Topologization of entire sequences spaces (Q2730531)

This article deals with the locally convex sequence spaces of entire functions. Denote by \(\omega\) the space of all sequences \(x=(x_{n}), n\in\{0,1,2,\ldots\}\), \(x_{n}\) is a complex number. Let \(\varphi\) be a space of all finite sequences. If \(E\) is a subspace of \(\omega\) and \(\varphi\subset E\), then the bilinear form \((x,y)\to\langle x,y\rangle=\sum x_{n}y_{n}, x\in\varphi, y\in E\) defines the dual pair \((\varphi,E)\). By \(\tau(\varphi,E)\) we denote the strongest topology which is consistent with duality \((\varphi,E)\), and by \(\beta (\varphi,E)\) we denote the strongest topology of uniform convergence. The author proves that for any space \(E\) such that \(\varphi\subset E\subset \omega\) the topological space \((\varphi,\beta(\varphi,E))\) coincides with the space \((\varphi,\beta(\varphi,\omega))=(\varphi,\tau(\varphi,\omega))\) and it is a complete space. The topology \(\beta(\varphi,\omega)\) is stronger then \(\sigma(\varphi,E)\). Here \(\sigma(\varphi,E)\) is the weakest topology in which the pre-norm \(x\mapsto|\langle x,y\rangle|\) is continuous for each \(y\in E\). Also the topologies \(\tau(E,F)\), \(\beta(E,F)\) are studied, where spaces \(E, F\) are such that \(\varphi\subset E\subset I\), \(\varphi\subset F\subset E^{*}\), \(I\) is a space of all entire sequences, \(E^{*}\) is a dual space for \(E\).