Motivation for condition of Normed Algebra

Question

Is there any particular motivation for the defining Normed Algebra with the condition $\|xy\| \leq \|x\|\|y\|$. Is there any Geometrical view of this condition?

Answer 1

I don't know about "geometrical view", but the condition essentially follows from assuming that multiplication is continuous. Note the "essentially"; we'll see below what I mean by that.

Say we have an algebra with a norm, and multiplication is (jointly) continuous. Continuity at $(0,0)$ shows that there exists $\delta>0$ so that $$||xy||\le1\quad(||x||,||y||\le\delta).$$Now given any $x\ne0$ and $y\ne0$ we have $$\left|\left|\frac{\delta x}{||x||}\right|\right|=\delta,\quad \left|\left|\frac{\delta y}{||y||}\right|\right|=\delta, $$and it follows that $$||xy||\le c||x||\,||y||,$$where $c=\delta^{-2}$. Now $|||x|||=c||x||$ defines an equivalent norm with $$|||xy|||\le|||x|||\,|||y|||.$$