i don't fully understand it and i'll try to write what i don't understand as a question.
for instance if we say that V is an unitarian space and $u,v \in V$, so is it possible that:
a)$||u+v||=||u-v||$ then $u \bot v$
b)if $u \bot v$, $||u+v||^2 = ||u||^2 + ||v||^2$
c)if V is unitarian space, can there be a $0 \neq v \in V$ so (v,v)=i?
because basically, regarding a) and b) the property of unitarian space that is relevant is (a+b,c)=(a,c)+(b,c), but when i tried to calculate, i saw that only b is true, am i right or wrong? and regarding c)the property of unitarian space that is relevant to the question is: if a≠0, then (a,a)>0, so i don't see how (v,v) can be equal to i.
so to put it short, i think that a)wrong b)right c)wrong.
could you please check and correct me if i'm wrong?