A Hille-Yosida Theorem is proved on convenient vector spaces, a class, which contains all sequentially complete locally convex spaces. The approach is governed by convenient analysis and the credo that many reasonable questions concerning strongly continuous semigroups can be proved on the subspace of smooth vectors. Examples from literature are reconsidered by these simpler methods and some applications to the theory of infinite dimensional heat equations are given.

This article is devoted to a study of locally convex topologies on $\mathbf{H}\left(U\right)$ (where $U$ is an open subset of the locally convex topological vector space $E$ and $\mathbf{H}\left(U\right)$ is the set of all complex valued holomorphic functions on $E$). We discuss the following topologies on $\mathbf{H}\left(U\right)$:(a) the compact open topology ${\mathbf{I}}_0$,(b) the bornological topology associated with ${\mathbf{I}}_0$,(c) the ported topology of Nachbin ${\mathbf{I}}_{\omega }$,(d) the bornological topology associated with ${\mathbf{I}}_{\omega }$ ; and(e) the ${\mathbf{I}}_{\omega }$ topological of Nachbin.For $U$ balanced we show these topologies are...

Let $E$ and $F$ be two complex Banach spaces, $U$ a nonempty subset of $E$ and $K$ a compact subset of $E$. The concept of holomorphy type $\theta$ between $E$ and $F$, and the natural locally convex topology ${𝒯}_{\omega ,\theta }$ on the vector space ${\mathscr{H}}_{\theta }(U,F)$ of all holomorphic mappings of a given holomorphy type $\theta$ from $U$ to $F$ were considered first by L. Nachbin. Motived by his work, we introduce the locally convex space ${\mathscr{H}}_{\theta }(K,F)$ of all germs of holomorphic mappings into $F$ around $K$ of a given holomorphy type $\theta$, and study its interplay with ${\mathscr{H}}_{\theta }(U,F)$ and some...