An example of scalar product

Let V be the space of continous real-valued functions on the interval [a,b]. If f, g \in V, we define

\langle f, g\rangle = \int\limits_a^b f(x)g(x)dx

It is easy to prove that this is a scalar product.

If V is the space of continous functions on the interval [-\pi, \pi]. Let f be the function given by f(x) = \sin{kx}, where k is some interger > 0. Then

\|f\| = \sqrt{\langle f, f \rangle} = \big( \int\limits_{-\pi}^\pi \sin^2{kx}\big)^{1/2} dx = \sqrt{\pi}.

If g if a continous function on [-\pi, \pi] then \langle g, f \rangle = \int\limits_{-\pi}^\pi g(x)\sin{kx} dx.

And Fourier coefficient of g with respect to f is \dfrac{\langle g, f \rangle}{\langle f, f \rangle} = \dfrac{1}{\pi}{\int\limits_{-\pi}^\pi g(x)\sin{kx} dx}.

References

[La] Lang. S. Linear Algebra, Springer, 2004.