MATHEMATICS 421/510, PROBLEM SET 4
Due on Thursday, March 8
Write clearly and legibly, in complete sentences. You may discuss the
homework with other students, but the final write-up must be your own. If
your solution uses any results not introduced in class, state the result clearly
and provide either a reference or a proof.
1. (10 points) Recall the definition of the Sobolev space W 1,p (0, 1): a
function f ∈ L1loc (0, 1) has a weak derivative f 0 ∈ L1loc if for every
φ ∈ Cc∞ (0, 1),
Z 1
Z 1
0
f (x)φ (x)dx = −
f 0 (x)φ(x)dx.
0
0
Then W 1,p (0, 1) is the space of all f ∈ L1loc (0, 1) such that f and its
weak derivative f 0 are in Lp (0, 1), with the norm
kf kW 1,p = kf kLp (0,1) + kf 0 kLp (0,1) .
Prove that if f ∈ W 1,p (0, 1) for some 1 ≤ p < ∞, then f is equal a.e. to
an absolutely continuous function, f 0 (in the conventional sense) exists
a.e. and f 0 ∈ Lp (0, 1).
2. (15 points) Let H be the collection of all absolutely continuous functions f : [0, 1] → C such that f (0) = f (1) = 0 and f 0 ∈R L2 (0, 1). Then
1
H is a Hilbert space with the inner product (f, g) = 0 f 0 (x)g 0 (x)dx.
(You do not have to prove this, since a very similar problem was included on HW 2.)
(a) Given a fixed a ∈ (0, 1), prove that the mapping `a : H → C given
by `a (f ) = f (a) is a bounded linear functional. What is the norm of
this functional? (Be careful.)
(b) By the Riesz-Frechet representation theorem (the characterization
of bounded linear functionals on Hilbert spaces), there is a unique ga ∈
H such that `a (f ) = (f, ga ). Find ga .
3. (15 points)R Let X = Cc∞ (0, 1), and let the linear functional ` be given
1
by `(f ) = 0 xf 0 (x)dx.
1
(a) Suppose that X is equipped with the L1 norm: kf k1 =
Is ` bounded with respect to this norm?
R1
0
|f (x)|dx.
(b) Is ` bounded if X is equipped instead with the norm k · kH defined
in Problem 2?
2