Given an orthogonal set of functions, how can you find an orthonormal set? I understand that you'd have to divide it by the norm of the function, but I'm not sure how to get the norm to be honest.
If I'm given the set $\{\sin(\frac{n\pi x}{L}) \}$ or $\{\cos(\frac{n\pi x}{L}) \}$, how would I find the corresponding orthonormal set of functions?
Could anyone give me a good starting point, or point me to a resource that'll explain how this works?
Thanks in advance!
EDIT: Can someone check if I am doing this correct??
With the limits $-L$ to $L$, I integrate the following: $$\int_{-L}^{L} \sin^2\bigg(\frac{n\pi x}{L}\bigg)dx$$ and get the norm as $$\sqrt{L - \frac{\sin(2n\pi)L}{2n\pi}}$$
So my orthonormal set will be $$\bigg\{\frac{\sin(\frac{n\pi x}{L})}{\sqrt{L - \frac{\sin(2n\pi)L}{2n\pi}}} \bigg\} $$
I feel like it doesn't seem right, could anyone let me know if I'm making any mistakes?