I am trying to prove the following:
Let $V$ be a vector space with an inner product, and $x,y\in V$. If $\langle x,z\rangle = \langle y,z\rangle$ for all $z\in V$, prove that $x = y$.
I started with the dot product as the inner product of a vector space:
$$\langle x,z\rangle = \sum\limits_{i=1}^nx_iz_i$$ $$\langle y,z\rangle = \sum\limits_{i=1}^ny_iz_i$$
but I am not sure how to prove that these sums must be equal.
Where do I go from here/is this the right first step?
For all $z \in V$ we have that \begin{equation} \label{1}\tag{1} \langle x , z \rangle = \langle y , z \rangle \Leftrightarrow \langle x - y , z \rangle = 0 . \end{equation} Suppose that $x \neq y$. Since \eqref{1} holds for all $z \in V$. Takes $z = x - y \in V$ then we have $$ \Leftrightarrow \langle x - y , z \rangle = 0 \Leftrightarrow \lVert x - y \rVert > 0 $$ which gives a contradiction.