Upper bound for Bilinear form

Question

Let $x,y\in\mathbb{R}^n$ and $A\in\mathbb{R}^{n\times n}$. It is clear (e.g., by Cauchy-Schwarz) that, $|\langle x,Ay\rangle|\leqslant \|A\|\cdot \|x\|\cdot \|y\|$.

Now, I'm interested in a slightly stronger version of this object. Is it true that, for every $x,y,A$: $$ |\langle x,Ay\rangle|\leqslant \|A\|\cdot |\langle x,y\rangle|? $$

Answer 1

No. For instance, consider $$ A = \pmatrix{0&1\\1&0}, \quad x = \pmatrix{1\\0}, \quad y = \pmatrix{0\\1}. $$