MATHEMATICS 421/510, PROBLEM SET 5
Due on Thursday, March 22
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) Let 1 < p < ∞. Prove that a sequence {fn } in Lp (R)
converges weakly to 0 if and only if supn kfn kp < ∞ and
Z
lim
fn (x)dx = 0 for all E ⊂ R of finite measure.
n→∞
E
2. (10 points) Let
1
` = {x = (x1 , x2 , . . . ) : xi ∈ R, kxk`1 =
∞
X
|xj | < ∞}.
j=1
Recall that the dual space to `1 is
`∞ = {y = (y1 , y2 , . . . ) : yi ∈ R, kyk`∞ = sup |yj | < ∞}.
j
Prove that if a sequence {x(n) } in `1 converges weakly, it converges in
norm. (Hint: it suffices to consider the case when x(n) converge weakly
to
PNzero.(n)Prove first that this implies that for any finite N we have
j=1 |xj | → 0 as n → ∞.)
3. (10 points) Let H be a Hilbert space. Suppose that {xn } is a sequence
in H such that xn converge weakly to x and kxn k → kxk. Prove that
xn converge to x in norm, i.e. kxn − xk → 0 as n → ∞.
1