What exactly is your question? I looked at the .gif and everything seemed to be correct (except for the last part with the ).
http://i36.tinypic.com/2ebh4hw.gif
why they remove the absolute value?
and norm by definition equals the root of inner product integral
why they put second power on both of them?
They must have forgotten to write it, but it should be there.
this is only true when you actually have an inner product defined, and it's not hard to see that in or the norm you are given does not come from an inner product (actually, precisely because the norm doesn't satisfy the parallelogram law).and norm by definition equals the root of inner product integral