= Solution
This is the <boundedness criterion for a bilinear form>. Since $B$ is <continuous> at $(0,0)$, there are $r,s>0$ such that $|B(x,y)|\leq1$ whenever $\lVert x\rVert<r$ and $\lVert y\rVert<s$. For nonzero $u,v$, apply this to
$$
x=\frac{r}{2\lVert u\rVert}u,
\qquad
y=\frac{s}{2\lVert v\rVert}v.
$$
<Linearity> in each argument then gives
$$
|B(u,v)|\leq\frac4{rs}\lVert u\rVert\lVert v\rVert.
$$
The same inequality is immediate if either vector is zero, so \b[one may take $M=4/(rs)$].
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