APPM5440 Applied Analysis: Section exam 2
17:15 18:30, Oct. 30, 2012. Closed books.
Please motivate all answers unless the problem explicitly states otherwise.
You may want to do Problem 5 last (it is only 20 points, and could be a lot of work).
Problem 1

Solutions to homework set 6 APPM5440 Fall 2012
2.10: Let A denote the set of functions in C(Rn ) that vanish at innity. That A = Cc (Rn ) follows
from the following two claims:
Claim 1: Cc (Rn ) is dense in A.
Claim 2: A is closed.
Proof of Claim 1: Fix

Homework 1 partial solutions APPM5440, Fall 2012
Problem 1.3: Note that the desired inequality is equivalent to the following pair of inequalities:
cfw_
d(x, z) d(y, z) d(x, y)
d(y, z) d(x, z) d(x, y)
Now prove each of the two inequalities in the pair abo

Solutions to homework set 7 APPM5440 Fall 2012
Problem 3.1: Suppose T (x) = x. Then /2 arctan(x) = 0 which clearly is impossible.
Set
d(T (x), T (y)
.
d(x, y)
x=y
The CMT holds only if (0, 1). In other words, there must be a single such that the relation

Homework set 4 APPM5440 Fall 2012 partial solutions
Solution for 2.4: Lets consider X = [1, 1] instead. Then set f (x) = |x|, and
1 + n x2
.
fn (x) =
n + n2 x2
Then fn f uniformly, fn C (X), and f is not dierentiable. (To justify the shift we made
initia

Homework set 2 APPM5440, Fall 2012
Problem 1.8: Let (xn ) be a sequence in R, let C denote the set of cluster points of (xn ). Set
M = sup C, yk = supcfw_xn : n k and recall that
lim sup xn lim yk .
k
n
Show that C is closed: We will prove that C is compl

Applied Analysis (APPM 5440): Section exam 3
8:30am 9:50am, Nov. 30, 2009. Closed books.
Problem 1: (24p) With X a Banach space, which statements are necessarily true (please motivate):
(a) If S, T B(X) and T is compact, then ST is compact.
(b) If S, T B(

Applied Analysis (APPM 5440): Section exam 2
8:30am 9:50am, Oct. 28, 2009. Closed books.
Problem 1: (24 points) For each of the statements below, state whether it is TRUE or FALSE.
(TRUE of course means necessarily true.) No motivation required.
2
(a) Den

Applied Analysis (APPM 5440): Section exam 1 8:30am 9:50am, Sep. 21, 2009. Closed books. Problem 1: In what follows, (X, dX ) and (Y, dY ) are metric spaces. (a) Define what it means for a function f : X Y to be continuous. (b) Define what it means for a

Applied Analysis (APPM 5440): Section exam 1
8:30am 9:50am, Sep. 21, 2009. Closed books.
The following problems are worth 20 points each.
Problem 1: In what follows, (X, dX ) and (Y, dY ) are metric spaces.
(a) Dene what it means for a function f : X Y to

Homework set 10 APPM5440
Problem 5.4:
You can compute the eigenvalues of A using standard techniques. This
directly leads to a formula for r(A).
Try to nd explicit formulas for A2n and A2n+1 . To do this, it might be
worth it to evaluate analytically A2 ,

Homework set 5 APPM5440 Spring 2009
2.7: Set I = [0, 1], and = cfw_f C(I) : Lip(f ) 1, f = 0.
We will use the Arzel`-Ascoli theorem, of course.
a
The Lipschitz condition implies that is equicontinuous. (To prove this,
x any > 0. Set = . Then for any f , a

Solutions to homework set 8 APPM5440 Fall 2012
3.6: Set X = C([a, a]) and dene on X the operator
1 a
1
[F u](x) =
u(y) dy + 1.
a 1 + (x y)2
Then the given equation can be formulated as a xed point problem u = F (u). We nd that
1
2 arctan(a)
1 a
(u(y) v(y

APPM 5440, Fall 2012.
Notes on Chapter 5 Banach Spaces
Most important topics: You are expected to know these denitions and results well.
k
Denition of a Banach space. The spaces p (N), Cb (I), Cb (I).
The space B(X, Y ). The operator norm (equations (5.

APPM 5440, Fall 2012.
Notes on Chapter 4 Point set topology
Core topics:
These topics may show up on an exam. You are expected to know these denitions and results well.
Denition of a topological space. Denition of open and closed sets.
Topologies generate

Applied Analysis (APPM 5440): Section exam 2 solutions
8:30am 9:50am, Oct. 28, 2009. Closed books.
Problem 1: (24 points) For each of the statements below, state whether it is TRUE or FALSE.
(TRUE of course means necessarily true.) No motivation required.

Applied Analysis (APPM 5440): Section exam 3
8:30am 9:50am, Nov. 30, 2009. Closed books.
Problem 1: (24p) With X a Banach space, which statements are necessarily true (please motivate):
(a) If S, T B(X) and T is compact, then ST is compact.
(b) If S, T B(

Homework set 8 APPM5440 Solutions
3.6: Set X = C([a, a]) and dene on X the operator
1
1 a
[F u](x) =
u(y) dy + 1.
a 1 + (x y)2
Then the given equation can be formulated as a xed point problem u = F (u).
We nd that
|F (u) F (v)| =
sup
x[a,a]
a
1
a
1
(u(y)

Solution for 2.4: Lets consider X = [1, 1] instead. Then set f (x) = |x|,
and
1 + n x2
fn (x) =
.
n + n2 x2
Then fn f uniformly, fn C (X), and f is not dierentiable. (To justify
the shift we made initially, simply note that if we dene gn C([0, 1]) by
gn

APPM5440 Applied Analysis: Section exam 2 Solutions
17:15 18:30, Oct. 30, 2012. Closed books.
Problem 3: (20p) Dene for n = 1, 2, 3, . . . the function fn : R R via
fn (x) = en(xn) .
2
Let N be a xed positive integer. In the table below, mark each box cor

Homework set 5 solutions APPM5440 Fall 2012
2.7: Set I = [0, 1], and = cfw_f C(I) : Lip(f ) 1,
f = 0.
We will use the Arzel`-Ascoli theorem, of course.
a
The Lipschitz condition implies that is equicontinuous. (To prove this, x any > 0. Set = .
Then for a

APPM5440 Applied Analysis: Section exam 3 Solutions
17:15 18:30, Dec. 4, 2012. Closed books.
Problem 1: (28p) No motivation requireds please just write the answers.
(a) Let X be a set, and let T1 and T2 be two topologies on X. Suppose that T1 is weaker th

APPM5440 Applied Analysis: Section exam 3
17:15 18:30, Dec. 4, 2012. Closed books.
WRITE YOUR NAME:
Please ll out your answers to problems 1, 2, 3 directly on the problem sheet, if possible.
Write your answer to problem 4 either on the exam, or on a separ

APPM5440 Applied Analysis: Section exam 1
17:15 18:30, Sep. 25, 2012. Closed books.
Please motivate all answers unless the problem explicitly states otherwise.
Problem 1: (24 points) The following questions are worth 8 points each.
(a) Specify which of th

APPM5440 Applied Analysis: Section exam 1
17:15 18:30, Sep. 25, 2012. Closed books.
Please motivate all answers unless the problem explicitly states otherwise.
Problem 1: (24 points) The following questions are worth 8 points each.
(a) Specify which of th

Homework set 3 APPM5440, Fall 2009
From the textbook: 1.17, 1.18, 1.20, 1.22, 1.27.
Solution for 1.27: Suppose xn does not converge to x. Then there exists an > 0
and a subsequence such that d(xnj , x) > . Since the space is compact, (xnj ) has a
converge

Applied Analysis (APPM 5440): Final exam
1:30pm 4:00pm, Dec. 14, 2009. Closed books.
Problem 1: (20p) Set I = [0, 1]. Prove that there is a continuous function u on I such that
)
1 x (
(1)
u(x)
sin u(t)2 dt = cos(x),
x I.
5 0
Solution: Dene the operator