On the subspace \(\Gamma (\lambda)\) of entire functions (Q1287052)

Let \(\Gamma\) be the topological space of entire functions \(a(z)=\sum_{n=0}^\infty a_nz^n\) with the topology \(T\) generated by the metric \(d(a,b)=\sup\{| b_0-a_0|, | b_n-a_n| ^{1/n}, n=1,2,\dots\}\). Let \(\lambda=\{l_0,l_1,\dots\}\) be a fixed sequence having no zero elements. Let \(T_\lambda\) be the topology generated by the metric \(d_\lambda(a,b)=\sup\{| l_0| | b_0-a_0| , | l_n| ^{1/n}| b_n-a_n| ^{1/n}, n=1,2,\dots\}\). The space \(\Gamma\) can be treated as the space of sequences \(\{a_0,a_1,\dots\}\). Let \(\Omega\) be the space of all complex sequences \(w=\{x_0,x_1,\dots\}\). Denote \(\lambda(\omega)=\{l_0x_0,l_1x_1,\dots\}\), \(\Gamma(\lambda)=\{a\in\Gamma :\lambda(a)\in\Gamma\}\), \(\overline{\Gamma}(\lambda)=\{\omega\in\Omega : \lambda(\omega)\in\Gamma\}\). The space \(\Gamma\) was considered by Ganapathy Iyer (1948), the space \(\Gamma(\lambda)\) was considered by Titus (1979). The author studies properties of the spaces \(\Gamma\), \(\Gamma(\lambda)\), \(\overline{\Gamma}(\lambda)\). In particular, he proves that the following properties are equivalent: 1) \(\Gamma=\Gamma(\lambda)\), 2) \(\Gamma=\overline{\Gamma}(\lambda)\), 3) the restrictions of the topologies \(T\) and \(T_\lambda\) to \(\Gamma(\lambda)\) are the same.