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MATHEMATICS 421/510, PROBLEM SET 4 Due on Thursday, March 8 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) Recall the definition of the Sobolev space W 1,p (0, 1): a function f ∈ L1loc (0, 1) has a weak derivative f 0 ∈ L1loc if for every φ ∈ Cc∞ (0, 1), Z 1 Z 1 0 f (x)φ (x)dx = − f 0 (x)φ(x)dx. 0 0 Then W 1,p (0, 1) is the space of all f ∈ L1loc (0, 1) such that f and its weak derivative f 0 are in Lp (0, 1), with the norm kf kW 1,p = kf kLp (0,1) + kf 0 kLp (0,1) . Prove that if f ∈ W 1,p (0, 1) for some 1 ≤ p < ∞, then f is equal a.e. to an absolutely continuous function, f 0 (in the conventional sense) exists a.e. and f 0 ∈ Lp (0, 1). 2. (15 points) Let H be the collection of all absolutely continuous functions f : [0, 1] → C such that f (0) = f (1) = 0 and f 0 ∈R L2 (0, 1). Then 1 H is a Hilbert space with the inner product (f, g) = 0 f 0 (x)g 0 (x)dx. (You do not have to prove this, since a very similar problem was included on HW 2.) (a) Given a fixed a ∈ (0, 1), prove that the mapping `a : H → C given by `a (f ) = f (a) is a bounded linear functional. What is the norm of this functional? (Be careful.) (b) By the Riesz-Frechet representation theorem (the characterization of bounded linear functionals on Hilbert spaces), there is a unique ga ∈ H such that `a (f ) = (f, ga ). Find ga . 3. (15 points)R Let X = Cc∞ (0, 1), and let the linear functional ` be given 1 by `(f ) = 0 xf 0 (x)dx. 1 (a) Suppose that X is equipped with the L1 norm: kf k1 = Is ` bounded with respect to this norm? R1 0 |f (x)|dx. (b) Is ` bounded if X is equipped instead with the norm k · kH defined in Problem 2? 2