Open sets with the Runge property in Banach spaces
Let \(E\) and \(F\) be complex Banach spaces. For an open subset \(U\) of \(E\), consider the vector space \({\mathcal H}(U;F)\) of all \(F\)-valued holomorphic mappings on \(U\) equipped with Nachbin's ported topology \(\tau _\omega \). If \(K\) is a compact subset of \(E\), then \({\mathcal H}(K;F)\) denotes the vector space of all \(F\)-valued holomorphic germs on \(K\). The topology \(\tau _\omega \) is defined by the locally convex inductive limit \[ ({\mathcal H}(K;F),\tau _\omega) = \mathop{\text{ind}}\limits_{U \supset K}({\mathcal H}(U;F),\tau _\omega), \] where \(U\) is open in \(E\). In [Topology on spaces of holomorphic mappings, Berlin-Heidelberg-New York: Springer-Verlag (1969; Zbl 0172.39902)], \textit{L. Nachbin} conjectured that the space \(({\mathcal H}(U;F),\tau _\omega)\) coincides with the locally convex projective limit of the spaces \(({\mathcal H}(K;F),\tau _\omega)\) with \(K\subset U\) compact. In 1971, \textit{S. B. Chae} [Ann. Inst. Fourier 21, No.~3, 107--141 (1971; Zbl 0222.46018)] proved Nachbin's conjecture under the hypothesis that the open set \(U\) has the \(F\)-valued Runge property. This means that each compact subset \(K\) of \(U\) is contained in some compact subset \({\tilde K}\) of \(U\) with the property that \({\mathcal H}(U;F)\) is sequentially dense in \(({\mathcal H}({\tilde K};F),\tau _\omega)\). \textit{M. Schottenloher} [Infin. Dim. Holom. Appl., Proc. int. Symp., Univ. Estadual de Campinas/Brasil 1975, 379--391 (1977; Zbl 0397.46044)] showed that every pseudoconvex open subset of a finite-dimensional Banach space has the \({\mathbb C}\)-valued Runge property. The first author has extended Schottenloher's result to the case of pseudoconvex open subsets of Fréchet-Schwartz spaces with a Schauder basis in 1984 and to the case of pseudoconvex Riemann domains over Fréchet-Schwartz spaces with a Schauder basis in 1985. In this paper, the authors improve Schottenloher's result by proving that every pseudoconvex open subset of a Banach space with an unconditional Schauder basis has the \(F\)-valued Runge property for every Banach space \(F\). The proof rests on an extension theorem of \textit{L. Lempert} and \textit{I. Patyi} [Ann. Sci. Éc. Norm. Supér. (4) 40, No.~3, 453-486 (2007; Zbl 1135.32008)], in tandem with an approximation theorem of \textit{L. Lempert} [Ann. Inst. Fourier 50, No.~2, 423--442 (2000; Zbl 0969.46032)].
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