Show that an integral is well defined

Hi

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?

well-defined integral revisited

Hello,

Let me try one more time:

$\displaystyle \int_{-\pi}^{\pi}|f(x)|dx= \int_{-\pi}^{\pi}|f(x)g(x)|dx \leq \left(\int_{-\pi}^{\pi}|f(x)|^{2}dx \right)^{\frac{1}{2}}\left(\int_{-\pi}^{\pi}|g(x)|^{2}dx \right)^{\frac{1}{2}} \leq $

$\displaystyle \left(\int_{-\pi}^{\pi}|f(x)|^{2}dx \right)^{\frac{1}{2}}\left(\int_{-\pi}^{\pi}|f(x)|^{2}dx \right)^{\frac{1}{2}} < \infty $

What do you think?