Prove/Disprove: $\forall u\in V: \langle v,u \rangle = \langle w,u \rangle \implies v=w$.

Question

Prove/Disprove: $\forall u\in V: \langle v,u \rangle = \langle w,u \rangle \implies v=w$.

I want to say "Yes", but couldn't formulate my intuition into a proof.

How to prove it?

Answer 1

$\forall u\in V: \langle v,u \rangle = \langle w,u \rangle \implies \forall u\in V: \langle v-w,u \rangle =0$. Now put $u=v-w$ to get $\langle v-w,v-w \rangle = 0 \implies v-w=0\implies v=w$