MATHEMATICS 421/510, PROBLEM SET 5
Due on Wednesday, March 31
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. 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 → ∞.
2. 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 there is a weakly
0
open set {y : aj < `j (y) < bj , `j ∈ X , j = 1, . . . , N } containing x and
disjoint from S. Prove that this is impossible.)
3. 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 from Lp [0, 1] to Lp [0, 1] with norm
1/q 1/p
= 1. (Hint: estimate |Kf (x)| by
Rbounded by C1 C2 ,Rwhere 1/p+1/q
|k(x, y)||f (y)|dy = |k(x, y)|1/p |k(x, y)|1/q |f (y)|dy, and use Hölder’s
inequality.)
4. Easter break: 10 marks for free!
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