About the continuity of multiplication in a normed algebra

Question

A normed linear space $(A,\|\cdot\|)$ over $\mathbb{F}$ is said to be a normed algebra if $A$ is an algebra and $$\|ab\|\leq\|a\|\|b\|\qquad (a,b\in A).$$ They say that multiplication in a normed algebra $A$ is a continuous mapping of $A\times A$ into $A$. I understand the proof of the statement (Proposition 2.4, Complete normed algebras by Bonsall and Duncun). But can anyone tell: Is the norm on $A\times A$ defined as $$\|(a,b)\|_{A\times A } = \max\{\|a\|,\|b\|\}?$$ If yes, then is it always defined like that?

Answer 1

No, you can define the norm in many other (equivalent) ways: $||(a,b)||=||a||+||b||$,$||(a,b)||=\sqrt {||a||^{2}+||b||^{2}}$ etc. For the continuity of multiplication it doesn't matter which of these norms you use. There may be situations where a particular norm may be helpful but for basic properties of Banach algebras you can take any of these.