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 solution by the spectral method discussed in Chapter 1 o
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 the norms also converges with respect
to the other. Sh
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 the proof.
2. Dene a linear operator A : C([0, 1]) C([0,
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 that fn
1 for n x n + 1
0 otherwise .
0, but that fn
0 str
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 that A : L2 () L2 (). (ii) Compute A . Hint: Apply the
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 examination. If a result is used from the lectures or t
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 appreciate any comments or questions,
which can be addressed to