Approximation of holomorphic functions of infinitely many variables II
Annales de l'Institut Fourier, Tome 50 (2000) no. 2, pp. 423-442.

Soit X un espace de Banach et B(R)⊂X la boule de rayon R centrée en 0. Étant donnés 0<r<R,ε>0 et une fonction f holomorphe dans B(R), existe-t-il toujours une fonction g, holomorphe dans X, telle que |f-g|<ε sur B(r) ? On démontre que c’est bien le cas pour une certaine classe d’espaces, en particulier pour la plupart des espaces de Banach classiques.

Let X be a Banach space and B(R)⊂X the ball of radius R centered at 0. Can any holomorphic function on B(R) be approximated by entire functions, uniformly on smaller balls B(r)? We answer this question in the affirmative for a large class of Banach spaces.

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     author = {Lempert, L\'aszl\'o},
     title = {Approximation of holomorphic functions of infinitely many variables {II}},
     journal = {Annales de l'Institut Fourier},
     pages = {423--442},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {50},
     number = {2},
     year = {2000},
     doi = {10.5802/aif.1760},
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Lempert, László. Approximation of holomorphic functions of infinitely many variables II. Annales de l'Institut Fourier, Tome 50 (2000) no. 2, pp. 423-442. doi : 10.5802/aif.1760. http://www.numdam.org/articles/10.5802/aif.1760/

[D1] S. Dineen, Cousin's first problem on certain locally convex topological vector spaces, An. Acad. Brasil. Cienc., 48 (1976), 11-12. | MR | Zbl

[D2] S. Dineen, Complex Analysis in Locally Convex Spaces, North Holland, Amsterdam, 1981. | MR | Zbl

[D3] S. Dineen, Complex Analysis on Infinite Dimensional Spaces, Springer, Berlin, 1999. | MR | Zbl

[DS] N. Dunford, T. Schwartz, Linear Operators I, John Wiley & Sons, New York, 1988.

[L1] L. Lempert, Approximation de fonctions holomorphes d'un nombre infini de variables, Ann. Inst. Fourier, 49-4 (1999), 1293-1304. | Numdam | MR | Zbl

[L2] L. Lempert, The Dolbeault complex in infinite dimensions, II, J. Amer. Math. Soc., 12 (1999), 775-793. | MR | Zbl

[L3] L. Lempert, The Dolbeault complex in infinite dimensions III, manuscript.. | Zbl

[M] P. Mazet, Analytic Sets in Locally Convex Spaces, North Holland, Amsterdam, 1984. | MR | Zbl

[MV] R. Meise and D. Vogt, Counterexamples in holomorphic functions on nuclear Fréchet spaces, Math. Z., 182 (1983), 167-177. | MR | Zbl

[N] P. Noverraz, Pseudo-convexité polynomiale et domaines d'holomorphie en dimension infinie, North Holland, Amsterdam, 1973. | MR | Zbl

[P] I. Patyi, On the ∂-equation in a Banach space, Bull. Soc. Math. France, to appear. | Numdam | Zbl

[R] R. A. Ryan, Holomorphic mappings in l1, Trans. Amer. Soc., 302 (1987), 797-811. | MR | Zbl

[S] I. Singer, Bases in Banach spaces I-II, Springer, Berlin, 1981. | MR | Zbl

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