The continuous linear image of a p-adic compactoid

The key result is the following. Let A be a closed (local) compactoid in a Banach space E over a non-archimedean valued field K. If F is a K-Banach space and T:E→F is a continuous linear map then $\overline{TA}\subset \lambda TA$ for each λ∈K, |λ|>1. As an application the existence of the finest admissible topology of countable type on a locally convex space is proved.