= Boundedness criterion for a bilinear map
{title2=$\|F(x,y)\|\leq C\|x\|\|y\|$}
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>.
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