MATH 303 Final, Fall 2002

Bete Math 303 - Final Exam (Fall 03) | Liv - yo B. Shayya OF BUKDT Problem 1 (20 pts) Let "OC A={feL(a):[" eo" s(e)dz = 1}, Let g be a nonnegative function in Co(R) with ||giln: = f%g(x)dz = 1, and for € > 0 define ge(z) = 29(2). (i) Prove that A is a closed and convex subset of L'(R). (ii) Let Find lim, 49 ®(ge). (iii) Let y = infyea || fl|z:. Prove that y = 1. (Hint: Ruy EA.) (iv) Conclude that A has no element of least length. Problem 2 (10 pts) Let A={feLl(R) 7 e-* f(z)dr = 1}. oO Then A, being a closed and convex subset of L7(R), has an element A of least length. Find h. (Recall that f° e7?" da = \/4/2.) Problem 3 (20 pts) Define » € M(R’) by [fdu= [ f (cos, sin 0)d0 (fF €C(R")). What is u({ (21,22) € Bs x2 > O}? Problem 4 (25 pts) Suppose 1 < p < oo, f, € L*(R) and ||frilze < 1 for n = 1,2,..., and fn(z) — f(x) for ae. € R as n — 00. Show that ia falc)g(adz + [~ f(e)g(xzjdt as noo for all g € L°(R), where ; + ; =1. Problem 5 (25 pts) Let f € Z'(R) and assume that 1 se lim — |f(z +h) — fla) |dz = 0. 0 fh Jase Show that f(z) =O forae. 2 eR.