Complex Inner Product Space always zero

Question

I'm trying to solve the problem below:

Let $(X,\langle\,\cdot\,\rangle)$ be a complex inner product space. Suppose $x \in X$ and $Y$ is a linear subspace of $X$ such that $\operatorname{Re} \langle x,y\rangle = 0$ for all $y\in Y$. Show that in fact $\langle x,y\rangle = 0$ for all $y \in Y$.

This is for an assignment so I'm not looking for a full solution as I want to understand how to do this in the future! A hint would be very welcome though.

Answer 1

Consider $\langle x, iy \rangle$. What's the real part of that expression?