Example 2

Let $K$ be a non-discrete valued field and let $M_n(K)$ be the ring of square matrices of order $n$ over $K$. Regarded as a vector space over $K$, $M_n(K)$ is isomorphic to $K^{n^2}$. If for each $X = (x_{ij}) \in M_n(K)$ we define $\|X\| = \sup_{i,j} |x_{ij}|$, then $\|X\|$ is a norm on $M_n(K)$, and the topology defined by this norm is the product topology on $K^{n^2}$ (Proposition 10); from this it follows (because of the continuity of polynomials in any number of variables over $K$) that this norm is compatible with the $K$-algebra structure of $M_n(K)$.