Bilinear operator norm question. So I wrote this down a week ago and cannot figure out what I was thinking. Not sure if this is correct. 
Context we have a bilinear operator $B:X\times Y\to \mathbb{K}$.
Is it true that
$$\sup_{x\in X, y\in Y} \|B(x,y)\|< \infty \implies |B(x,y)|\leq K \|x\|\|y\|$$
So the absolute value is a norm on the reals, complex in one dimension so that's not a problem because the norms are equivalent. 
So the implication works for the $x,y$ supremum case. But does it immediately follow for all other $(x,y)$? I cannot remember what I was thinking. Can anyone help? 

$$\sup_{x\in X, y\in Y} \|B(x,y)\|< \infty \implies B=0$$
