Spaceability and algebrability in the theory of domains of existence in Banach spaces (Q496385)

Let \(U\) be an open subset of a separable infinite-dimensional complex Banach space \(E,\) let \(\mathcal H(U)\) denote the algebra of \(\mathbb{C}\)-valued holomorphic functions on \(U,\) and let \(\mathcal E(U) = \{f \in \mathcal H(U) \;| \;U\) is the domain of existence of \(f \}\). The author's principal interest lies in studying algebraic and topological properties of \(\mathcal E(U)\). For this, following an argument of \textit{J. Mujica} and \textit{D. M. Vieira} [Rev. Mat. Complut. 23, No. 2, 435--452 (2010; Zbl 1211.46036)] (see also [\textit{Y. Hervier}, Lect. Notes Math. 364, 157--167 (1974; Zbl 0284.46017)]), he shows that if \(U\) is a domain of existence, then whenever \((x_n)_n \subset U\) is a sequence of distinct points whose boundary distance tends to \(0\) and whenever \((\alpha_n)_n\) is an arbitrary sequence in \(\mathbb C,\) there is \(f \in \mathcal H(U)\) such that \(f(x_n) = \alpha_n\) for all \(n\). In addition, two other algebrability properties are described, where \(U \subset E\) is a connected domain of existence. First, the set \(\mathcal E(U) \cup \{0\}\) contains a closed subalgebra which, in turn, contains an infinite algebraically independent set of generators. Second, the set \(\mathcal E(U) \cup \{0\}\) contains a dense subalgebra which, in turn, contains an infinite algebraically independent set. Reviewer's remark. See also the work of \textit{M. Valdivia} for domains of existence in \(\mathbb{C}^n\), such as [``On certain spaces of holomorphic functions'', in: Proceedings of the 2nd international school ``Advanced courses of mathematical analysis 2'', Granada, Spain, September 20--24, 2004. Hackensack, NJ: World Scientific. 151--173 (2007; Zbl 1146.46015)].