Upper bound of a bilinear form

Question

Suppose I have a form $|X^TBY|$ where $X \in R^n, Y \in R^m$ and $B \in R^{n \times m}$ is a matrix whose elements are bounded. Is there an upper bound for the whole expression of the form $|X^TBY|\le \lambda ||X||||Y||$?

Answer 1

By the Cauchy-Schwarz inequality, we have $|X^TBY| \leq \|X\|\|BY\|$. Also, $\|BY\| \leq \|B\|\|Y\|$, where $\|B\|$ is the Euclidean operator norm of $B$ (i.e., the largest singular value of $B$). So together this gives $$|X^TBY| \leq \|B\|\|X\|\|Y\|.$$