Hi,

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

,

so

for some , and

I do not understand the last two equalities. Would be great if someone could help me out a bit.

Thanks.