Open AccessCHARACTERIZATION OF SEMINORMABILITY OF A TOPOLOGICAL ALGEBRA
Author(s) -
Pammy Manchanda
Publication year - 1993
Publication title -
tamkang journal of mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.324
H-Index - 18
eISSN - 0049-2930
pISSN - 2073-9826
DOI - 10.5556/j.tkjm.24.1993.4477
Subject(s) - mathematics , hausdorff space , bounded function , norm (philosophy) , combinatorics , topological algebra , open set , locally nilpotent , characterization (materials science) , topological ring , discrete mathematics , nilpotent , algebra over a field , topological space , pure mathematics , topological vector space , nilpotent group , mathematical analysis , physics , optics , political science , law
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.