MATHEMATICS 421/510, PROBLEM SET 4
Due on Wednesday, March 17
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. All page and section numbers below
refer to the textbook, “Functional Analysis” by Peter D. Lax.
1. (a) Let X be a reflexive Banach space, and let Y be a closed linear
subspace of X. Prove that Y is reflexive.
(b) Why is it necessary to assume that Y is closed?
2. 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
3. The functions {e2πinx }∞
n=−∞ form an orthonormal basis in L ([0, 1]),
hence in particular e2πinx converge weakly to 0 in L2 ([0, 1]) as n → ∞.
Prove that they also converge weakly, but not strongly, to 0 in L1 ([0, 1]).
What about other p in (1, ∞)?
4. (Exercise 2, page 101) 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 → ∞.)
1