On some vector valued sequence spaces defined using a modulus function
Let \(\{E_k\}\) be a sequence of complex Banach spaces with norms \(\|\cdot\|_{E_k}\) \((k=1,2,3,\dots)\) and \(F\) a normal sequence space with monotone paranorm \(g_F\) such that coordinatewise convergence implies convergence in paranorm and let \(f\) be a modulus function. Define \[ F(E_k, f)= \{(x_k): x_k\in E_k\;\forall k\text{ and }(f(\|x_k\|_{E_k}))\in F\}. \] The authors, among other results, prove the following theorem: \(F(E_k, f)\) is a complete paranormed space under the paranorm \(g(x)= g_F[(f(\|x_k\|_{E_k}))]\) for \(x= (x_k)\) belonging to \(F(E_k, f)\).
scientific article; zbMATH DE number 1359899