.T.)
- know the statement of the Mean value theorem and be ready to apply
- know Minkowski’s inequality (See HW Set 10 solutions, first page)
- prove Theorem 4.8 (a useful characterization for continuity)
- know when the inverse image of an open set is open, when the image of a compact set is compact, when the extreme value theorem (Theorem 4.16) holds, and demonstrate related counter examples
- statement of Theorem 6.6 (criteria for Riemann integrability) and related definitions
- statement of Theorem 6.17 and how to use it to prove the F.T.C. (See HW Set 10 solutions, 6.5:1)
- statement of Theorem 7.12 (the uniform limit of continuous functions is continuous)
- examine the necessity of the hypotheses of Theorem 7.16
- know the statement of Stone-Weierstrass and how to apply it
- know related counter examples: a differentiable function whose derivative has a discontinuity, a function which is not continuous, but is the limit of continuous functions
- prove the Weirstrass Theorem (every bounded infinite subset of R^d has a limit point in R^d)
- be familiar with concepts presented in class on the last week (we will discuss equicontinous families of functions)
- here are some extra practice problems: practice final problems . You may also look at the problems worked in recitation for practice.
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