On a function space as another generalization of Iyer's space of integral functions (Q1090893)

We define and study the function space \[ E(X,{\mathbb{C}},s)=\{\alpha: X\to {\mathbb{C}}| \sum_{x\in X}| \alpha (x)| R^{s(x)}<\infty \text{ for each }R,\quad 0<R<\infty\}, \] wherein X is a set, N is the set of non-negative integers and \({\mathbb{C}}\) is the field of complex numbers and \(s: X\to N\), as a generalization of the space of integral functions studied by \textit{V. Ganapathy Iyer}; J. Indian Math. Soc., II. Ser. 12, 13-30 (1948; Zbl 0031.12802), Quart. J. Math., Oxford II. Ser. 1, 86-96 (1950; Zbl 0037.205). [cf. the generalization studied by \textit{R. S. Kankurikar}, Math. Forum, (Dibrugarh) India 1, 18-21 (1978), J. Indian Math. Soc. 44, 379-393 (1980)]. By considering the natural generalization of linear topology on E(X,\({\mathbb{C}},s)\) of Iyer's space, we characterize normability, continuous linear functionals, weak and strong convergences in this space and in its dual space.