05/18/05: Here are some statistics for the results of the final. The
mean is 44.8 out of 80. The top score is 73, while the bottom one is
14. It was a pleasure to teach you all!
05/18/05: Here are some statistics for the results of the final. The
mean is 44.8 out of 80. The top score is 73, while the bottom one is
14. It was a pleasure to teach you all!
05/09/05: Office hours for the finals week: Monday 3-4 p.m., Tuesday
and Wednesday 2-3 p.m.
05/05/05: Hint to Problem 5 on p. 681. For each measurable set A,
approximate in L^2 the characteristic function chi_A by a continuous
function g, whose absolute value does not exceed 1, see the proof of
Theorem 14.4.5. Now, if we prove that for any continuous function g,
the integral from -pi to pi int fg = 0, it will imply that int_A f =
0, which will imply that f = 0 a.e. by 14.3.6.
Given a continuous g, approximate it with a sequence of trig. polynomials T_n uniformly. We can assume |T_n| <= |g|+1 uniformly on [-pi,pi]. Then |f T_n| <= |f| (|g|+1), and thereby, the sequence fT_n is dominated by an integrable function. The dominated convergence theorem implies lim_n int fT_n = int fg. But int fT_n =0 by the assumption. Thus int fg = 0.
05/05/05: Hint to Problem 5 on p. 559. Use the following lemma for a =
-pi, b = pi, which is the same as Problem 7 on p. 681, except that it
is done for L^1.
Lemma. If f is Riemann integrable on [a,b], then lim_{y --> 0} int_a^b |f(x-y) - f(x)| dx = 0, where we extend f to the real line from [a,b) by periodicity with period (b-a).
Sketches of proof.
1st way. f is Riemann integrable, therefore, integrable, i.e., in L^1. Approximate it by a continuous function g with error epsilon/3 for a given epsilon. g is uniformly continuous, therefore, for small enough y, the integral int_a^b |g(x-y) - g(x)| dx will be less than epsilon/3 for small enough y. Now, int_a^b |f(x-y) - f(x)| dx <= ||f-g||_1 + int_a^b |g(x-y) - g(x)| dx + ||f-g||_1 <= epsilon.
2nd way. If f is Riemann integrable, then the limit of Osc (f,P) is zero as the maximal interval length ||P|| of partition P tends to zero. Given epsilon > 0, let delta be such that if ||P|| < delta, then Osc(f,P) < epsilon/2. We claim that int_a^b |f(x-y) - f(x)| dx < epsilon whenever |y| < delta/3.
Take a partition P={x_0, x_1, ..., x_n} of [a,b] into intervals of equal length < delta/3. Then for each x in [x_{k-1}, x_k], |f(x-y) - f(x)| <= sup f - inf f, where the sup and inf are taken over [x_{k-2}, x_{k+1}].
Now, we always have int_a^b |f(x-y) - f(x)| dx <= S^+(|f(x-y)-f(x)|,P). Because of the above remark, the latter is <= Osc (f,P_1) + Osc(f,P_2) + Osc(f,P_3) < 3 epsilon/3 = epsilon, where P_1, P_2 and P_3 are the partitions going over every third point of the partition P, starting from x_0, x_1, and x_2, respectively.
05/04/05: Coverage for the final, May 12: Sections 7.5.1-3,
7.6, 8.1.1, 9.1, 9.2, 9.3.1-4, 10.1, 10.2, 11.1.1-3, 11.1.5, 12.2.1-3,
13.1.1, 13.3.1, 14.1.1-2, 14.1.4-5, 14.3, 14.4, class notes.
05/04/05: The final exam will be at 01:30pm - 03:30pm Thursday, May
12, in Vincent Hall 113 (our regular classroom).
05/04/05: A sample final is posted. Make sure you have the last
version, the one which has 9 problems.
05/04/05: In Problem 5 on p. 681, you may use the fact that continuous
periodic functions are dense among the L^1-periodic ones, see the
paragraph before Theorem 14.4.5 on p. 678. Then just repeat the
argument given at the end of the class for L^2-periodic functions.
05/04/05: In Problem 7 on p. 559, the Fourier cosine series is defined
on p. 523. You can guess what the Fourier sine series is from there.
04/28/05: Hint: Problem 2 on p. 559 is just Dirchlet's Theorem proven
in class on Wednesday. But there are some details to attend to!
04/27/05: The current homework due date is postponed till Monday, May
2.
04/27/05: Correction: At the end of the class today, I should
have used h(y) = 2g(2y) instead of g(y/2)/2.
04/21/05: Correction: As the homework indicates, L^1
intersected with bounded functions is a subspace of L^2. I forgot the
boundedness condition when I talked about that in class Wednesday.
04/20/05: Hints for HW 11. Problem 2: The integral of f:
{naturals} --> {reals} with respect to the counting measure is just
the sum of the series \sum_{n=1}^\infty f(n). Problem 10: Yes, you can
make the functions f_n continuous. So, just think about continuous
functions for that problem.
04/18/05: I have changed the homework due April 22 to avoid problems
which use Fourier theory.
04/18/05: Here are some statistics of the results of the second
midterm. The mean is 27.5 out of 40. The top score is 37, while the
bottom one is 18.
04/14/05: Hint for HW 10, the problem on the Riemann integral
(perhaps, #7): As hinted in the text, the idea is to construct a
sequence of measurable functions whose limit is a function equal to
the given f almost everywhere (a.e), i.e., everywhere but on a set of
measure zero. This would imply that f is measurable, because the
(pointwise) limit of measurable functions is measurable (proved in
class) and a function different from a measurable function on a set of
measure zero is measurable: it is obvious the sets {f>a} and {g>a}
will differ by a set of measure zero, and the measurability of one
will then imply the measurability of the other. Then the point is to
take a sequence of partitions, for example, dividing the intervals
into halves successively, and construct a sequence of simple functions
f_n^+ whose Lebesgue integrals are the same as the upper Riemann sums
for f. Then define also simple functions f_n^- related to the lower
Riemann sums. Show each of the two sequences have pointwise limits,
say, f^+ and f^-. Then these limits are measurable as limits of simple
functions and the Lebesgue integral of g : = f^+ - f^- is zero. Since
this function is nonegative, this implies that f^+ and f^- are the
same almost everywhere: indeed, {g>0} = \cup_n A_n, where A_n := {g>
1/n}, and 0 <= |A_n|/n <= \int_{A_n} f <= \int f = 0, which yields
|A_n| = 0 for all n and thereby |{g>0}| = 0. Note that f is between
f^+ and f_- ...
04/06/05: There is an error on page 474 (Section 11.1.3): right
above Corollary 11.1.1, it should be I included in the finite union of
compact intervals, rather than equal to it. Thanks to Meijuan.
04/04/05: I have reduced coverage for Midterm 2 even more than
announced in class today: the exam will not cover Section 14.3 at
all. Problem Set 10 now has to be done after the exam by April 15 and
handed in.
03/31/05: Coverage for the second midterm, April 8: Sections
10.2.1-4, 11.1.1-3, 11.1.5-6, 13.1.1, 13.3.1, 14.1.1-2, 14.1.4-5.
03/25/05: More hints for HW 8: To do Problem 6.b on p. 581 honestly,
you may use Problem 6.c. If you do 6.b just at the identity matrix I,
i.e., compute the derivative of the matrix square root function at I,
then it is fine, too.
03/23/05: More hints for HW 8: In Problem 4 on p. 580, you do not need
to solve the equations. Just set them up.
03/23/05: Hints for HW 8: To avoid a mess when working in the space of
matrices R^{n^2}, for Problem 6 on p. 580, guess what the derivative
of the map f(A) = A^2 is and prove from scratch that it is what you
think at the point A=I. When a max/min problem is given in terms of an
m-dimensional surface in R^n, think of it as a max/min problem with
constraints g_1 (x) = ... = g_{n-m} (x) = 0, with the condition that
the rank of the differential of the map G(x) = (g_1(x), ...,
g_{n-m}(x)) is equal to n-m at all points x; this condition means that
the surface defined by G(x)=0 is of dimension m. For Problem 6 on
p. 609, first estimate graphically whether the maximum or the minimum
values will occur in the interior of the disk or on the boundary and
for each of the two cases, figure out what the (equality-type)
constraints must be, if any.
03/07/05: Here are some statistics of the results of the first
midterm. The mean is 29.96 out of 50, the median is 32. The top score
is 49, while the bottom one is 5.
03/02/05: As Alex Miller pointed out to me, there is a misprint on
p. 444 of the text: In the last three lines of the proof that ends on
that page, it should be the function f used instead of g.
02/25/05: I have to change my office hours Thursday permanently to
1:25-2:15 pm, because of a seminar I have to attend.
02/22/05: I have to change my office hours this Thursday, the day
before the first midterm, to 12:15-1:15 pm, because of a seminar I
have to attend.
02/16/05: Coverage for the first midterm, February 25: Sections
7.5.1-3, 7.5.5, 7.6, 8.1.1, 8.1.4, 9.1, 9.2, 9.3.1-4, 9.3.7, 10.1.
2/11/05: For those of you who got hooked on math as a result of taking
this class (or earlier). The School of Mathematics has a Math Club. Club meetings
take place Mondays, from 12:20 to 1:10 pm, in Vincent Hall 502. The
focus of a meeting is usually the discussion of a mathematical
problem. Students and guest faculty take turns in leading the
meeting. Pizza and drinks are provided.
2/07/05: Our grader Na Yuan will be available for any kind of help,
including grading questions, weekly in 150 Lind Hall, according to the
following schedule (military time). Monday: 9:05-9:55; Tuesday:
12:20-14:15; Wednesday: 9:05-9:55; Thursday: 12:20-13:10; Friday:
9:05-9:55.
2/6/05: I have to be at a meeting Friday, February 11, at 1:25
p.m. and therefore am moving my office hours that day to
11-noon.
2/2/05: Correction to Problem 2 on p. 335. It must be x --> 0+
instead of x --> 0.
1/26/05: Hint to Problem 1 on p. 314. This problem is not as
simple as the other ones, but it has a few different solutions. Here
is a hint, leading to one of them. Take the partition a = x_0 < x_1 <
... < x_N = b of [a,b] into N equal intervals for N large enough. Then
use the triangle inequality to estimate |f_j(x) - f_k(x)|. i.e.,
notice that f_j(x) will be close enough to f_j(x_p) for some p, while
f_j(x_p) close enough to f_k(x_p) for j and k large enough, and
f_k(x_p) close enough to f_k(x). This verifies the conditions of the
Cauchy test for uniform convergence.
1/24/05: My being stuck at the end of the class in relating the
integral over |x| > 1/n to that over |x| > 1/m for n > m had a good
reason behind it. And actually, I appreciate it that Julia (if not
others) apparently noticed that problem in the textbook. The matter is
that Strichartz uses a wrong definition of an approximate
identity. Believe me, it does not make Strichartz a bad guy. Such
lapses are unavoidable in the first edition. The most reliable way to
avoid such mistakes is to engage good students to study the subject
using the new text and ask them to find errors. It is also believed to
be a very efficient way of learning. (Unfortunately, I overlooked that
error earlier and repeated it in class.) Under Strichartz's
definition, the sequence g_n(x) constructed in the proof of the
Weierstrass Approximation Theorem is not at an approximate identity,
as it does not satisfy Property 3.
Correction: Property 3 in the definition of an approximate identity should be replaced with lim_{n --> infty} int_{x > 1/m} g_n(x) dx = 0 for any fixed natural m > 1. The end of Section 7.5.2, which shows that f*g_n converges uniformly to f, must be redone to use this relaxed Property 3. The same argument used there does go through, it is just that you have to change the intervals of integration to [-1/m,1/m] and its complement. Then it is not surprising why we only care about n --> infty for a fixed m at the end of the proof of the Weierstrass Approximation Theorem.
1/21/05: In the homework Problem 10 from 7.5.5, assume f(x) to be
continuous. Otherwise, the problem would not be correct. Thanks to
Devin for bringing this to my attention.
1/19/05: The first homework is already posted on the class web
page. Note that it is posted in a different style as compared to that
of the last term.
1/18/05: Welcome to the new semester! I hope it will be full of math
discoveries and adventures for you. I have changed slightly the old
syllabus posted on the class web page. The changes concern the new
room, VinH 113, the exam schedule and other important dates, and
coverage for the second semester, see also below. In the first few
classes, we will study approximation and equicontinuity, which are
topics of the hardcore real analysis, i.e., do not belong to calculus,
unlike most of the things we studied in the first term.
I will also try to post the class outlines before classes. We will see if I will be able to adhere to those plans, because it might be confusing if I am not. I am looking forward to seeing you tomorrow in class!
12/24/04: In the second term, we will finish Chapter 7, cover Chapters
8-10, selections from Chapters 12 and 13, and then Chapters 14 and
15. Have a merry Christmas and a happy New Year! I will be looking
forward to seeing you after the Break.
12/24/04: I have decided to stay with Strichartz as the required
text. Overall, I like Rudin better, just like most of you. Here are
the reasons for my choice of continuing with Strichartz: (1) It may be
not as bad as you think it is: perhaps, I projected my personal
preference on you, Rudin's text had the psychological advantage of
being a supplementary text, and Strichartz is more modern (in terms of
subject choices and style); (2) Some of you did object switching to a
new textbook in the middle of the year; (3) I did not want to
interrupt the flow, i.e., the course was designed based on Strichartz,
and if we chose Rudin for the second term, we would have to go in a
quite a zig-zag way through Rudin, while some of the things we did in
the first term would have to be wasted and redone using a different
approach.
12/24/04: The mean on the final was 44.51 out of 70. I have just
submitted your grades to the registrar. Sorry about the delay. Grades
will be available to you online in 24 hours, i.e., 5 pm, December
25. Otherwise, you are welcome to ask me via e-mail.
12/17/04: Reminder: the final will take place Saturday,
December 18, 10:30 am - 12:30 pm in our regular classroom, VinH
1. Good luck!
12/17/04: Coverage for the final: Sections 1.2, 1.4, 2.1, 2.2.,
2.3, 2.4.1, 2.5, 3.1, 3.2, 3.3, 3.4, 4.1, 4.2, 4.3, 5.1, 5.2, 5.3, 5.4
(omitting the starred sections), 6.1.1, 6.1.2, 6.2.1, 6.2.2, 6.3 (even
though it is starred), 7.1, 7.2, 7.3, 7.4 (omitting the starred
sections).
12/17/04: Your last homeworks are graded and available. If you are an
Economics graduate student, you should find your homework in your
mailbox in your department. Otherwise, you may claim your homework
from me while I am still here, till 6:10 pm.
12/15/04: Solutions to the problems of the sample final final not
discussed in class are posted.
12/15/04: I am in the process of posting solutions to the problems of
the sample final final not discussed in class. Prof. Keel is also
posting solutions to all homeworks.
12/14/04: A final version of the sample final is posted on our class
web page.
12/13/04: I have just realized that I will have to attend a seminar
tomorrow (Tuesday, December 14) from 10 to 11 am, just when I
scheduled my extra office hours previously. I am moving my Tuesday
office hours then to 11:00 to noon. I am sorry.
12/13/04: A preliminary version of the sample final is posted on our
class web page.
12/13/04: I am posting that sample final on the web page, if you did
not get it in class today. We will continue discussing that sample
final on Wednesday.
12/11/04: Extended office hours for the finals week, December
13-17: Mon 11-noon, Tue 11-noon, Wed 11-noon, Fri 1:30-3:30 p.m.,
and by appointment. Note that I will be out of town Thursday and have
to cancel my regular office hours on Thursday.
11/22/04: As Adil Ali pointed out to me, there is a misprint in the
textbook in Problem 7 on p. 250. It should read: Show that z^2 = i has
solutions z = +-(1/sqrt{2} + i/sqrt{2}).
11/19/04: F_sigma in Problem 12 on p. 232 means a set which is a
countable union of closed sets, as defined on the last line of
p. 229. G_delta is defined in the very Problem 12.
11/17/04: The mean on the second midterm was 23.4 out of 40.
11/09/04: There was a note I posted here yesterday, which wrongly
corrected Strichartz's definition of an inflection point. His
definition is correct: an inflection point is a point where f''
changes sign.
11/09/04: The following correction was brought to my attention by
Meijuan Li and Alex Miller. If you ever look at Problem 14 on p. 219,
the estimate there should read as |f(x)| <= M_2 (x-a)(b-a).
11/09/04: I wanted to announce extra office hours for today before the
test, but got a pretty bad cold and am staying home. You are welcome
to contact me via e-mail or call me up at home today: (651) 224-5634.
11/03/04: The first midterm exam with solutions from Prof. Keel's
class is available from his class web
page. The best use of the exam will be as a practice exam, i.e.,
cover the solutions!
11/01/04: Coverage for the second midterm, November 10:
Sections 3.3, 3.4, 4.1, 4.2, 4.3, 5.1, 5.2, 5.3, 5.4 (omitting the
starred sections), 6.1.1, and 6.1.2.
11/01/04: The U of M's team needs more students to compete in the
North Central Team Competition Saturday, November 13. It is a contest
for undergraduates in colleges and universities in Minnesota, the
Dakotas, and Manitoba. The 10 problems in this 3-hour competition are
interesting but below the level of the Putnam Competition. If this
might be a challenging but fun three hours for you, please contact
Prof. Bert Fristedt: VinH 252, 612-625-5081, fristedt@math.umn.edu.
10/26/04: In Problem 6 on p. 153, Strichartz assumes that the
function's value at x=0 is defined as 0, that is, x sin (1/x) is
extended to x=0 by continuity.
10/19/04: An interval in Strichartz might be open, closed, half-open,
half-closed, or even unbounded. When solving problems from Problem Set
6, you may denote an interval by I and use the defining property of an
interval: if real numbers a < b are in I, then [a,b] is contained in
I.
10/14/04: The next, short problem set, due Friday, October 22, is
posted.
10/13/04: Problem Set 5's due date is put off till Monday, October 18.
10/12/04: The mean on the first midterm was 17.9 out of 30.
10/08/04: The next problem set, due Friday, October 15, is
posted. Refresh your browser, if you do not see the problem set on our
class page. The problem set has been there since Wednesday.
10/06/04: My office hours tomorrow, Thursday, are from 11 a.m. till
noon.
10/04/04: Thinking it will be more convenient for you, given the
change of the exam date to Friday, I would like to change my office
hours this week back to the regular schedule: Wed 11-noon, Thu
11-noon, Fri 1:30-2:30 p.m.
10/04/04: I found an error in the page number in Problem Set 4:
it must be page 98 rather than 84. I have corrected it in the current
version of Problem Set 4. For that reason, I decided to put off the
first midterm exam. It will now be on Friday, October 8, in class.
10/01/04: Our grader Na Yuan will be available for any kind of help,
including grading questions, weekly in 150 Lind Hall, according to the
following schedule. Tuesday: 13:25-14:15 pm, 14:30-15:20 pm,
15:35-16:25 pm; Thursday: 14:30-15:20 pm, 15:35-16:25 pm; Friday:
9:05-9:55 am.
09/29/04: Coverage for the first midterm, October 6: Sections
1.2, 1.4, 2.1, 2.2., 2.3, 2.4.1, 2.5, 3.1, 3.2.
09/27/04: I have posted the contact information about our
grader on the main class web page. You are welcome to get in touch
with her, when you have a question related to grading. You will also
be able to ask her questions related to the class material soon. She
will have 6 office hours per week at the Tutorial Center at Lind
Hall. When I will know her schedule there, I will let you know.
09/27/04: Changes in office hours for the two coming weeks, Sep 27
- Oct 8. This (Sep 27) week's office hours (to cover for the
canceled ones on Thursday): Tuesday 12:20 - 1:20, Wednesday 11 - noon,
Friday 1:30-2:30. Next (Oct 4) week's office hours (10/04/04: changed
back to regular schedule, because of the change of the exam date):
Wednesday 11 - noon, Thu 11-noon, Friday 1:30-2:30.
09/24/04: Next Thursday, September 30, I will be out of town and have
to cancel my new Thursday office hours (11 to noon).
09/24/04: If you like mathematics, you may like joining the Math Club. The first
meeting of this semester will take place next Monday, September 27 at
3:35 pm, in 570 Vincent. There will be a guest speaker to talk about
research and careers in math. The Math Club will also attend the public
lecture by Sir Roger Penrose on October 5.
09/23/04: I had to change my office hours for today to 2:00 to 3:00
pm. Sorry about the late notice.
09/22/04: I have moved my office hours from Tuesday to Thursday
11-noon. Unfortunately, tomorrow, September 23, I will not be in my
office at that time. However, I will be available from 2:00 to 3:00 pm
tomorrow.
09/22/04: A hint to Problem 2.3.3.5: write x_n as p_n/q_n,
where p_n is a nonnegative integer and q_n is a positive integer. Do
it in a specific way to have p_n and q_n defined uniquely, rather than
up to a common factor. You may do this inductively for each fraction
x_n: if you have already done that for k_1 + 1/(k_2 +...) = p'/q',
then x_{n} = k_0 + q'/p' = (k_0 p'+q')/p' and take p_n = k_0 p' + q',
q_n = p'. Then show by induction that p_n = k_n p_{n-1} + p_{n-2} and
q_n = k_n q_{n-1} + q_{n-2}. Call these the recurrence
formulas. Derive that x_{n-1} - x_n = (-1)^n/(q_n q_{n-1}) and
x_{n-2} - x_n = (-1)^{n-1} k_n /(q_n q_{n-2}). From these equations
show that {x_2n} is an increasing subsequence and {x_{2n+1}} is a
decreasing one and x_{2k} < x_{2n+1} for any k, n. Also, note that
from the second recurrence formula yields q_n > q_{n-2} > 0. Conclude
that the sequence q_n q_{n-1} tends to infinity, i.e., may be made
greater than any given natural number for all n from some point
on. Combined with the increasing/decreasing statements above, this
implies that {x_n} is Cauchy.
To show that any positive real number x may be represented as the limit of a sequence of continued fractions, construct one as follows. Take the greatest natural k_0 not exceeding x. If x is an integer, take k_1 = k_2 = ... = 0. If x is not an integer, find r_1 from the equation x = k_0 + 1/r_1. Note that r_1 > 1. In general, if r_n is not an integer, take k_n to be the greatest integer not exceeding r_n and define r_{n+1} from r_n = k_n + 1/r_n. This way you get a sequence of nonnegative integers k_n. Show that |x - p_n/q_n| < 1/(q_n)^2, using ideas similar to those used in the first part of the problem, such as the equations x = (p_{n-1}r_n + p_{n-2}) / (q_{n-1}r_n + q_{n-2}) and x_n = p_n/q_n = (p_{n-1}k_n + p_{n-2}) / (q_{n-1}k_n + q_{n-2}). This will imply that the sequence of continued fractions converges to x.
09/22/04: An improved hint to Problem 2.2.4.2 (this is not the
only way to solve this problem): show that every real number may be
represented by an infinite binary fraction, moreover, in a unique way,
if you exclude binary fractions ending with a string of 1's (see 2.4.1
as a guideline). Then show that such fractions are in one-to-one
correspondence with a subset of the set of subsets of the
integers. Show that this subset has the same cardinality as 2^Z.
09/21/04: The students noticed two misprints in the textbook. One is
on page 51 in the first formula: it must be a "+" between 1/n and
1/n. The other one is on page 55, Problem 5, which is a homework
problem. In the bottom fraction, it should be k_{n-1} instead of
k_{n+1}. I am grateful to those who pointed these misprints out to
me.
09/20/04: This is an announcement for those who are interested in
undergraduate math contests. You should have received e-mails
from Prof. Fristedt, a copy of which you may also see on his office
door, VinH 252. The next practice is on Tuesday, Sep 21, at 3:35 in
VinH 203B, with further practices scheduled to best fit schedules.
09/20/04: The due date for Problem Set 2 is put off till Friday,
September 24.
09/17/04: Solutions to selected homework problems are available from
Prof. Keel's
5615 web page.
09/17/04: The second homework was posted Thursday.
09/16/04: In addition to Strichartz's textbook, there is another text
on reserve in the library for our class: Principles of Mathematical
Analysis by W. Rudin.
09/15/04: I will have to drive to Chicago Friday, September 15, and
cancel my Friday office hours, which are normally from 1:30 through
2:25 pm. If you would like to talk to me, please, do that sometime
after 2 pm Thursday. Professor Ezra Miller will substitute for me in
the Friday class, which will be about the ordered field axioms for our
system of real numbers.