Holomorphic germs on Banach spaces

Chae Soo Bong

Annales de l'Institut Fourier, Tome 21 (1971), p. 107-141 / Harvested from Numdam

Résumé

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 ${\cal T}_{\omega ,\theta }$ sur l’espace vectoriel ${\cal H}_\theta (U,F)$ 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 ${\cal H}_\theta (K,F)$ de tous les germes d’applications holomorphes autour de $K$ dans $F$, d’un type d’holomorphie donné $\theta $, en étudiant ses rapports avec ${\cal H}_\theta (U,F)$, et quelques unes des propriétés de la topologie ${\cal T}_{\omega ,\theta }$.

Abstract

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 ${\cal T}_{\omega ,\theta }$ on the vector space ${\cal H}_\theta (U,F)$ 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 ${\cal H}_\theta (K,F)$ of all germs of holomorphic mappings into $F$ around $K$ of a given holomorphy type $\theta $, and study its interplay with ${\cal H}_\theta (U,F)$ and some other properties of the topology ${\cal T}_{\omega ,\theta }$.

Détail

Publié le : 1971-01-01

MR 49 #9627 | Zbl 0222.46018

DOI : https://doi.org/10.5802/aif.381

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