Normed algebra

Remarks

The inequality makes the product $A \times A \to A$ continuous. A normed algebra that is complete as a normed vector space is a Banach algebra.

Example: the algebra $C([0,1])$ of continuous real functions with $\norm{f}_\infty = \sup_t \abs{f(t)}$. For each $t$, $\abs{f(t) g(t)} \le \norm{f}_\infty \norm{g}_\infty$, so $\norm{fg}_\infty \le \norm{f}_\infty \norm{g}_\infty$. The polynomial functions form a subalgebra, which is a normed algebra that is not complete.

Non-example: $M_2(\R)$ with the norm $\norm{a} = \max_{i,j} \abs{a_{ij}}$. For the matrix $a$ with all entries $1$, $a^2$ has all entries $2$, so $\norm{a^2} = 2 > 1 = \norm{a}^2$. The operator norm on $M_2(\R)$, in contrast, does make it a normed algebra.