ECE 580: Optimization by Vector Space Methods Fall 2013 H.W. II R. Srikant Due: September 18 1. For 1 ≤ p, q ≤ ∞. Show that lp ⊆ lq if and only if p ≤ q. Note: You have to consider the cases when p, q are finite as well as infinite. 2. (a) Let p < q. Show that lp lq . (b) Let p < q. and consider lp as a subspace of lq . Is lp a closed subspace with respect to the norm k.kq ? 3. Prove Hölder’s inequality for p = 1, ∞. [This is Problem 11 on page 44 of the text.] 4. Obtain a continuous function x(·) on the interval [−1, 1] which maximizes the integral Z 1 t3 x(t) dt −1 subject to the constraint Z 1 |x(t)|3 dt ≤ 2 . −1 Hint: Use Hölder’s inequality applied to a specific normed linear space. 5. Consider the same problem as in 4 above, but now we want to minimize the integral Z 1 t3 x(t) dt −1 What is the optimal (minimizing) solution? 6. Let (X, k · k) be a normed linear space, and x1 , x2 , . . . , xn be linearly independent vectors from X. For fixed y ∈ X, show that there exist coefficients a1 , a2 , . . . , an minimizing the quantity ky − a1 x1 − a2 x2 · · · − an xn k. Hint:: First show that the search for the minimizing coefficients can be restricted to a closed and bounded finite-dimensional set, and then use the Weierstrass theorem (p. 40 of the text). [This is Problem 9 on page 44 of the text.] 7. In Problem 6 above, let X = L2 [0, 1] , n = 2, x1 (t) = 1, x2 (t) = t, y(t) = t3 , 0 ≤ t ≤ 1. Obtain the optimum values of a1 and a2 .