If $\Re e\langle Ax,y\rangle \leq M$, why $|\langle Ax,y\rangle| \leq M$?

Question

Let $E$ be a complex Hilbert space and $A\in \mathcal{L}(E)$.

Assume that $\Re e\langle Ax,y\rangle \leq M$ for all $x,y\in E$ and some constant $M$. Why $$|\langle Ax,y\rangle| \leq M,\;\forall x,y\in E.$$

Answer 1

In light of Sangchul Lee's hint, you can perform a proof like this:

Consider the polar decomposition $\langle Ax,y\rangle=re^{i\theta}$ and then you'll have $$ \mathfrak Re\langle Ax,e^{i\theta}y\rangle=r=|\langle Ax,y\rangle|. $$ Thus $|\langle Ax,y\rangle|\leqslant M$.