### Holomorphic germs on Banach spaces

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 ${\mathcal{T}}_{\omega ,\theta}$ 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...