Hilbert space problem

Question

Let $(H, (\cdot,\cdot))$ be a Hilbert space over complex numbers and $x,y\in H$. Suppose $$|(z,x)|\leqslant |(z,y)|$$ for all $z\in H$. Is it true that $x=\lambda y$ for some $\lambda\in\mathbb C$?

Any help is appreciated.

Answer 1

This is not true. Put $x=\alpha y$, where $|\alpha|\leq1$.

Given the edit to the question, observe that if $(z,y)=0$, then $(z,x)=0$. So $\{y\}^\perp\subset\{x\}^\perp$, and use this to show that $x\in\text{span}\{y\}$.