#### MAT 357H: Real Analysis I

Professor: Mary Pugh
Contact information: mpugh@math. utoronto.ca
Class times: MWF 12:10-1:00
Office hours: Tu 5:00-6:00
Office location: room 3141, Earth Sciences Centre, 22 Russell Street (To find my office, enter the building on the Northwest corner of Huron and Russell. Get to the third floor. Walk as far south as you can. Now walk as far west as you can.)

Wanna send me email? If you're going to send me email, please include "mat357" in the subject line. Otherwise, my spam filter will catch your email. (Note: do not include a space between "mat" and "357", the subject line should contain a six-symbol string.)

Homework Policy
• In working on your homework, you can invoke anything from earlier in the book, both theorems and exercises. But do make sure to include an opening clause on the lines of, "By exercise xx on page nn," You don't need to reprove/prove the thing you're invoking.
• Starting on Wednesday March 2, homework is due in class by 12:10. If you can't come to class, give it to a classmate to hand in. If it arrives after 12:10 it will be counted as late and your mark will be docked by five points. (To date, each problem has been worth 5 points. By handing something in late you'll be losing a full problem's worth of credit.) You are actively encouraged to work in groups on the homework.

Is there something in the notes that confuses you? It might be a mistake on my part. Check the current version of the notes on the web --- sometimes I find mistakes and correct them. If the current version has the same point of confusion, please email me and we can figure out what's going on.

• Lecture Notes: Feb 21, 2005.
HW 6: Due in class before 12:10 on Weds March 2.
page 157: 4, 6, 9
section 12A: "Prove or disprove: if A is measurable and B agrees with A up to a set of measure zero, then B is measurable." Agreeing up to a set of measure zero means that the measure of A Delta B is zero where A Delta B is defined in problem 1, page 165.
page 165: 3, 4, 5
• Lecture Notes: Feb 23, 2005.
• Lecture Notes: Feb 25, 2005.

• Lecture Notes: Feb 28, 2005.
HW 7: Due in class before 12:10 on Weds March 9.
• Let X be a finite dimensional real vector space. Let ||.|| and |||.||| be two norms on X. Prove that these norms are equivalent, using the definition on page 6 of the Feb 28 lecture notes.
• Consider the space C^1(R), defined on page 2 of the Feb 28 lecture notes. Consider the norms ||.||_{inf,0} and ||.||_{inf,1} defined as: ||f||_{inf,0} := ||f||_{inf} and ||f||_{inf,1} := ||f||_{inf}+||f_x||_{inf}. (Note that "inf" means "infinity". And that if f(x) = 2 + 4 cos(9 x) then ||f||_inf = 6, ||f_x||_inf = 36, hence ||f||_{inf,0} = 6 and ||f||_{inf,1} = 42.) First prove that these norms are not equivalent by showing directly that one can never find a constant C so that the definition of norm equivalence on page 6 holds. Now prove that these norms are not equivalent by proving that (C^1(R),||.||_{inf,0}) is not complete while (C^1(R),||.||_{inf,1}) is complete. (Make sure to explain why this disparity then implies that no C can exist.)
• The proof of Lemma 12.6 in the book is very sketchy. Present a full proof. For example, make it clear how you used that B is bounded, why you had to introduce a new open set A, and prove that g_0 really is continuous.
• page 168: 2, 9, 12. Note that in problems 2 and 9 you're expected to assume that f_n, f, g_n, and g are all in L^2 and the convergence is in L^2 as opposed to pointwise or some other type of convergence. (I.e. ||f_n - f||_2 goes to zero.)
• page 172: 2-6.
• Lecture Notes: Mar 2, 2005.
• Lecture Notes: Mar 4, 2005.

• Lecture Notes: Mar 7, 2005.
HW 8: Due in class before 12:10 on Weds March 16.
• Let X be a real inner product space. A subspace of X is a subset of X that is closed under vector addition and scalar multiplication. An orthonormal collection of vectors is an orthonormal basis for X if X equals the closure of the subspace spanned by the orthonormal collection. I.e. given x in X you can find a sequence x_n such that each x_n is a (finite) linear combination of vectors in the orthonormal collection and ||x-x_n|| -> 0 as n -> infinity. Prove that if X is separable (has a countable dense subset) then X has an orthonormal basis.
• Consider the Hardy-Littlewood maximal function defined on page 169. Prove or disprove: "If h is in L^1 then h* is a measurable function."
• In the proof of the covering lemma on pp 170-171, the author uses a factor of 1/2, ultimately leading to the constant 5 in the statement of the theorem. If one used a factor of lambda, where 0 < lambda < 1, what would be the 5 be replaced by? (Present your reasoning; don't just give an answer w/o explaining your logic.) This suggests that this method of proof can never result in a constant less than or equal to what number?
• Given an integrable function h, we define the function A_delta h by |Ix|*(A_delta h)(x) = int_{Ix} h where Ix is an interval of length 2*delta, centered at x. I.e., (A_delta h)(x) is the average value of h, as averaged over the interval Ix. Prove that (A_delta h)(x) is a continuous function of x.
• page 171: 1-3.
• Lecture Notes: Mar 9, 2005.
• Lecture Notes: Mar 11, 2005.

• Lecture Notes: Mar 14, 2005.
HW 9: Due in class before 12:10 on Weds March 23.
• page 179: Problem 1: find the Fourier coefficients for the functions given in a, b, d, and for f(x) = |x|x. Discuss how the smoothness of f appears to be reflected in the large-k behaviour of the Fourier coefficients.
• page 179: Problems 5, 6, 8, 9
• page 184: 1, 2
• The following theorem about convolutions is true: "If f is in L^1(R) and g is in L^p(R) then f*g is in L^p(R) and ||f*g||_p <= ||f||_1 ||g||_p." (We can't discuss the proof because it depends on Fubini's theorem, which is slightly beyond the reach of the course.) Prove the following: "Assume g is in L^1(R). If f and f_x are in C_o(R) then both f*g and (f*g)_x are continuous." Now prove: "If f and its first k derivatives are in C_o(R) then f*g is continuous and has k continuous derivatives."
• If f and f_x are continuous then what additional assumption would imply that if g is in L^1 then f*g and (f*g)_x are continuous? (Please find an assumption that's weaker than having compact support.)
• Prove or disprove: the above statements are true for g in L^p(R) where 1 < p < infinity.
Here's an interesting theorem that you can't use in your HW: "If 1/p + 1/q = 1 and 1 <= p <= infinity and f is in L^p and g is in L^q then f*g is bounded and uniformly continuous."
Sad news: my note-taking computer has died. There will be no more online notes this semester.
Here is a *.ps file of the graph that I'm handing out in class on Friday. And here is the matlab *.m file that I used to make the graph. In case you have access to matlab.

HW 10: Due in class before 12:10 on Weds March 30.
• page 189: Problem 2
• page 191: Problem 2
• page 193: Problems 1, 3, 5, 8, 9, 10