Let be the space of continous real-valued functions on the interval . If , we define
It is easy to prove that this is a scalar product.
If is the space of continous functions on the interval . Let be the function given by , where is some interger . Then
If if a continous function on then .
And Fourier coefficient of with respect to is .
[La] Lang. S. Linear Algebra, Springer, 2004.