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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...