# 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.