CHARACTERIZATION OF SEMINORMABILITY OF A TOPOLOGICAL ALGEBRA

Let $\mathcal{A}$ be an algebra over a field $F$ and let $N$ be a norm on $F$. A seminorm (norm) on $\mathcal{A}$ associated with $N$ is defined. It is proved that if $(\mathcal{A}, \mathcal{J})$ is a proper topological algebra over a proper topological field $(F,T)$, then $T$ is defined by a norm $N$ and $\mathcal{J}$ is defined by a seminorm $||\cdot ||$ associated with $N$ (a norm $||\cdot ||$ associated with $N$ if $\mathcal{J}$ is Hausdorff) if and only if the following three conditions are satisfied. (i) $(F,T)$ has a nonempty open bounded set.(ii) $(F,T)$ has a nonzero topological nilpotent element. (iii) $(\mathcal{A},\mathcal{J})$ has a nonempty open bounded set.