Math 201b: Analysis
Homework 4, Solutions
HN7.2. Suppose that f : T C is a continuous functions, and
1
SN =
2
N
fn einx
n=N
is the Nth partial sum of its Fourier series.
(a) Show that SN = DN f , where DN is the Dirichlet kernel
1 sin[(N + 1/2)x]
.
2
sin
UNIVERSITY OF TORONTO
Faculty of Applied Science and Engineering
Term Test II
First Year
MAT185H1 S
Program 5
Linear Algebra
Examiners: G S Scott & G M T D'Eleuterio
1 March 2016
Student Name:
Fair Copy
Last Name
Student Number:
First Names
Tutorial Secti
Math 201b: Some notes on measures and transportation
Greg Kuperberg
1
Borel measures on compact spaces
If K is a compact Hausdorff space, then the Riesz representation theorem (or a theorem by that
name) says that C(K) is the vector space of signed, nite
Math 201b: Analysis
Homework 3 (Solutions)
GK3.1. Decomposing a closed interval into a meager set and a null set.
(a) Recall that a Cantor set C in R is made in the following way: Start with an interval I and remove a
centered open interval to obtain two
Math 201b: Analysis
Homework 6
This problem set is due Friday, February 28. Do problem *7.6 in the book, in addition to the following
problems. (I added a star for this book problem.)
GK6.1. In class I said that if K is a compact metric space, then the we
Math 201b: Analysis
Homework 5
This problem set is due Friday, February 21. Do problem 7.5 in the book, in addition to the following:
GK5.1. Prove that C([0, 1]) with its Banach norm contains an isometric copy of 1 ([0, 1]). (This is not a
typo; I mean 1
Math 201b: Analysis
Homework 3
This problem set is due Friday, January 31. Do the following problems.
GK3.1. Decomposing a closed interval into a meager set and a null set.
(a) Recall that a Cantor set C in R is made in the following way: Start with an in
Math 201b: Analysis
Homework 7
This problem set is due Friday, March 7. Do problems 9.6, 9.7, and 9.8 in the book, in addition to the
following.
Note: I give a star to problem 9.7(b).
HN9.6. Let G be a multiplication operator on L2 (R) dened by
G f (x) =
Math 201b: Analysis
Homework 7
This problem set is due Friday, March 7. Do problems 9.6, 9.7, and 9.8 in the book, in addition to the
following.
Note: I give a star to problem 9.7(b).
GK7.1. As mentioned in class, we can dene the fractional derivative of
Problem 1. Suppose that X is a vector space with two dierent Banach norms 1 and 2 . Suppose
that 2 is ner than 1 . Show (using the open mapping theorem) that the two norms are equivalent.
Solution. The fact that 2 generates a ner topology than 1 implies t
Problem 1. Let X and Y be two Banach spaces, and let X Y denote their algebraic direct sum. We can
combine the norms on X and Y with a p-norm-type formula
xy
p
=
p
x
p
+ y
p
to obtain a normed linear space that I will call X p Y .
(a) Show that the norms
Math 201b: Analysis
Homework 8
This problem set is due Monday, March 17. Do problems 5.9, 9.11, and 9.18, in addition to the following
problems:
HN5.9. Let X,Y, Z be Banach spaces. Prove that the following statements are true:
(a) If S, T B(X,Y ) are comp