Let $V$ (which is infinitely dimensional) be the set of all continuous functions $\Bbb{S}^1 \to \Bbb{R}$. Show that $V$ is a vector space. Define $\langle-,-\rangle: V\times V\to \Bbb{R}$ by $$\langle f,g\rangle=\dfrac1\pi\int_{-\pi}^\pi f(x)g(x)dx.$$
Using the above integral. Show that $\{\frac1{\sqrt2},sin(x),cos(x),sin(2x)cos(2x),...,sin(kx)cos(kx)\}$ such that k is an element of Z positive, is an orthonormal set.
End question
I know to find orthornormal vectors I am suppose to normalize vectors but I am not sure how to do this in an infinite dimensional way, especially when there is an integral involved.
The definition of "orthonormal set" is that, for any functions $f$,$g$ in the set, $$ \langle f,g\rangle = \begin{cases} 1 &\text{if $f=g$,} \\ 0 &\text{if $f\ne g$.} \end{cases} $$ So, prove that.