I am trying to show that for f belonging to L^2(-pi;pi) the integral that defines the complex Fourier Coefficients is well defined. In other words what I need to show is that
int_from -pi to pi(|f(x)*exp(-i*k*x)|dx) < infinity (limited)
I was thinking that since f belongs to L^2(-pi;pi) then the integral of this will be finite. Further more since the inteval is limited (-pi;pi) and k belonging to the set of integers the complex exponential function would also be finite.
Am I right?