Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 4 / 23H / a

This is the boundedness criterion for a bilinear form. Since $B$ is continuous at $(0,0)$, there are $r,s>0$ such that $|B(x,y)|\le 1$ whenever $\Vert x\Vert <r$ and $\Vert y\Vert <s$. For nonzero $u,v$, apply this toLinearity in each argument then givesThe same inequality is immediate if either vector is zero, so one may take $M=4/(rs)$.