Nickname | Points |

Mikhail | 30* |

Sonia | 24* |

Alex | 20 |

Evan | 15 |

Anonymous | 14 |

Chris | 9 |

Mavricks | 5 |

Wayne | 4 |

Josh | 4 |

Greg | 2 |

Chris#2 | 2 |

Jared | 1 |

**Corrections to Homework**

- (Jun 6) Sonia noted that in Q1(b) of Assignment 8, "an decreasing" should read "a decreasing".
- (Jun 5) Josh noted that in Q2 of Assignment 8, the condition 0 <= j <= n should read 1 <= j <= n instead.
- (May 31) Evan noted that in Proposition 7(b) of page 13 of Week 10 notes, the right-hand side should be "integral_Omega f + integral_Omega g" and not just "integral_Omega g". (Thanks to Sonia for clarifying this correction).
- (May 29) Evan noted that in Corollary 9 of Week 8/9 notes (Q9(b) of assignment 8), m(A/B) = m(A) - m(B) should instead read m(B/A) = m(B) - m(A).
- (May 28) Alex noted that in Q8 of Assignment 9, "Corollary 4" should read "Corollary 5".
- (May 28) Evan noted that in Q7(b) of Assignment 8, A should be a box in R^n, not in R.
- (May 25) Evan noted that in Assignment 8, and in Weeks 8-9 notes, Lebesgue measure m is often written as mu instead (and outer measure m^* written as mu^*). One should replace all occurrences of mu with m in the notes.
- (May 24) Evan noted that in Q1 of Assignment 8, the limits should be indexed by j, not by n. In the definition of box in Q2, x_j should range between 0 and 1/q, not between 0 and 1/m.
- (May 21) Alex noted a spacing problem in Assignment 8 Q2 in the definition of an open and closed box.
- (May 21) Mikhail noted that in Assignment 6, Q8(a), the sine and cosine should be enclosed in parentheses to emphasize that both terms are being summed in n. In Q10(a), the sum inside the integral should range from -N to N, not from -infinity to +infinity.
- (May 20) Sonia noted that in Assignment 6, Q6, "to use" appears twice.
- (May 18) I observed that in Assignment 8, Q1(b), one needs the additional hypothesis that m(A_1) is finite (otherwise the statement is false, take for instance A_n = [n,infinity) in
**R**). - (May 15) An anonymous student noted that Lemma 7 of Week 6 notes (i.e. Assignment 6, Q7), e^{pi i N x} should instead read e^{pi i (N-1) x}.
- (May 14) An anonymous student noted that in Assignment 6, Q8, all occurrences of nx should instead read 2 pi nx.
- (May 13) An anonymous student noted that in Assignment 6, Q4, C(R;Z) should read C(R/Z;C).
- (May 11) Alex noted that in Assignment 6, "Weeks 6" should read "Week 6", and the parentheses do not match up in the hints for Q3 and Q6.
- (May 11) Mikhail noted that in the hint for Q8(b) of Assignment 4, [-1,delta] should be [-1,-delta].
- (May 8) Chris noted that the method sketched to prove Q7(b) of Assignment 5 will not work, and Q7(b) should thus be omitted. Since Q7(c) depends on Q7(b), this problem should also be omitted.
- (May 5) I noted a missing right parenthesis in Q7(a) of Assignment 5.
- (May 3) Evan noted that Q9 of Assigment 5 should refer to Theorem 19 instead of Theorem 18.
- (May 1) Greg noted that the inequality in Q3(a) of Assignment 4 does not appear to be provable by the means suggested. To fix this problem, one would have to replace all occurrences of "8" in this problem with a much larger number, such as "32".
- (Apr 23) Greg noted that the hint in Q6(b) of Assignment 3 is unhelpful and should in fact be ignored.
- (Apr 23) Evan noted that in Q1 of Assignment 4, in the sketch of proof given in page 2 of Week 4/5 notes, the integrals of f_n on [x,x_0] and [x_0,x] should instead read f'_n.
- (Apr 20) Chris and Evan both noted that in Q6(b) of Assignment 3, "Proposition 5" should read "Proposition 6".
- (Apr 18) Chris noted that in Q4 of Assignment 3 (i.e. in Proposition 4 of Week 3 notes), it must be assumed that Y is complete (otherwise the statement is not true). Also, the conclusion should state that the limit "exists" rather than the limit "converges".
- (Apr 11) Evan and Chris both noted that in the hint for Q6 of Assignment 2, the infinite union should instead read infinite intersection.
- (Apr 11) Mavricks noted that Q3(d) of Assignment 1 is false; this question should be ignored in the homework. (More information is available in the virtual office hours).
- (Apr 9) Mikhail noted that all occurrences of R^+ in the homework and notes should read [0, infinity) for clarity (in case R^+ is confused with (0, infinity)). (Thanks to Mikhail for spotting a typo in this correction, now fixed).
- (Apr 5) Josh noted that in Q2(b), the metric d should instead read "d_{l^2}".
- (Apr 1) Wayne pointed out that on pages 13-14 of Week 1 notes, all occurrences of "(integral sign)(E)" should read "int(E)" - the interior of E.
- (Mar 31) Alex noted that in proposition 3 on page 9 of Week 1 notes, "for all x >= N" should read "for all n >= N".

- (Jun 10) Mikhail noted that in Lemma 4 on page 9, the proof should refer to Lemma 24 from last week's notes rather than Lemma 23. In the proof of the monotone convergence theorem on page 7, the proof should refer to Proposition 3(c) rather than Proposition 3(d). (Thanks to Sonia for clarifying this correction).
- (Jun 6) I discovered that F_- should be F^- in several places in the proof of Lemma 8. Also, on page 1, in the definition of Lebesgue integral, the last appearance of the word "such" should be replaced by "thus" or "e.g.". Sonia also noticed that f_- and f_+ should similarly be f^- and f^+ on page 20.
- (Jun 5) Sonia noted that on page 20, "no-negative" should be "non-negative"; on page 21, "and obtain, we also have" should just read "and obtain".

- (Jun 10) Mikhail noted that the word "we" should be dropped from the definition of measurable function immediately following Lemma 20 on page 23. (Thanks to Sonia for clarifying this correction).
- (Jun 7) Mikhail noted that on the first line of page 6, "of a open box" should be "of an open box"; on page 12, "b_i = e_i" should be "b_i = a_i", while "From Theorem 2" should be "From Proposition 2", and in page 14, the word "and" should be deleted from the sentence which ends "x+q and." Thanks to Sonia to clarifying these corrections.
- (Jun 7) Sonia noted that on page 8, last sentence, "This to" should read "Thus to". In Lemma 10 on page 18, A_j should be E_j (this was also observed by Mikhail).
- (Jun 5) Sonia noted that on page 17, "imply that thee" should be "imply that there"; on page 20 inside the parenthetical remark in the proof of Lemma 12, "this is a rational box" should read "this is since a rational box"; in the first sentence of page 21, "which is contains" should be "which contains", while "E as the countable union" should be "E is the countable union". I also clarified that Corollary 18 follows immediately from Lemma 17 by using the functions g(x) = |x|, g(x) = max(x,0), and g(x) = min(x,0). In Lemma 11, the word "intersection" should appear before the intersection of the intersection of the Omega_j's (in both the second and third bullet), for consistency's sake.
- (May 28) Evan noted that Lemma 11 on page 20 does not have a proof. (The proof is assigned as Q10 of Assignment 8).
- (May 25) Evan noted that in the definition of measurability on page 16, E and A should be subsets of R^n, not of R.
- (May 21) Alex noted a right parenthesis was missing in the bottom paragraph of page 24.
- (May 19) I noted that the second occurence of Lemma 23 should instead be labeled Lemma 24. Also, in the first Lemma 23, I intended the sets E_j to be disjoint, but I will not require you to prove this in the homework.
- (May 16) Jared noted that in the description of the Boolean algebra property on page 4, the second union should instead be an intersection.

- (May 23) Mikhail noted that the row and column vectors in the example on page 19, in the equation computing D(k o h), should be interchanged.
- (May 22) Sonia noted that "Contraction mapping theorem" should be in boldface on page 23.
- (May 22) An anonymous student noted that in Lemma 4 on page 10, the hypothesis that E is a subset of R^n is missing. In the first bullet of page 11, "necessarily true of" should read "necessarily true if". In page 24, "x+g(x)=x+g(y)" should read "x+g(x)=y+g(y)".
- (May 22) Mikhail noted that in the example of page 9, the two occurrences of "x" in the numerator and denominator of the first displayed equation should instead read "(x,y)".
- (May 19) Evan noted that in the first bullet point of the proof of the inverse function theorem, f^{-1} should be (f^{-1})' at two places near the bottom (after the chain rule is applied).
- (May 12) Alex pointed out that a norm sign || was missing at the top of page 10 (in the numerator of the first expression).

- (May 18) Mikhail noted that in Lemma 1(iii) on page 4, "converge" should read "converges". Also the definition of C(R/Z;C) was clarified to indicate that the continuity is expected on all of R, and not just on [0,1). Also, on page 5 parentheses should be placed around the integrand (1+ix) for clarity.
- (May 5) Chris#2 noted that on page 8, sin(2 pi i n x) should read sin(2 pi n x).
- (May 5) I noted that a semicolon at the end of the section on page 7 should be a period.
- (May 4) Chris#2 noted that on page 2, in the definition of a periodic function, "for every real number L" should read "for every real number x".
- (May 2) Alex observed that in Lemma 1(i) on page 4, a closing parenthesis is missing. There is a superfluous closing parenthesis in page 15, in the expression || f - sum_n hat{F}(n) e_n ||_2.

- (May 24) Alex noted that a right parenthesis was missing on page 28, in the first item on trigonometric functions.
- (May 16) Mavricks noted that on page 30, in the proof of Lemma 21, "cos(0) = 0" should read "cos(0) = 1".
- (May 11) Mikhail noted that in Corollary 10 (and also immediately afterwards), f(0)=f(1)=1 should instead read f(0)=f(1)=0.
- (May 9) Mavricks noted that on page 3, in Corollary 2, the phrase "is the sup norm of f_n" should instead read "is the sup norm of f'_n".
- (Apr 29) An anonymous student noted that on page 3, in the proof of Theorem 1, "for all x in [0,1]" should read "for all x in [a,b]".
- (Apr 28) I observed that a period was missing at the end of the second definition on page 14.
- (Apr 28) Chris observed that on the last equation on page 8, x^n should read x^k.
- (Apr 21) I observed that it was not stated how Lemma 3 would be proven on page 5. Lemma 3 is assigned as homework.
- (Apr 19) An anonymous student noted that on page 1, "and does f'_n also converge to f?" should read "And does f'_n also converge to f'?". On page 2, in Theorem 1, the word "exists" is missing from the end of the third sentence, and "Suppose also there exists" should be "Suppose also that there exists" (these changes were also noted by Chris). On page 3, there is a space missing between "Corollary 3" and "from", and "the functions f'_n is not" should read "the functions f'_n are not".On page 6, the expression f*g(x) should read (f*g)(x) for clarity.

- (Apr 25) Alex noted that on page 10, in the definition of B(X;Y), the colon : is used both as a separator for the notion of a set, and a separator for the notion of a function. To reduce ambiguity I have replaced the first separator with a vertical bar |.
- (Apr 21) Josh noted that on page 11, it should be pointed out that d_{B(X;Y) is synonymous with d_infinity.
- (Apr 21) Mikhail noted that on page 2, in the definition of continuity, d(x,x_0) should read d_X(x,x_0).
- (Apr 16) An anonymous student noted that on page 14, the phrase "epsilon = 0" should read "epsilon > 0".
- (Apr 13) I noticed that on page 14, the italicized phrase "int" should instead represent an integral sign.

- (Apr 16) An anonymous student noted that on page 1, "so does all its subsequences" should read "so do all its subsequences".
- (Apr 13) Alex noted that the word "Then" was missing at the start of the second sentence of Lemma 2 on page 2.
- (Apr 11) Sonia noted that in the definition of connectedness on page 12, V and W need to be specified as open sets (as well as being disjoint and non-empty).
- (Apr 4) Wayne noted that on the parenthetical remark in the definition on page 12, "X contains a non-empty set" should read "X contains a non-empty proper subset".
- (Apr 4) Evan noted that in the definition of continuity on page 8, "x in E" should read "x in X".

- (Apr 10) Mikhail noted that in the first paragraph of the proof of Proposition 7 on page 16, the second occurence of B_{Y, d|_{YxY}}(x,r) should read B_{Y, d|_{YxY}}(x,r_x); also, on page 17, "B_{X,d}(x,r) intersect y" should read "B_{X,d}(x,r) intersect Y". Also I added a remark that the proof of this Proposition requires the axiom of choice.
- (Apr 9) Mikhail pointed out that in page 5 in the example of the taxicab metric, 3+4=7 should read 5+2=7, and pointed out a typo in the correction below (now fixed).
- (Apr 1) Mavricks pointed out that on page 5 in the example of the taxicab metric,
(
**R**^d, d_{l^1}) should instead read (**R**^n, d_{l^1}). On page 7, in the definition of convergence, in the parenthetical comment, x_n should read x^{(n)}. - (Apr 1) Wayne pointed out that the role of the boundary of [1,2) in determining that [1,2) is neither open nor closed was clarified (this set is neither open nor closed because it contains one boundary point 1, but does not contain the other boundary point 2).
- (Mar 31) Alex pointed out that on the first bullet in page 6, d_{l^1}(10010, 10101) should equal 3, not 2. A right parenthesis is missing at the bottom of page 8. A right parenthesis is missing after "(Y, d|_{Y x Y}" on the second-to-last bullet on page 16.

- (Jun 11) Chris noted that in the final, the definition of connectedness is incorrect; the sets V and W should be assumed to be open (in addition to the other properties listed).
- (May 29) Sonia noted that in the solutions to the second midterm, e^0 is mistakenly equated with 0 instead of 1; this is now fixed.
- (May 26) Josh pointed out an incorrect date for the revised due date of Assignment 8 (it should be Jun 6).
- (May 24) Alex pointed out that the title and dating of the announcement page had some errors (now fixed).
- (May 24) Alex pointed out that in Q3 of the second midterm, "let R -> R" should have read "let r: R -> R".
- (May 23) Sonia pointed out that the second midterm reference sheet referred to Weeks 1-4 when it should have referred to Weeks 4-7.
- (May 20) Sonia pointed out superfluous dollar signs in the Weeks 8/9 corrections.
- (May 18) Mikhail pointed out an incorrect header on this page (now fixed).
- (May 15) Sonia pointed out a typesetting problem with the bonus point page, which is now fixed.
- (Apr 23) Mikhail pointed out a repeated answer and a notational inconsistency in the metric spaces java quiz (now fixed).
- (Apr 21) Mikhail and an anonymous student both pointed out that the corrections to Week 4/5 notes were labeled incorrectly (this is now fixed).
- (Apr 16) Mikhail noted that the title to the syllabus page was incorrectly labeled as 131AH instead of 131BH; this is now fixed.
- (Apr 9) Wayne pointed out an erroneous link to the 131B textbook and schedule page.
- (Apr 7) Mikhail pointed out some bad links to the logic and set theory handouts, which are now fixed.
- (Apr 2) Evan pointed out that the due dates for the homework on the homework web page were incorrect; this is now fixed.
- (Mar 31) Alex pointed out some bad links in the main class web page and on the bonus point page which have now been fixed. (Update, Apr 2. No, really. They are now fixed.)

Rules and regulations

- It is possible for students to earn bonus points to improve their grade. You can earn a bonus point whenever you discover an error (or have a good suggestion to make) in any of my printed notes, homework, exams, solutions, class web page material, java applet questions, or anything I say in Virtual Office Hours. You can also earn bonus points by participating in Virtual Office Hours or suggesting a question for the Java quiz; see below.
- Each of the first 20 points earnt in this course adds 0.1% to the final grade, however at most 20 bonus points per student may be applied to the grade (i.e. it is possible to earn more than 20 bonus points, but after 20 there will be no further effect on your grade). Thus the theoretical maximum score for the class is 102%.
- Bonus points can still be awarded after Wednesday, June 11 (the day of the final), but they will no longer count toward the grade (i.e. you cannot, after receiving your grade, try to suddenly earn a large number of bonus points to try to alter it).
- The bonus point system is designed to encourage class
participation, but should not be viewed as a way to salvage a bad grade
or as a substitute for doing homework or reading the textbook - this is
why your bonus grade is limited to 2%!

- Errors can be typographical (i.e. spelling or grammar errors), factual (e.g. concerning the date of the final), mathematical, logical, or occasionally even aesthetic. Basically, the rule is that if it warrants a change to the printed notes (or other web page material), it deserves a bonus point.
- A group of related errors will be considered as a single error for the purposes of awarding bonus points. (This is to avoid a single, widely spread, error overwhelming the system, or to avoid a single error being "sliced" into many smaller errors in order to stretch the bonus point value).
- Detection of verbal mis-steps, for instance in lecture or office hours, will not earn bonus points, due to unavailability of a reliable written record.
- You can also earn a bonus point by correctly answering any question posed by another student in the Virtual Office Hours (as long as the question has some relationship to the course). Again, once an answer is used by one student to earn a bonus point, it cannot be re-used by another student (unless he or she comes up with a very different, but still correct, answer).
- You may also be able to earn a bonus point for asking a particularly good question in Virtual Office Hours. (Questions such as "How do you do Question 4 of Homework 2?" or "Will Section 4.3 be on the final?" are generally not considered good questions for this purpose). Last, but not least, you can earn bonus points by suggesting a new question for the Java quiz. Each question suggested is worth one bonus point. To qualify, each question must be original (i.e. not a minor modification of an existing question), relevant to the course, educational (i.e. the correct answer should teach you something about the concepts covered by the question) and should come with at least two incorrect but plausible answers. Contributors will be acknowledged in the data files of the java quiz as well as on the bonus point system.
- To qualify for a bonus point you must notify me of the error, either by e-mail or in Virtual Office Hours (you can use your exam nickname instead of your real name in Virtual Office Hours if you wish).
- Once an error is pointed out by one student (or by myself), it cannot be re-used by another student to earn bonus points (unless I judge the error notifications to be independent and nearly simultaneous).
- Any attempt to abuse the bonus point system may result in forfeiture of all bonus points by the abuser.
- These rules and regulations are subject to change without notice.