The second-to-last equality is just a complex number written as its magnitude times a phase.
The last equality is just the expression
substituted into the denominator of
I learned Gram-Schmidt a few years ago, and now that I'm reading about it from another source I've having some problems with it.
Suppose that is an orthonormal basis for , and consider the problem of finding one additional vector such that is an orthonormal basis for .
For this to hold, the Fourier expansion of with respect to must be,
which in turn implies that
Since | , we have that
for some , and
I do not understand the last two equalities. Would be great if someone could help me out a bit.