Quick Question: If $\varphi$ is a linear map and inner product $\langle \varphi x, y\rangle = 0$, why is $\varphi \equiv 0$?

Question

The title says it all. If $\varphi$ is a linear map, and we have that the inner product is $\langle \varphi x, y\rangle = 0$ for all $x,y$ in the domain of $\varphi$, how does this imply that $\varphi$ is the zero map? In particular, I was told that this equation implies that $\varphi \equiv 0$ by setting $y = \varphi x$, but how does this give us the desired conclusion? I'm missing something trivial. Thank you!

Answer 1

For any $x$ we get $\langle \phi x, \phi x \rangle = 0$ taking $y = \phi x$ in the condition. The positive definiteness of the inner product implies $\phi x = 0$ and so $\phi \equiv 0$ as $x$ was arbitary.