Problem 2. let v = c([−π, π]) and define hf, gi = r π −π f(x)g(x)dx . as in the latest lab, we can actually prove that the set { √ 1 π sin(kx)}k∈n is an orthonormal set in the infinite dimensional space v (you do not have to prove any of these). use the exercise above and the function f(x) = x to prove x[infinity] k=1 1 k 2 ≤ π 2 6 hint: let ek = √ 1 π sin(kx), and calculate ck = hf, eki = √ 1 π r π −π x sin(kx)dx. remark: as noted above, we can actually have equality!