On the algebraic property of locally convex topological algebras

By a Fréchet algebra, we mean a complete metrizable locally convex topological algebra. The boundedness of a set in Fréchet algebras is of course a topological property, but the uniform bound of a uniformly bounded set is an algebraic property, since it depends on the choice of seminorms generating the same topology on Fréchet algebra. In this paper, we show that if S is a bounded subsemigroup of a Fréchet algebra (A; (pn)n∈N), then there is an equivalent family of seminorms (tn)n∈N on A, such that tn(s)≤1 (s∈S; n∈N). In the rest of this paper, we get a result by using this fact, and we also have a discussion on continuous inverse algebras.