Holomorphic germs on Banach spaces

Chae Soo Bong

Annales de l'Institut Fourier, Tome 21 (1971) no. 3, pp. 107-141.

Soient $E$ et $F$ des espaces de Banach complexes, $U$ un ouvert non-vide de $E$ et $K$ un compact de $E$. La notion de type d’holomorphie $\theta$ de $E$ dans $F$ et la topologie localement convexe naturelle ${𝒯}_{\omega ,\theta }$ sur l’espace vectoriel ${\mathscr{H}}_{\theta }(U,F)$ de toutes les applications holomorphes de $U$ dans $F$, d’un type d’holomorphie donné $\theta$, ont été considérées d’abord par L. Nachbin. C’est le motif pour lequel nous introduisons l’espace localement convexe ${\mathscr{H}}_{\theta }(K,F)$ de tous les germes d’applications holomorphes autour de $K$ dans $F$, d’un type d’holomorphie donné $\theta$, en étudiant ses rapports avec ${\mathscr{H}}_{\theta }(U,F)$, et quelques unes des propriétés de la topologie ${𝒯}_{\omega ,\theta }$.

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 other properties of the topology ${𝒯}_{\omega ,\theta }$.

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