Math 677
Homework #1
Solutions
1. We wish to show that the Cauchy-Schwarz inequality and the triangle
inequality follow from the properties of the inner product. To prove the
Cauchy-Schwarz inequality, consider the nonnegative quadratic polynomial
P ( ) =
Math 677
Homework #5
Due Friday, October 5
P
1
1
1. Let fck gk=1 be a sequence of positive numbers with c2 < 1, let fek gk=1
k
be an orthonormal basis for a Hilbert space H, and let
nX
o
A=
ak ek : jak j ck :
1
We claim that A is a compact subset of H. Le
Math 677
Homework #6
Due Friday, October 12
1. (2 points; one point each)
(a) Compute
Z
fh (y )e
2iy
dy =
b
f(
Z
f (y )e
2 i(
h) y
dy
h)
(b) Compute (changing variables to y 0 = y h)
Z
Z
2iy
gh (y )e
dy = g (y h)e 2 i y dy
Z
0
= g (y 0 )e 2 i (y +h) dy 0
Math 677
Homework #7
Solutions
1. (4 points) Suppose that M is a collection of subsets of X closed under
nite intersections, complements, and countable unions of disjoint sets.
01
0
Let fEk gk=1 be a sequence of sets in M. We wish to show that [1 Ek
k=1
a
Math 677
Homework #8
Solutions
1. (Stein and Shakarchi, p. 313, problem 5)
R2 R1
R2 R1
2
2
e jxj dx = 0 0 e r r dr d = 21 0 0 e u du = 1 where
R1
2
we set u = r2 . From this, it follows that I =
e jxj dx =
1
R
2
1 since I 2 = R2 e jxj dx = 1. We can now o
Math 677
Homework #4
Solutions
1. (a) Weak convergence does not imply strong convergence. Consider the
shift operators
1
0
C
B
Rn (fan g) = @0; : : : ; 0; a1 ; a2 ; : : :A
| cfw_z
n places
We claim Rn ! 0 weakly but not strongly.
Strong convergence does
Math 677
Homework #3
Solutions
1. First, if a and b are nonnegative, since f (x) = xp is a convex function
p
a+b
2
1p 1p
a+ b
2
2
p
p
p
p
so (a + b)
2p 1 (ap + bp ). Thus jf + g j
2p 1 (jf j + jg j ) which
shows that f + g 2 Lp whenever f; g 2 Lp . Since
Math 677
Real Analysis II
Take-Home Final Exam
Due Monday, December 10, 5:00 PM
Instructions: This is an open-book, open-notes nal exam. You are to
work on this examination alone (i.e., you may not discuss your work with other
students), but you may conta
Math 677
Midterm Exam
Please work all of the following problems. You may ask me about any of the
problems and refer to other sources. However, you should not discuss the exam
with anyone but the instructor. These problems are due in-class on Monday,
Octob
Math 677
Midterm Exam Solutions
1. (30 points) If f is a measurable, extended real-valued function, we dene
the essential supremum by
ess sup f = sup f : m fx : f (x) > g > 0g :
(a) (5 points) If f (x) = 1 for x 2 Q and f (x) = 0 for x 2 Q, nd ess sup f .
Math 677
Homework #2
Due Wednesday, September 12
1. Let fxn g be a Cauchy sequence from X . For every positive integer n
we can nd yn 2 A so that d(xn ; yn ) < 1=n. We claim that fyn g is
Cauchy. Given " > 0, there is an N so that d(xn ; ym ) < "=3 for n;
Math 677
Homework #9
Due Monday, November 19
Solutions
1. Suppose that ` is a continuous linear functional on C ([a; b]). We assume
also that ` is positive in the sense that `(f ) 0 for f 0, which implies
that ` is order-preserving: Rif f
g on [a; b], the