Let $C[0,2π]$ be the vector space of continuous functions defined on $[0,2π]$
For $n = 1,2,\ldots $, let $f_n ,g_n ∈ C[0,2π]$ be given by
$$f_n(x) = \cos(nx)$$
$$g_n(x) = \sin(nx)$$
(i) Show that $\{1, f_1 ,g_1 ,\ldots, f_n ,g_n,\ldots\}$ is an orthogonal set
(ii) Turn the above set into an orthonormal set
I am having trouble on how show that the set for (i) is orthogonal
I did: $$\langle f,g\rangle = \int_0^{2\pi} \cos(nx)\sin(nx)\,dx = \frac{\sin^2(2\pi n)}{2n}$$
Now for each value of $n$, this evaluates to $0$, and we know that $u$ if orthogonal to $v$ if $\langle u,v\rangle = 0$
Would this be enough to show that the set if orthogonal?
For (ii) would multiplying each vector $v$ in the orthogonal set by $1/\|v\|$ give me an orthonormal set?
Found a very similar question http://ms.mcmaster.ca/courses/20102011/term4/math2zz3/Lecture1.pdf