Montel space

Remarks

A locally convex space in which closed bounded sets are compact, without the barrelled condition, is called semi-Montel. Example: for an open set $\Omega \subseteq \C$, the space $H(\Omega)$ of holomorphic functions with the seminorms $p_K(f) = \sup_{z \in K} \abs{f(z)}$, $K \subseteq \Omega$ compact, is a Montel space. It is a Fréchet space, hence barrelled. A bounded set is a family that is uniformly bounded on each compact set, so Montel's theorem gives the compactness of closed bounded sets. Non-example: an infinite-dimensional Banach space is barrelled, but its closed unit ball is closed and bounded and not compact, by Riesz's lemma. Every Montel space is reflexive, and its strong dual is again a Montel space.