Exam 1
Math 556, Fall 2013
Zhiyuan Wang
(1) (a) To check k k is really a norm, we have:
k
(i) If kAk = 0, then supf 6=0 kAf
kf k = 0. Now since f 6= 0, we
get supf 6=0 kAf k = 0, which implies kAf k =
MATH 556: PROBLEM SET 1
DUE SEPTEMBER 16, 2014
1. Consider the dierential equation
u + k 2 u = sin x ,
u(0) = u(1) = 0 ,
where k is constant. (i) Find the solution by elementary means. (ii) Find
the s
MATH 556: PROBLEM SET 2
DUE SEPTEMBER 30, 2014
1. Prove Proposition 3.3 in the notes.
2. Two norms on a linear space X are said to be equivalent if every sequence
that converges with respect to one of
MATH 556: PROBLEM SET 3
DUE OCTOBER 16, 2014
1. Show that p is a Banach space for 1 p . Note that this is a theorem
in every textbook on functional analysis. Try to resist the temptation to look
up th
MATH 556: PROBLEM SET 4
DUE NOVEMBER 6, 2014
1. Let H be a Hilbert space and xn a sequence in H. If xn
x lim inf n xn .
x show that
2. Consider the sequence fn (x) in L2 (R) dened by
fn (x) =
Show tha
MATH 556: PROBLEM SET 5
DUE NOVEMBER 25, 2014
1. Let A : H H be a compact operator on a Hilbert space H. Show that
A is also compact.
2. Let = [0, 1]. Consider the operator
x
A:f
f (y)dy .
0
(i) Show
MATH 556: MIDTERM EXAM
DUE OCTOBER 23, 2014
This is a take-home examination. It is due in class on the above date.
Please answer every question. You are not permitted to work with anyone
else on this
Lectures on Applied Functional Analysis
John C. Schotland
October 1, 2014
Preface
These notes were prepared for a one-semester graduate course on applied
functional analysis. I would greatly appreciat