MATH 766 Spring 12
Additional practice problems for Exam 2
The exam will be based on the material in the book form Sections 10.1, 10.2, 10.3, 10.4, 10.5, and
10.6 that we covered in class. (Note that some of the results in 8.3, 8.4, 9.3, 9.4 are particul
Bonus work (worth 20 points)
1) For E1 and E2 , two sets in Rn , dene
E1 + E2 = cfw_x + y : x E1 , y E2 .
a) Prove that if E1 and E2 are compact then E1 + E2 is also compact.
b) Give an example of a closed set E in R such that E + N is not closed (here N
MATH 766 Spring 2012
Additional practice problems for Exam 1
The exam will be based on the material in the book form Sections 7.1, 7.2, 7.3, 8.1, 8.2, 8.3, and
10.1, that we covered in class. (Note that some of the results in 9.1 are particular case of t
Homework 6
Math 766
Spring 2012
10.4.2 Let A and B be compact subsets of X . Prove that A B and A B are compact.
Proof: Let U = cfw_u be an open cover of A B. Then U is also an open cover of A and B since
A AB
U
B AB
U .
Since A and B are compact, ther
Bonus Homework
Math 766
Spring 2012
1) For E1 , E2 Rn , dene
E1 + E2 = cfw_x + y : x E1 , y E2 .
(a) Prove that if E1 and E2 are compact, then E1 + E2 is compact.
Proof: Since E1 + E2 Rn , it is sufcient to prove that E1 + E2 is closed and bounded.
E1 + E
Homework 2
Math 766
Spring 2012
k
7.2.3 Let E (x) = =0 x !
k
k
a) Prove that the series dening E (x) converges uniformly on any closed interval [a, b].
Proof: Let [a, b] R be a closed interval and dene M = max(|a|, |b|). Then
|x|k
k!
M
k!
and
M
k!
k=0
co
Homework 7
Math 766
Spring 2012
10.6.6 Suppose that H is a nonempty compact subset of X and that Y is a Euclidean space.
a) If f : H Y is continuous, prove that
| f |H := sup | f (x)|Y
xH
is nite and there exists x0 H such that | f (x0 )|Y = | f |H .
Proo
Homework 4
Math 766
Spring 2012
8.2.11 Fix T L (Rn ; Rm ). Set
M1 := sup |T (x)|
|x|=1
M2 :=
cfw_C > 0 : |T (x)| C|x| for all x Rn .
a) Prove that M1 |T |.
Proof: Let x Rn and note that x/|x| = 1. Then
|T (x)|
|T (x)|
sup
= |T |.
|x|=0 |x|
|x|=1 |x|
M1 =
Homework 5
Math 766
Spring 2012
10.3.8 Let Y be a subspace of X .
a) Show that V is open in Y if and only if there is an open set U in X such that V = U Y .
Note that Y is given a topology in two different ways here: One topology is the metric space
topol
Homework 3
Math 766
Spring 2012
7.3.2 Find the interval of convergence of the following power series.
b)
(1)k + 3)k (x 1)k .
k=0
Solution: First compute the radius of convergence for this power series
R=
1
1
=.
k
lim sup(1) + 3 4
k
3
So the power series