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 .