In this problem, we will take a look at different norms on spaces of functions f : N - R. Such functions might arise from, e.g. the Fourier sine cosine series of some periodic function g : [0, 2x] - R. Given some f : N - R and o 2 0, we say that f e Ca if the set {ko |f (k) | : ke N} is bounded. This is called the ko weighted (" space. When fe Ce, we can define the C norm of f as IIflla := sup {ke If (k) | : KEN} = [|koflly where || ||, is the uniform norm. Note that IIfllo = IIflly; so these norms generalize the uniform norm. (a) Prove the triangle inequality for the C2 norm: if f, ge ex, then IIf + glla S lIflla + Ilglla (b) Let m E N. Show that I'm Cem-1. (c) We say a collection of functions S C ex is bounded in (x norm if there exists some B E R such that for all f E S, IIfll. 0, there exists some C' E N such that for all k 2 C and for all fe K, If ( 1:) |