MATHEMATICS 421/510, PROBLEM SET 6
Due on Thursday, April 5
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 X be an infinite-dimensional Banach space, and let
S = {x ∈ X : kxk = 1}. Prove that the weak closure S of S is equal
to {x ∈ X : kxk ≤ 1}. (Hint: Suppose that |x| < 1, x ∈
/ S. Then
0
there is a weakly open set {y : aj < `j (y) < bj , `j ∈ X , j = 1, . . . , N }
containing x and disjoint from S. Prove that this is impossible.)
2. (10 points) Let k(x, y) be a Lebesgue-measurable function from [0, 1]2
to R such that
Z 1
|k(x, y)|dy ≤ C1 for a.e. x ∈ [0, 1],
0
Z
1
|k(x, y)|dx ≤ C2 for a.e. y ∈ [0, 1].
0
Let 1 ≤ p ≤ ∞. For f ∈ Lp [0, 1], define
Z 1
(Kf )(x) =
k(x, y)f (y)dy.
0
Prove that K is a bounded operator on Lp [0, 1] with norm bounded by
R
1/q 1/p
where 1/p+1/q = 1. (Hint: estimate |Kf (x)| by |k(x, y)||f (y)|dy =
RC1 C2 , 1/p
|k(x, y)| |k(x, y)|1/q |f (y)|dy, and use Hölder’s inequality.)
3. (10 points) Let H be a Hilbert space, {xn }∞
n=1 ⊂ H, kxn k ≤ 1 for all
n. Prove that {xn } has a weakly convergent subsequence.
4. (10 points) Let A be a compact and symmetric operator on a Hilbert
space H. Let {xn }∞
n=1 ⊂ H be a sequence such that xn → x weakly.
Prove that kAxn − Axk → 0.
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