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.

Résumé
Abstract

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 $𝒯_{ω, θ}$ .

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 $𝒯_{ω, θ}$ .

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},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {21},
     number = {3},
     year = {1971},
     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. https://www.numdam.org/articles/10.5802/aif.381/

Bibliographie
Cité par

[A] H. Alexander, Analytic functions on a Banach space, Thesis, University of California at Berkeley (1968).

[Ch] S.B. Chae, Sur les espaces localement convexes de germes holomorphes, C.R. Ac. Paris, 271 (1970), 990-991. | MR | Zbl

[Ar] R.M. Aron, Topological properties of the space of holomorphic mappings, Thesis, University of Rochester (1970).

[B] J.A. Barroso, Topologia em espacos de aplicações holomorfas entre espaços localmente convexos, Thesis, Instituto de Matematica Pura e Aplicada, Rio de Janeiro (1970).

[C] G. Coeure, Fonctions plurisousharmoniques sur les espaces vectoriels topologiques et applications à l'étude des fonctions analytiques, Thèse, Université de Nancy (1969).

[D1] S. Dineen, Holomorphy type on a Banach space, Thesis, University of Maryland (1969).

[D2] S. Dineen, Holomorphic functions on a Banach space, Bulletin of American Mathematical Society (1970). | MR | Zbl

[D3] S. Dineen, The Cartan-Thullen theorem for Banach spaces, to appear Annali della Scuola Normale Superiore de Pisa. | Numdam | Zbl

[D4] S. Dineen, Bounding subsets of a Banach space (to appear). | Zbl

[DS] J. Dieudonne, L. Schwartz, La dualité dans les espaces (F) et (LF), Annales de l'Institut Fourier, Grenoble, t. 1 (1949), 61-101. | EuDML | Numdam | MR | Zbl

[GJ] L. Gillman, M. Jerison, Rings of continuous functions, Van Nostrand, Princeton (1960). | MR | Zbl

[Gr] A. Grothendieck, Sur les espaces (F) et (DF), Summa Brasiliensis Mathematicae, v. 3 (1954), 57-122. | MR | Zbl

[Gr2] A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Memoirs of American Mathematical Society, n° 16 (1955). | MR | Zbl

[G] C.P. Gupta, Malgrange's theorem for nuclearly entire functions of bounded type on a Banach space, Thesis, University of Rochester (1966). Reproduced in Notas de Matematica, n° 37 (1968), Instituto de Matematica Pura e Aplicada, Rio de Janeiro. | MR | Zbl

[H] J. Horvath, Topological vector spaces and distributions, v. 1., Addison-Wesley, Mass. (1966). | MR | Zbl

[Hr] L. Hörmander, Introduction to complex analysis in several variables, Van Nostrand, Princeton (1966). | MR | Zbl

[L] P. Lelong, Fonctions et applications de type exponentiel dans les espaces vectoriels topologiques, C.R.A.c Paris 169 (1969). | MR

[M1] A. Martineau, Sur les fonctionnelles analytiques et la transformation de Fourier-Borel, Journal d'Analyse Mathématique, v. 11 (1963), 1-164. | MR | Zbl

[M2] A. Martineau, Sur la topologie des espaces de fonctions holomorphes, Mathematische Annalen, v. 163 (1966), 62-88. | EuDML | MR | Zbl

[Mt] M.C. Matos, Holomorphic mappings and domains of holomorphy, Thesis, University of Rochester (1970). | Zbl

[N1] L. Nachbin, Topological vector spaces of continuous functions, Proc. Nat. Acd. Sci. USA. v. 40 (1954), 471-4. | MR | Zbl

[N2] L. Nachbin, Lectures on topological vector spaces, Lecture note, University of Rochester (1963).

[N3] L. Nachbin, Lectures on the theory of distributions, University of Rochester (1963), Reproduced by Universidade do Recife (1964) ; North-Holland Publishing Company (1970). | Zbl

[N4] L. Nachbin, On the topology of the space of all holomorphic functions on a given open subset, Indagationes Mathematicae, Koninklijke Nederlandse Akademie van Wetenschappen, Proceedings, Series A 70 (1967), 366-368. | MR | Zbl

[N5] L. Nachbin, On spaces of holomorphic functions of a given type, Proceedings of the Conference on Functional Analysisis, University of California at Irvine (1966), 50-60. Thompson Book Company (1967). | Zbl

[N6] L. Nachbin, Topology on spaces of holomorphic mappings, Ergebnisse der Mathematik und ihrer Grenzgebiete, v. 47 (1969), Springer-Verlag, Berlin. | MR | Zbl

[N7] L. Nachbin, Convolution operators in spaces of nuclearly entire functions on a Banach space, Proceedings of the Symposium on Functional Analysis and Related Fields, University of Chicago (1969), Springer-Verlag, Berlin (in press). | Zbl

[N8] L. Nachbin, Holomorphic functions, domains of holomorphy and local properties, North-Holland Publishing Company (1970). | MR | Zbl

[N9] L. Nachbin, Concerning holomorphy types for Banach spaces, Studia Mathematica, Proceedings of the Colloquim on Nuclear Spaces and Ideals in Operator Algebras held in Warsaw, Poland, June 18-25, 1969.

[NG] L. Nachbin, C.P. Gupta, On Malgrange's theorem for nuclearly entire functions (to appear).

[Nr] P. Noverraz, Fonctions plurisousharmonique et analytiques dans les espaces vectoriels topologiques complexes, Annales de l'Institut Fourier, Grenoble, 19,2 (1969), 419-493. | EuDML | Numdam | MR | Zbl

[P] H.R. Pitt, A note on bilinear forms, Journal London Math. Society, v. 11 (1936), 174-180. | JFM | Zbl

[R] H.P. Rosenthal, On quasi-complemented subspaces of Banach spaces, with an appendix on compactness of operators from Lp (µ) to Lr (v), Journal of Functional Analysis 4 (1969), 176-214. | MR | Zbl

[T] F. Treves, Topological vector spaces, distributions and kernels, Academic Press, New York and London (1967). | MR | Zbl

[Z] M.A. Zorn, Characterization of analytic functions in Banach spaces, Annals of Mathematics, 12 (1945), 585-593. | MR | Zbl

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