Holomorphic functions on locally closed convex sets and projective descriptions
A subset \(Q\) of \({\mathbb C}^N\) is locally closed if for each \(z\in Q\) there is a closed neighborhood \(U\) of \(z\) in \({\mathbb C}^N\) such that \(Q\cap U\) is closed. Every open subset and every compact subset of \({\mathbb C}^N\) is locally closed. A convex locally closed set \(Q\) admits an increasing fundamental sequence \((Q_n)_n\) of convex compact sets. Let \({\mathcal V}=(v_{n,k})_{n,k\in\mathbb N}\) be the double sequence of strictly positive upper semicontinuous functions on \({\mathbb C}^N\) given by \(v_{n,k}(z):=\exp (-H_n(z)-| z| /k)\), \(n,k\in{\mathbb N}\), \(z\in {\mathbb C}^N\), where \(H_n\) is the support function of the convex compact set \(Q_n\), \(n\in{\mathbb N}\). The weighted inductive limit of Fréchet spaces \( VH({\mathbb C}^N)\) of entire functions associated with \(V\) is defined by \[ VH({\mathbb C}^N):=\text{ ind}_n\text{ proj}_kH(v_{n,k},{\mathbb C}^N), \] where the steps \(H(v,{\mathbb C}^N)\) are defined, for a positive function \(v\) on \({\mathbb C}^N\), as the Banach space of entire functions \[ H(v,{\mathbb C}^N):=\{f\in H({\mathbb C}^N):\;\| f\| _v:=\sup_{z\in {\mathbb C}^N}v(z)| f(z)| <\infty\}. \] The space \( VH({\mathbb C}^N)\) is a Hausdorff (LF)-space. Its projective hull \(H\bar{V}({\mathbb C}^N)\) is defined by \[ H\bar{V}({\mathbb C}^N):=\{f\in H({\mathbb C}^N):\;\| f\| _{\bar{v}}:=\sup_{z\in {\mathbb C}^N}\bar{v}(z)| f(z)| <\infty \;\forall \bar{v}\in \bar{V}\}, \] where \(\bar{V}\) is the system of all those functions \(\bar{v}\colon {\mathbb C}^N\to [0,\infty)\) which are upper semicontinuous on \({ \mathbb C}^N\) and have the property that for each \( n\) there are \( k=k(n)\) and \(\alpha_n>0\) such that \(\bar{v}\leq \alpha_n v_{n,k}\) on \({\mathbb C}^N\). The projective hull \(H\bar{V}({\mathbb C}^N)\) is a complete locally convex space with respect to the Hausdorff locally convex topology defined by the system of seminorms \(\{\| \;\| _{\bar{v}}:\;\bar{v}\in \bar{V}\}\), and \(VH({\mathbb C}^N)\) is contained in its projective hull with continuous inclusion. In the paper under review, the authors consider the problem of projective description of the (LF)-space \(VH({\mathbb C}^N)\), i.e., the problem to determine conditions under which the spaces \(VH({\mathbb C}^N)\) and \(H\bar{V}({\mathbb C}^N)\) coincide algebraically, or the space \(VH({\mathbb C}^N)\) is a topological subspace of its projective hull \(H\bar{V}({\mathbb C}^N)\). Precisely, they show that: (1) If a convex locally closed set \(Q\) in \({\mathbb C}^N\) is bounded and strictly convex at \(\partial_r\omega\) (i.e., the intersection of \(Q\) with each supporting hyperplane to \(\bar{Q}\) is compact), then the spaces \(VH({\mathbb C}^N)\) and \(H\bar{V}({\mathbb C}^N)\) coincide algebraically and topologically. (2) Suppose that a convex locally closed set \(Q\) in \({\mathbb C}^N\) is bounded, nonpluripolar, and that it has a neighborhood basis of domains of holomorphy. If \(VH({\mathbb C}^N)\) is a topological subspace of \(H\bar{V}({\mathbb C}^N)\), then \(Q\) is strictly convex at \(\partial_r\omega\). (3) Let \(Q\) be a bounded convex locally closed subset of \({\mathbb C}^N\). If there is a supporting hyperplane \(\Pi\) to \(\bar{Q}\) such that \(\Pi\cap Q\not =\emptyset\) and there exists \(z_0\in (\Pi\cap \bar{Q})\setminus Q\) which is a smooth point of \(\partial Q\), then \(VH({\mathbb C}^N)\not = H\bar{V}({\mathbb C}^N)\) and the converse also holds in the case \(N=1\).
scientific article; zbMATH DE number 2118938