Boundedness criterion for a bilinear map ($\Vert F(x,y)\Vert \le C\Vert x\Vert \Vert y\Vert$)

A bilinear map between normed vector spaces is jointly continuous exactly when it satisfies the displayed product bound. Continuity at zero gives a bound on a product of balls; rescale each nonzero input separately to obtain the inequality. Conversely, expand the increment as $F(x+h,y+k)-F(x,y)=F(h,y)+F(x,k)+F(h,k)$ and apply the product bound to every term. Thus continuity at zero already implies joint continuity everywhere for a bilinear map.