which statement is true? (both can be true\false)
1)if $u,v$ are vectors in Inner product space $V$ and $V$ is an Euclidean space and $||u+v||^2=||u||^2+||v||^2$ then $u \perp v$
2)If $V$ is an unitary space then the statement above is not true
I am self studying this so my approach is probably wrong but:
I think the first statement is true because we can show that $u \perp v$ if we suppose that $u,v$ are vectors in an Euclidean space then $(u,v)= \frac{1}{4}(||u+v||^2 -||u-v||^2)$ therefore if $||u+v||=||u-v||$ we will get that $(u,v)=0$ so $u \perp v$
and if $||u+v||^2=||u||^2+||v||^2$ then $||u+v||^2=||u||^2 +2Re(u,v)+||v||^2=||u||^2+||v||^2$ since it is a Euclidean space it means we are in $\Bbb R$ so $2Re(u,v)=2(u,v)$ from here $||u||^2 +2(u,v)+||v||^2=||u||^2+||v||^2$ $\iff$ $(u,v)=0$ meaning $u \perp v$
The second statement is also true ( if it is an unitary space then the first statement is not true) because for example $z_1 =i , z_2 =1$ we get $||z_1+z_2||^2=2$ and $||z_1||^2=1$ , $||z_2||^2=1$ but $(z_1,z_2)= i\cdot 1 = i \not = 0$
is my way correct that both statements are true?
Use the bilinear form $$\langle u+v,u+v\rangle = \langle u,u\rangle + 2\langle u,v\rangle + \langle v,v\rangle$$
Your restriction gives $\langle u,v\rangle=0$ which is equivalent to $u\perp v$.