EXAM
Takehome Exam
Math 5319, Spring 2010
April 14, 2010
• This is a takehome exam.
• Work independtly. If you need help getting started
on a problem, see the Instructor.
• You can use the book and your class notes, and notes
posted on the Instructor’s website.
• This exam has 5 problems. There are 500 points
total.
Good luck!
100 pts.
Problem 1. Let X be a compact metric space. If p ∈ X define Jp ⊆ C(X; R)
by
Jp = {f ∈ C(X; R) | f (p) = 0}.
A. Show that Jp is a closed subalgebra of C(X; R) that separates points.
B. Suppose A ⊆ C(X; R) is a closed subalgebra that separates points, but the
condition “A does not vanish at any point” fails. Show that A = Jp for
some point p ∈ X. Hint: consider
A1 = {f + c | f ∈ A, c=const.}.
100 pts.
Problem 2. Two real numbers p, q > 1 are called conjugate exponents if
1 1
+ = 1,
p q
(of course, one determines the other).
A. Prove the inequality
up
vq
+ ,
p
q
uv ≤
u, v ≥ 0,
where p and q are conjugate exponents. Hint: one way to do it is to use the
fact that f (x) = − ln(x) is convex.
B. If f ∈ R on [a, b], the p-norm of f is defined to be
b
Z
1/p
|f (x)| dx
.
p
kf kp =
a
Let p and q be conjugate exponents. Show that if f and g are two integrable
functions such that
kf kp = 1,
kf kq = 1,
then
Z
a
b
f (x)g(x) dx ≤ 1.
C. If p and q are conjugate exponents and f and g are integrable functions,
show that
Z b
f (x)g(x) dx ≤ kf kp kgkq .
a
This is called Hölder’s Inequality.
1
D. Prove that if f and g are integrable functions on [a, b] and p > 1 then
kf + gkp ≤ kf kp + kgkp
This is called Minkowski’s Inequality. Hint: Let q be the conjugate exponent
to p. Write
|f + g|p = |f + g| |f + g|p−1 ≤ [|f | + |g|]|f + g|p−1 .
Integrate and use Hölder’s Inequality.
100 pts.
Problem 3. Fill in the following omitted details in Rudin’s discussion of the
Gamma function on Page 192ff.
A. Show that the integral
∞
Z
tx−1 e−t dt,
Γ(x) =
x>0
0
converges.
A possible hint for showing convergence at infinity is to show that
lim tp e−t/2 = 0
t→∞
for suitable (not necessarily integer!) powers p.
B. Show that Γ(x) is a continuous function.
C. Carry out the omitted details in the proof of Theorem 8.18.
100 pts.
100 pts.
Problem 4. Carry out the omitted details in the proof of Theorem 8.20 and
in the discussion following the theorem, down to equation (102)
on page 194.
You can use Theorem 8.19 without proof.
Problem 5. Do Problem 8.6 in the book, page 197.
2