Some new sequence spaces defined by a modulus function (Q2708397)

A modulus is a non-negative increasing function \(f\) defined on \([0,\infty)\) which satisfies: \(f(x)=0\) iff \(x=0,\) \(f(x+y)\leq f(x)+f(y),\) and \(f\) is right continuous at 0. Let \(p=\{p_{k}\}\) be a bounded sequence of positive real numbers and \(s\geq 0\). The author defines the following sequence spaces which generalize sequence spaces introduced by I. Maddox and M. Basarir: \(\ell_{\infty }(p,f,s)=\{\{t_{k}\}:\sup_{k}k^{-s}f(|t_{k}|)^{p_{k}}<\infty \},\) \(c_{0}(p,f,s)=\{\{t_{k}\}:k^{-s}f(|t_{k}|)^{p_{k}}\rightarrow 0\},\) \(c(p,f,s)=\{\{t_{k}\}:\exists \) \(L\) such that \(k^{-s}f(|t_{k}-L|)^{p_{k}}\rightarrow 0\}\). The author shows that each of these spaces is a linear space which carries a natural paranorm under which they are complete. Some inclusion results are also obtained.