Holomorphic germs on Banach spaces

Chae Soo Bong

Annales de l'institut Fourier (1971)

  • Volume: 21, Issue: 3, page 107-141
  • ISSN: 0373-0956

Abstract

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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 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 .

How to cite

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Chae Soo Bong. "Holomorphic germs on Banach spaces." Annales de l'institut Fourier 21.3 (1971): 107-141. <http://eudml.org/doc/74041>.

@article{ChaeSooBong1971,
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 $\{\cal T\}_\{\omega ,\theta \}$.},
author = {Chae Soo Bong},
journal = {Annales de l'institut Fourier},
language = {eng},
number = {3},
pages = {107-141},
publisher = {Association des Annales de l'Institut Fourier},
title = {Holomorphic germs on Banach spaces},
url = {http://eudml.org/doc/74041},
volume = {21},
year = {1971},
}

TY - JOUR
AU - Chae Soo Bong
TI - Holomorphic germs on Banach spaces
JO - Annales de l'institut Fourier
PY - 1971
PB - Association des Annales de l'Institut Fourier
VL - 21
IS - 3
SP - 107
EP - 141
AB - 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 ${\cal T}_{\omega ,\theta }$.
LA - eng
UR - http://eudml.org/doc/74041
ER -

References

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  1. [A] H. ALEXANDER, Analytic functions on a Banach space, Thesis, University of California at Berkeley (1968). 
  2. [Ch] S.B. CHAE, Sur les espaces localement convexes de germes holomorphes, C.R. Ac. Paris, 271 (1970), 990-991. Zbl0201.15603MR45 #876
  3. [Ar] R.M. ARON, Topological properties of the space of holomorphic mappings, Thesis, University of Rochester (1970). 
  4. [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). 
  5. [C] G. COEURE, Fonctions plurisousharmoniques sur les espaces vectoriels topologiques et applications à l'étude des fonctions analytiques, Thèse, Université de Nancy (1969). Zbl0187.39003
  6. [D1] S. DINEEN, Holomorphy type on a Banach space, Thesis, University of Maryland (1969). 
  7. [D2] S. DINEEN, Holomorphic functions on a Banach space, Bulletin of American Mathematical Society (1970). Zbl0237.46027MR41 #4216
  8. [D3] S. DINEEN, The Cartan-Thullen theorem for Banach spaces, to appear Annali della Scuola Normale Superiore de Pisa. Zbl0235.46037
  9. [D4] S. DINEEN, Bounding subsets of a Banach space (to appear). Zbl0202.12803
  10. [DS] J. DIEUDONNE, L. SCHWARTZ, La dualité dans les espaces (F) et (LF), Annales de l'Institut Fourier, Grenoble, t. 1 (1949), 61-101. Zbl0035.35501MR12,417d
  11. [GJ] L. GILLMAN, M. JERISON, Rings of continuous functions, Van Nostrand, Princeton (1960). Zbl0093.30001MR22 #6994
  12. [Gr] A. GROTHENDIECK, Sur les espaces (F) et (DF), Summa Brasiliensis Mathematicae, v. 3 (1954), 57-122. Zbl0058.09803MR17,765b
  13. [Gr2] A. GROTHENDIECK, Produits tensoriels topologiques et espaces nucléaires, Memoirs of American Mathematical Society, n° 16 (1955). Zbl0064.35501MR17,763c
  14. [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. Zbl0182.45402
  15. [H] J. HORVATH, Topological vector spaces and distributions, v. 1., Addison-Wesley, Mass. (1966). Zbl0143.15101MR34 #4863
  16. [Hr] L. HÖRMANDER, Introduction to complex analysis in several variables, Van Nostrand, Princeton (1966). Zbl0138.06203
  17. [L] P. LELONG, Fonctions et applications de type exponentiel dans les espaces vectoriels topologiques, C.R.A.c Paris 169 (1969). Zbl0179.19001
  18. [M1] A. MARTINEAU, Sur les fonctionnelles analytiques et la transformation de Fourier-Borel, Journal d'Analyse Mathématique, v. 11 (1963), 1-164. Zbl0124.31804MR28 #2437
  19. [M2] A. MARTINEAU, Sur la topologie des espaces de fonctions holomorphes, Mathematische Annalen, v. 163 (1966), 62-88. Zbl0138.38101MR32 #8109
  20. [Mt] M.C. MATOS, Holomorphic mappings and domains of holomorphy, Thesis, University of Rochester (1970). Zbl0233.32004
  21. [N1] L. NACHBIN, Topological vector spaces of continuous functions, Proc. Nat. Acd. Sci. USA. v. 40 (1954), 471-4. Zbl0055.09803MR16,156h
  22. [N2] L. NACHBIN, Lectures on topological vector spaces, Lecture note, University of Rochester (1963). 
  23. [N3] L. NACHBIN, Lectures on the theory of distributions, University of Rochester (1963), Reproduced by Universidade do Recife (1964) ; North-Holland Publishing Company (1970). Zbl0135.16401
  24. [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. Zbl0147.11402MR35 #5910
  25. [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). Zbl0212.14604
  26. [N6] L. NACHBIN, Topology on spaces of holomorphic mappings, Ergebnisse der Mathematik und ihrer Grenzgebiete, v. 47 (1969), Springer-Verlag, Berlin. Zbl0172.39902MR40 #7787
  27. [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). 
  28. [N8] L. NACHBIN, Holomorphic functions, domains of holomorphy and local properties, North-Holland Publishing Company (1970). Zbl0208.10301MR43 #558
  29. [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. 
  30. [NG] L. NACHBIN, C.P. GUPTA, On Malgrange's theorem for nuclearly entire functions (to appear). 
  31. [Nr] P. NOVERRAZ, Fonctions plurisousharmonique et analytiques dans les espaces vectoriels topologiques complexes, Annales de l'Institut Fourier, Grenoble, 19,2 (1969), 419-493. Zbl0176.09903MR42 #537
  32. [P] H.R. PITT, A note on bilinear forms, Journal London Math. Society, v. 11 (1936), 174-180. Zbl0014.31201JFM62.0209.01
  33. [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. Zbl0185.20303MR40 #3277
  34. [T] F. TREVES, Topological vector spaces, distributions and kernels, Academic Press, New York and London (1967). Zbl0171.10402MR37 #726
  35. [Z] M.A. ZORN, Characterization of analytic functions in Banach spaces, Annals of Mathematics, 12 (1945), 585-593. Zbl0063.08407MR7,251e

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