Inner Product Same for all Inputs

Question

I'm working on the following problem:

Prove: If $<x,u> = <x,v>$ for all $x$ then $u=v$.

The solution in the book says:

$<x,u-v>=0$; take $x=u-v$

This was confusing to me -- the only work I have for this problem is:

$$<x,u> - <x,v> = 0 \\ <0,u-v> = 0$$

Then I get stuck! $u$ and $v$ can be different and this will still be true.

Any ideas on how to make progress on this? I suspect I'm bungling something trivial.

Answer 1

1. $u$ and $v$ are fixed throughout the problem.
2. The condition in the question allows us to take any $x$, so that $\langle x,u\rangle = \langle x,v\rangle $.
3. Note the equivalence between $\langle x,u\rangle = \langle x,v\rangle \iff \langle x,u\rangle - \langle x,v\rangle =\langle x,u-v\rangle =0$.
4. This suggests us to fix $x=u-v$, so that $\langle x,u-v\rangle =\langle u-v,u-v\rangle =||u-v||^2=0$.
5. From the norm property, $u-v=0\iff u=v$.