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 HoĢlder’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. 1