Boundedness criterion for a bilinear form
= Boundedness criterion for a bilinear form
A <bilinear form> on a <normed vector space> is <continuous> exactly when it is bounded. For the nontrivial direction, continuity at $(0,0)$ gives $r,s>0$ such that $|B(x,y)|\leq1$ whenever $\lVert x\rVert<r$ and $\lVert y\rVert<s$; scaling nonzero $u,v$ into these balls gives $|B(u,v)|\leq4(rs)^{-1}\lVert u\rVert\lVert v\rVert$.