How do I need to solve this question? Do I need to use the Gram-Schmit orthogonalization process to transform the basis into an orthogonal basis, and then normalize the orthogonal basis vectors to obtain an orthonormal basis?
I mean letting then
Then we normalize each element like this
and so on...
But I doubt this is the correct method of doing this...
What does this have to do with Fourier series? You only have to check whether the vectors you got, after you normalized to vectors of length 1, are pairwise orthogonal...and they must be, since normalizing vectors don't change orthogonality. Gram-Schmidt process usually gives orthoNORMAL basis, which is orthogonal and of length 1.
When I wrote "check it" I meant you check the above, which you can.