1. ## Inner product proof

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 $\displaystyle a\in F$.

2. Originally Posted by zebra2147
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 $\displaystyle a\in F$.
What do you think? And, I assume that the norm is the natural one induced by the inner product.

3. 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???