Thank you!

Problem 2. Let CJa, b be the vector space of continuous functions on a closed interval [a, b] C R and fix a function g E Cla, b]. Prove that (,g) : C[a, bj - R f -> (f,g) = / f(t)g(t) dt is an element of Ca, b]*, that is, a linear functional on Cla, b]. Notice that this is one part of the definition of an inner product on Cla, b. Important consequence (you don't have to prove this): It follows that when [a, b] = [0, 2x], for every integer n, the Fourier coefficients of f, an (f ) = f (t) cos(nt) dt and Bn.(f) = f (t) sin(nt) dt, define linear functionals, i.e. an, b,. ( Cla, b]*