MATH 7280 Operator Theory-Spring 2006
Instructor: Marian Bocea
Homework Assignment # 3
(Due April 24, 2006)
1. Let p ∈ (1, +∞), Ω ⊂ RN be open and bounded, and let {un } ⊂ Lp (Ω). Prove
that the following statements are equivalent
(i) un * u weakly in Lp (Ω);
Z
p
(ii) {un } is bounded in L (Ω) and
Z
un (x)dx →
I
interval I contained in Ω.
u(x)dx, for any N -dimensional
I
Hint: Use the fact that the set of step functions
m
P
k=1
αk χIk (m ∈ N, Ik ⊆ Ω are
intervals in RN , and αk ∈ R for any k ∈ {1, 2, · · · , m}), is dense in Lp (Ω) when
1 ≤ p < +∞.
2. Let p ∈ (1, +∞), and define un : (0, 1) → R by
 ¡
¢1
¡ k
k

 21 n2 p
if x ∈ n+1
− n13 , n+1
+
un (x) =

 0
elsewhere
1
n3
¢
, k = 1, 2, · · · , n
Show that un → 0 in measure, un * 0 weakly in Lp ((0, 1)), and that un does not
converge to 0 strongly in Lp ((0, 1)). Is the sequence {|un |p } equiintegrable?
3. Let E, F be two Banach spaces, T ∈ L(X, Y ) be a compact operator, and
{un } ⊂ E. Show that
un * u weakly in E ⇒ T un → T u strongly in F.
4. Let (E, k · k) be a Banach space.
(i) Show that that if
un → u strongly in E
(1)
un * u weakly in E and kun k → kuk.
(2)
then
(ii) If E is a Hilbert space (or, more generally, a uniformly convex space) then
(1) ⇔ (2).
1