PMath/AMath 331, Fall 2017 - Assignment 4. Posted on Friday October 6; due on Friday, October 13. Topics: Abstract normed vector spaces, convergence and completeness in Rn . Practice problems. (Not to be submitted for marking. These are to help students learn and better understand the concepts. If you are stuck I will gladly help you by pointing you in the right direction.) Problem P1. a) Consider the vector space, C[a, b], of all continuous functions on the interval [a, b]. In 11.2 of the posted lecture notes we defined the function, k k1 : C[a, b] R, as kf k1 = Z b a |f (x)|dx Show that k k1 is a valid norm on C[a, b]. b) Let {fn : n N, n 6= 0} be a sequence of constant functions defined as fn (x) = 2 + n1 for each n. That is, f1 (x) = 3, f2 (x) = 5/2, f3 (x) = 7/3, and so on. If {fn } is viewed as a subset of C[1, 2] equipped with the norm k k1 determine, lim n Z 2 1 |fn (x)| dx Justify all your steps carefully. Problem P2. a) In example 9.2.2 in the posted lecture notes we stated that if f and g belong to the vector space, C[a, b], of all continuous functions on the interval [a, b] then < f, g >= Rb a f (x)g(x) dx is a valid inner product. Prove that this is so. b) Let k k be the norm on C[0, 1] which is induced by the inner product given in part a) of this question. Compute kex k. Problem P3. a) In 11.1 of the posted lecture notes we defined the \"p-norm\