If $\langle x,y\rangle = \langle x,z\rangle$ , then $y=z$ PROOF

Question

I'm trying to prove a property of inner products. The property is: $$\langle x,y\rangle=\langle x,z\rangle \forall x \in V\Rightarrow y=z$$

My proof:

Let $y=z+w$ then $$\langle x,z+w\rangle = \langle x,z\rangle $$ $$\Rightarrow \langle x,z\rangle + \langle x,w\rangle =\langle x,z\rangle $$ $$\Rightarrow \langle x,w\rangle =0$$ Hence, $w=0$ so $y=z$.

My question is are we allowed to set $y=z+w$ i.e. is this true for all $y,z \in V$?

Answer 1

It is certainly true that $y-z$ is something in the vector space; call that something $w$.

Answer 2

Yes, you can set $y=z+w$, because $V$ is a linear space. From $\langle x,w\rangle =0,\,\forall x\in V$ follows that you can take $x=w$ and so $\|w\|^2=0\Rightarrow w=0$

A more straight forward approach is just to write $\langle x,y\rangle =\langle x,z\rangle\Leftrightarrow \langle x,y-z\rangle =0,\,\forall x\in V$ so you can take $x=y-z$ and get $\|y-z\|^2=0\Rightarrow y=z$