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Zbl 0969.46032
Lempert, László
Approximation of holomorphic functions of infinitely many variables. II. (Approximation de fonctions holomorphes d'un nombre infini de variables. II.)
(French)
[J] Ann. Inst. Fourier 50, No.2, 423-442 (2000). ISSN 0373-0956; ISSN 1777-5310/e

Let $X$ be a Banach space. An unconditional basis of $X$ is a subset $(e_\gamma)_{\gamma \in \Gamma}$ with the property that any $x \in X$ can be represented uniquely as $x = \sum_{\gamma \in \Gamma} x(\gamma) e_\gamma$. Typical examples of spaces with an unconditional basis are $\ell^p(\Gamma)$, $1 \leq p < \infty$. The main result of the paper under review is a solution of the following approximation problem for a Banach space $X$ with a countable unconditional basis or a space $\ell^p(\Gamma)$, $\Gamma$ arbitrary: \par Let $V$ be a sequentially complete locally convex space and $f : B(R) \to V$ a holomorphic function on the open ball $B(R)$ of radius $R$. Then for each $r \in ]0,R[$, $\epsilon > 0$ and each continuous seminorm $\psi$ on $V$, there exists a holomorphic function $h : X \to V$ such that $\psi(f - h) < \varepsilon$ on $B(r)$. \par As a consequence of this result, it follows from results in [{\it L. Lempert}, Invent. Math. 142, No. 3, 579-603 (2000)] that if $V$ is a Fréchet space and ${\cal V}$ the sheaf of germs of $V$-valued holomorphic functions on an open pseudoconvex subset $\Omega \subseteq X$, then $H^q(\Omega, {\cal V}) = 0$ for $q \geq 1$. \par [For part I see Ann. Inst. Fourier 49, No.~4, 1293-1304 (1999; Zbl 0944.46046)].
[Karl -Hermann Neeb (Darmstadt)]
MSC 2000:
*46G20 Infinite dimensional holomorphy
46E50 Spaces of holomorphic functions on infinite-dimensional spaces
32A05 Power series, etc. (several complex variables)

Keywords: holomorphic functions; Banach space; pseudoconvex domain; approximation problem; unconditional basis; Fréchet space; sheaf of germs

Citations: Zbl 0944.46046

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