Definition 9

If A is an algebra over a non-discrete valued field K, a norm $p(x)$ on A (A being considered as a vector space over K) is said to be compatible with the algebra structure of A if the topology it defines is compatible with the ring structure of A. An algebra over K, endowed with the structure defined by a norm compatible with the algebra structure, is called a normed algebra.

If A is a normed algebra over K, and if $p(x)$ is the norm on A, the bilinear mapping $(x, y) \to xy$ of $A \times A$ into A is continuous, by hypothesis; hence by Theorem 1 of no. 5 there exists a real number $a > 0$ such that $p(xy) \leq a \cdot p(x)p(y)$. Replacing $p(x)$ by the equivalent norm $a \cdot p(x)$, we may therefore always assume that the norm $||x||$ on a normed algebra A is such that

It follows from (5) by induction that, for each integer $n > 0$, we have