Some new sequence spaced defined by a modulus function (Q1327740)

Let \(\sigma: \mathbb{N}\to \mathbb{N}\) be an one-to-one mapping such that \(\sigma^ k (n)\neq n\) for all \(n\) and \(k\). Let \(f\) be a modulus function, and let \(A\) be a nonnegative regular matrix. The authors prove that \(w(A_ \sigma, f):= \{x= (x_ k)\mid \lim_ m \sum_ k a_{mk} f(| x_{\sigma^ \kappa(n)} -L|) =0\), for some \(L\) uniformly in \(n\}\) and \(w_ 0(A_ \sigma, f):= \{x\in w(A_ \sigma, f)\mid L=0\}\) are complete linear topological spaces paranormed by \(g(x):= \sup_{n,m} \sum_ k a_{mk} f(| x_{\sigma^ \kappa(n)} |)\).