I could use some help on this proof...
Suppose $\displaystyle u,v\in V$. Prove that $\displaystyle <u,v>=0$ if and only if $\displaystyle ||u||\leq ||u+av||$ for all $\displaystyle a\in F$.
Well, I'm guess that we are going to use the fact that since $\displaystyle <u,v>=0$ then $\displaystyle u,v$ are orthogonal. Then, we can evaluate the norm of $\displaystyle u+v$. That is, $\displaystyle ||u+v||$. Then, we can use Pythagorean Theorem to show that $\displaystyle ||u||^2\leq ||u+v||^2$. Then, the proof the other way would probably be similar???