Bounded bilinear map between normed spaces

A bilinear map $B:X\times Y\to Z$ between normed spaces is bounded if

\|B(x,y)\|_Z\le C\|x\|_X\|y\|_Y

for a fixed finite $C$ . This is equivalent to joint continuity. The least such constant is its bilinear operator norm.

Difference estimate

Bilinearity gives

B(x,y)-B(\widetilde x,\widetilde y) =B(x-\widetilde x,y)+B(\widetilde x,y-\widetilde y).

Thus the quadratic map $Q(x)=B(x,x)$ satisfies $\|Q(x)-Q(y)\|\le C(\|x\|+\|y\|)\|x-y\|$ . It is Lipschitz on every norm ball. When $B$ is an algebra multiplication, this is a bounded-multiplication estimate; associativity and completeness are separate requirements for a Banach algebra.