Confusion about inner product spaces

Question

I have an exercise like below

Let $X$ is an inner product space and $\forall u,v \in X$. Show that if $<x,u>=<x,v>$ then $u=v$ for $\forall x \in X$

What did I do

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

But if $x$ is unit of $X$, $u$ and $v$ don’t have to be equal. In addition, cannot both of $u$ and $v$ be orthogonal to $x$ when they’re not equal? Is there some missings?

Thanks in advance

Answer 1

You have $\langle x, u - v\rangle = 0$ for all $x\in X$. Now take $x = u - v$.