Let $L^{\varphi }$ be an Orlicz space defined by a convex Orlicz function $\varphi$ and let $E^{\varphi }$ be the space of finite elements in $L^{\varphi }$ (= the ideal of all elements of order continuous norm). We show that the usual norm topology ${𝒯}_{\varphi }$ on $L^{\varphi }$ restricted to $E^{\varphi }$ can be obtained as an inductive limit topology with respect to some family of other Orlicz spaces. As an application we obtain a characterization of continuity of linear operators defined on $E^{\varphi }$.

Let $K$ be a compact subset of an hyperconvex open set $D\subset {ℂ}^n$, forming with D a Runge pair and such that the extremal p.s.h. function ω(·,K,D) is continuous. Let H(D) and H(K) be the spaces of holomorphic functions respectively on D and K equipped with their usual topologies. The main result of this paper contains as a particular case the following statement: if T is a continuous linear map of H(K) into H(K) whose restriction to H(D) is continuous into H(D), then the restriction of T to $H\left(D_{\alpha }\right)$ is a continuous...