Bounding a bilinear map

Question

Let $B$ be a bilinear map from $R^p \times R^q$ to $R^N$. Prove that there exists a $C>0$ s.t. $\left\lvert\lvert B(u,v)\right\rvert\rvert \le \lvert\lvert u\rvert\rvert \cdot\lvert\lvert v\rvert\rvert\cdot C$. I am having trouble proving this. I know that $B(u,\cdot): R^q\to R^N$ is linear and linear maps are bounded. But I don't know how to continue from there.

Answer 1

You can think of $B$ as a linear map $u \in \mathbb R^p \mapsto B(u,-) \in \hom(\mathbb R^q,\mathbb R^N)$. Then there exists $C > 0$ such that

$$ \|B(u,-)\| \leq C\|u\| $$

for all $u$, because as you say linear maps are bounded. Now, we have:

$$ \|B(u,v)\| = \|B(u,-)(v)\| \leq \|B(u,-)\|\|v\| \leq C\|u\|\|v\|. $$