Math 4317B: Analysis I Welcome to Math 4317B: Analysis I with Prof. Heitsch!
Times & Places Lectures are in Skiles 170 on Tuesdays and Thursdays from 12:05 -- 1:25.
Office hours are currently scheduled for Monday from 2 - 3pm and Tuesday/Thursday from 1:30 - 2pm in Skiles 211B. (Meeting at other times is possible by appointment.)
Office hours are now scheduled for Monday from 4:30 - 5:30pm, Wednesday from 2 - 3pm, and Tuesday/Thursday from 1:30 - 2pm in Skiles 211B.
Handout from Tuesday, August 23rd: Course Syllabus
Extra office hour on Wednesday, August 24 from 2 - 3pm.
Logic review on Tuesday, August 30th from 5:30 - 6:30pm.
Extra office hour on Wednesday, Sept. 7th from 2 - 3pm.
Assignment 1 is now due on Thursday, Sept. 8th.
Typos in #4(c) and (d) of Assignment 1. #4(c) should read "f f^{-1} is the identity on P(Y) if f is onto" and #4(d) should read "f^{-1} f is the identity on P(X) if f is one-to-one."
A clarification/hint for #4(c) and (d) of Assignment 1. For #4(c), consider the map h: P(Y) -> P(Y) defined by h (C) = f(f^{-1}(C)) for C a subset of Y. Show that h is the identity function. #4(d) is similar.
REMINDER: Bring a copy of Assignment 1 (as well as the original!) to class on Thursday, Sept. 8th.
Second round for Assignment 1, #4, 5, 6. Choose exactly one of the following: Option 1 -- Submit a sheet of paper (with your name) that says that you want the original solutions (from last week) graded. Option 2 -- Submit (possibly revised) solutions for all three problems (#4, 5, 6). Remember that solutions must be easily readable to be graded.
Correction to Assignment 2, #6. The last line should read, "Also, give examples to show that each inequality can be strict." That is, "is" -> "can be."
The first midterm exam will be on Thursday, October 6th.
Some problems from the textbook to consider when preparing for the exam: pages 61 - 62, #5, 8, 9, 10, 14, 15, 16.
Solutions for Exam 1 from Thursday, Oct. 6th.
Alternative grading scheme: 0% Midterm 1, 20% Midterm 2, 50% Final exam, and 30% Homework.
Correction to Assignment 4, #1. The first line should read "Let (E,d) be a metric space with subsets S and T."
Assignment 4 is due on *Tuesday, Oct. 25th.* (Right day, wrong date on assignment.)
Hint for #1(a) and others on Assignment 4: Let T be the union of all open sets contained in S. (Compare with the definition of closure of S.) Can you prove that T equals the interior of S? If so, then you can use this alternative definition of the interior of S when helpful.
Office hours on Monday, Oct. 24th (only) will be 2 - 3pm, not the usual 4:30 - 5:30pm.
First hint for Assignment 5, #2. Let E be the real numbers with the usual metric. S = (0,1] and A = (0,1/2]. You should see (if not, then prove it!) that A is both open and closed in S, but A is neither open nor closed in E.
Second hint for Assignment 5, #2. CORRECTION FROM LECTURE. Show that if A is open (respectively closed) in (E,d), then A \cap S (read "A intersect S") is open (resp. closed) in (S,d). The first hint shows that the converse is not true!
Hint for Assignment 5, #3. These problems should remind you of proofs from the lectures/textbook. Compare and contrast.
The second midterm exam will be on Tuesday, Nov. 15th. It will cover up to and including section 4.3.
Correction to Assignment 6, #1(c): the limit should be h -> 0, not infinity.
Clarification to Assignment 6, #2: use the Euclidean metric on r^2 and R.
A reminder about the practice problems below when preparing for the exam.
Solutions for Exam 2 from Tuesday, Nov. 15th.
Correction to Assignment 7, #3(b): "Prove that if h is another continuous extension of f...".
Email all questions, requests, suggestions, etc. for topics, problems, etc. for next week by 11pm on Sunday, Dec. 4th.
There will be no office hours on Monday, Dec. 12th or Tuesday, Dec. 13th.
Due to unexpected scheduling changes, office hours on Wednesday, Dec 7th have been moved from 2 - 3pm to 4:30 - 5:30pm. Please email if you were planning to come during the original time and cannot come during the new time instead.
Correction to Assignment 6, #4. Assume that the limit function f is also continuous.
Final exam is on Thursday, Dec. 15th from 11:30am - 2:20pm in Skiles 170.
You may bring one (1) sheet (8.5" x 11" paper) of notes to the final exam. Typed is OK, writing on front and back is OK. Assume nothing else is OK, unless you check with me first.
There will be office hours on Wednesday, Dec 14 from 3pm - 5pm.
Scores on the final exam are available in T-Square. The maximum score was 149/150, and the average was 97/150 = 65%.
Recall that the adjusted average on Midterm 1 was 71% and on Midterm 1 was 71%. The homework average was 80%.
Numerical grades were calculated as the maximum of three possible numbers: the original grading scheme given in the course syllabus, the alternative grading scheme given above, and a third scheme (M1 25%, M2 25%, Final 25%, Homework 25%).
Readings Week 1: Review propositional logic and basic set theory. See links below and/or library reserves.
Week 2: Chapt. 1 of the textbook (_Introduction to Analysis_ by M. Rosenlicht).
Week 3: Chapt. 2 of the textbook.
Week 4: Chapt. 2 of the textbook.
Week 5: Chapt. 3, Sections 1 & 2 of the textbook.
Week 6: Chapt. 3, Sections 2 & 3 of the textbook.
Week 7: Chapt. 3, Section 3 of the textbook.
Week 8: Chapt. 3, Section 4 of the textbook.
Week 9: Chapt. 3, Section 5 of the textbook.
Week 10: Chapt. 3, Sections 5, 6 of the textbook.
Week 11: Chapt. 4, Sections 1, 2 of the textbook.
Week 12: Chapt. 4, Sections 2, 3 of the textbook.
Week 13: Chapt. 4, Section 4 of the textbook.
Week 14: Chapt. 4, Sections 4, 5 of the textbook.
Week 15: Chapt. 4, Section 6 of the textbook.
Week 16: Review.
Homework Assignment 0, Part 1 due on Tuesday, August 30th.
Assignment 0, Part 2, will be completed in class on Thursday, August 25th.
Assignment 0, Part 3, will be completed in class on Tuesday, August 30th. (Actually, collected in class on Thursday, Sept. 1st.
Assignment 1 originally due on Tuesday, Sept. 6th, now due on Thursday, Sept. 8th.
Assignment 1, #4, 5, 6 are due on Thursday, Sept. 13th. See announcements above for more information.
Solutions for Assignment 1.
Assignment 2 due on Thursday, Sept. 22nd. (Revised version fixes a typo in #4.)
Solutions for Assignment 2.
Assignment 3 due on Thursday, Sept. 29th.
Solutions for Assignment 3.
Assignment 4 due on TUESDAY, Oct. 24th. CORRECTION: Oct. 25th.
Solutions for Assignment 4.
Assignment 5 due on Thursday, Nov. 3rd.
Solutions for Assignment 5.
Assignment 6 due on Thursday, Nov. 10th. CORRECTION for 1(c): the limit should be h -> 0, not infinity.
Solutions for Assignment 6.
Assignment 7 due on Thursday, Dec. 1st.
Solutions for Assignment 7.
Assignment 8 due on Thursday, Dec. 8th.
Solutions for Assignment 8.
Practice Problems From Discrete Mathematics and Its Applications by K. H. Rosen, Fourth Edition. See links below and/or library reserves:
Sec 1.1, pgs 11 - 14, #4, 6, 8, 16, 18, 20, 21, 23, 25.
Sec 1.2, pgs 19 - 20, #6, 8, 12, 14, 16, 18.
Sec 1.3, pgs 33 - 37, #6, 8, 10, 12, 14, 20, 22, 24, 33.
Sec 1.4, pg 45, #7, 11, 12, 16, 21, 24.
Sec 1.5, pg 54 - 56, #6, 7, 9, 10, 12, 13, 14, 15, 19, 20, 21, 25, 26, 27, 28.
Sec 1.6, pg 67 - 69, #3, 4, 5, 12, 13, 22, 23, 28, 29.
From the textbook:
Chapt. 1, pg 12 - 13, #2, 3, 4, 5, 6, 9, 10.
Chapt. 2, pg 29 - 31, #2, 3, 10, 13, 14.
Prove that there does not exist a rational number s such that s^2 = 6.
Trying proving some of the algebraic or order properties of the real numbers on your own.
Prove that a non-empty finite set of real numbers has a l.u.b. and a g.l.b.
Give an example of a set of rational numbers that is bounded but does not have a rational supremum.
Prove that the union of two bounded sets is bounded.
Chapt. 3, pg 61 - 65, #1, 2, 3, 4, 6.
Chapt. 3, pg 61 - 65, #9, 10 11, 12, 13, 25.
Chapt. 3, pg 61 - 65, #24, 26, 27, 30, 32, 33.
Chapt. 4, pg 90 - 95, #1, 3, 4, 6.
Chapt. 4, pg 90 - 95, #7, 8, 9, 10, 11.
Chapt. 4, pg 90 - 95, #14, 17, 18, 19, 20, 26, 27, 28. If feeling ambitious, try #13, 24, and 25 also.
Chapt. 4, pg 90 - 95, #33, 34, 35, 36, 39, 40. If feeling ambitious, try #44, 45, 46 also.
Challenge Problems From Discrete Mathematics and Its Applications by K. H. Rosen, Fourth Edition. See links below and/or library reserves:
Sec 1.2, pgs 19 - 20, #28, 29, 35.
Sec 1.3, pgs 33 - 37, #42, 43.
Sec 1.4, pg 45, #25, 26.
Sec 1.5, pg 54 - 56, #30, 31.
Sec 1.6, pg 67 - 69, #57.
Chapt. 3, pg 61 - 65, #6.
Student are expected to follow the Georgia Tech Academic Honor Code and Student Code of Conduct.
Additional Resources Writing Proofs by Prof. Chris Heil, Georgia Tech and How to write proofs: a quick guide by Dr. Eugenia Cheng, U Sheffield.
Library Reserves
Logic
Propositional Equivalences
Predicates and Quantifiers
Solutions to Odd-Numbered Exercises (1.1 - 1.3)
Sets
Set Operations
Functions
Solutions to Odd-Numbered Exercises (1.4 - 1.6)
Countable Sets from Set Theory and Metric Spaces by Irving Kaplansky
C. E. Heitsch
Fall 2011