Holomorphic germs on Banach spaces

Chae Soo Bong

doi:10.5802/aif.381

Chae Soo Bong

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

Abstract (VO)
Résumé

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 $θ$ between $E$ and $F$ , and the natural locally convex topology $𝒯_{ω, θ}$ on the vector space $ℋ_{θ} (U, F)$ of all holomorphic mappings of a given holomorphy type $θ$ from $U$ to $F$ were considered first by L. Nachbin. Motived by his work, we introduce the locally convex space $ℋ_{θ} (K, F)$ of all germs of holomorphic mappings into $F$ around $K$ of a given holomorphy type $θ$ , and study its interplay with $ℋ_{θ} (U, F)$ and some other properties of the topology $𝒯_{ω, θ}$ .

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 $θ$ de $E$ dans $F$ et la topologie localement convexe naturelle $𝒯_{ω, θ}$ sur l’espace vectoriel $ℋ_{θ} (U, F)$ de toutes les applications holomorphes de $U$ dans $F$ , d’un type d’holomorphie donné $θ$ , ont été considérées d’abord par L. Nachbin. C’est le motif pour lequel nous introduisons l’espace localement convexe $ℋ_{θ} (K, F)$ de tous les germes d’applications holomorphes autour de $K$ dans $F$ , d’un type d’holomorphie donné $θ$ , en étudiant ses rapports avec $ℋ_{θ} (U, F)$ , et quelques unes des propriétés de la topologie $𝒯_{ω, θ}$ .

MR Zbl

DOI : 10.5802/aif.381

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     author = {Chae Soo Bong},
     title = {Holomorphic germs on {Banach} spaces},
     journal = {Annales de l'Institut Fourier},
     pages = {107--141},
     year = {1971},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {21},
     number = {3},
     doi = {10.5802/aif.381},
     mrnumber = {49 #9627},
     zbl = {0222.46018},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/aif.381/}
}

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Chae Soo Bong. Holomorphic germs on Banach spaces. Annales de l'Institut Fourier, Tome 21 (1971) no. 3, pp. 107-141. doi: 10.5802/aif.381

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