In this problem, we will take a look at different norms on spaces of functions f: NR. Such functions might arise from, e.g. the Fourier sine/cosine series of some periodic function g: [0,2] R. Given some f: N R and a 0, we say that fe ex if the set {k|f(k): ke N} is bounded. This is called the ka weighted l space. When fl, we can define the norm of f as ||f||= sup{k|f(k)| : k = N} = ||kf|| where is the uniform norm. Note that ||f|| = ||||,,, so these norms generalize the uniform norm. (a) Prove the triangle inequality for the norm: if f, g lo, then (b) Let me N. Show that ||f9|||||| +9a C-1 is bounded in ex norm if there exists (c) We say a collection of functions SC some BR such that for all fS, ||f|| B. Show that the set of functions S = {gn: ne N} given by 1 k = n In (k): 0 kn is bounded in norm, but not bounded in lo norm for any m = N. (d) We say a collection of functions KCl is uniformly small at infinity (USI) if for all > 0, there exists some C EN such that for all k > C and for all f K, |f(k) < E |f(k)| < Show that if K is bounded in lo norm for some m N, then K is uniformly small at infinity. (Hint: If B is the bound of K in ex norm, try picking M such that B/km M (why can you do this?). Then, what happens to |f(k)|?) (e) Let {f} be a sequence of functions in converging pointwise to some f loo. Show that if the collection {fn n N} is uniformly small at infinity, then fr converges to f uniformly. (Hint: Use the USI condition to find some cutoff C such that |fn(k) - f(k)| C. Then there are only a finite number of sequences {fn(k)}, k