414 Math Spring 2005 Homework Assignment no. 9 Due Thursday 14 April Section 12.2.6: 10, 11, 12, 15. 1. Let f(x) = ( - |x|)2 for - x . Prove that 2 4 f(x) = + cos nx. 3 n2 n=1 In what norm does the series converge? Now use Parseval to evaluate
n1
1/n4 .
2. A subset A of a real vector space E is convex if for every x and y in A the line segment joining x to y is contained in A, i.e. tx + (1 - t)y A for 0 t 1. Let a be norm on E. (a) Suppose that A is convex. Prove that A, the closure of A, is convex. (b) Let A = B (x) be an open ball in E. Show that A is convex. 3. Recall that for p 1 the Lp -norm of f C([0, 1], R) is defined by f p = 1/p 1 |f(x)|p dx . For n...