Let V be an inner product space, let be an orthonormal subset of V. Prove that for any , we have

proof so far.

by a theorem, there exist unique vectors u in span(S) and v in the orthogonal complement of span(S) such that x = u + v.

Now, since u is orth. to v, meaning

Since , write

then

Since they are all orthogonal to one another, we can write

Now, I'm a bit stuck... Am I at least doing this right so far?