A question involving inner product on a Hilbert Space

Question

Let $H$ be a hilbert space, and let $x,y \in H$. If $\langle x,z \rangle=\langle y,z\rangle$ for all $z \in H$, then $x=y$.

Is this statement true or false? and why?

Answer 1

This statement is true. Intuitively, this says that all "coordinates" of $x$ and $y$ are equal. A formal proof would proceed as follows:

From the given condition: $\langle(x-y),z\rangle=0$ for all $z \in H$. Now, take $z=(x-y)$ to get $||x-y||^2=0$ where $||\cdot||$ is induced by the inner product.

Now, one of the axioms that a norm has to satisfy (to be induced from a inner product on a Hilbert space) is that $||z||=0$ if and only if $z=0$. So from above we get that $x-y=0$, i.e. $x=y.$