Inner product proof

**zebra2147** · November 1st 2010, 05:49 AM

I could use some help on this proof...

Suppose $u,v\in V$ . Prove that $<u,v>=0$ if and only if $||u||\leq ||u+av||$ for all $a\in F$ .

**Drexel28** · November 1st 2010, 01:48 PM

Originally Posted by zebra2147

I could use some help on this proof...

Suppose $u,v\in V$ . Prove that $<u,v>=0$ if and only if $||u||\leq ||u+av||$ for all $a\in F$ .

What do you think? And, I assume that the norm is the natural one induced by the inner product.

**zebra2147** · November 3rd 2010, 06:18 AM

Well, I'm guess that we are going to use the fact that since $<u,v>=0$ then $u,v$ are orthogonal. Then, we can evaluate the norm of $u+v$ . That is, $||u+v||$ . Then, we can use Pythagorean Theorem to show that $||u||^2\leq ||u+v||^2$ . Then, the proof the other way would probably be similar???

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