On some new sequence spaces
Let \((p_i)\) be a bounded sequence of positive real numbers. Let \(\sigma\) denote a mapping of the set of positive integers into itself. The author defines the sequence spaces \[ \text{ces}^\sigma(p_i):= \Biggl\{ x= (x_i): \sum_m z_{mn}\text{ converges uniforly in } n\Biggr\} \] and \[ \text{ces}^{\sigma\sigma}(p_i):= \Biggl\{x= (x_i) : \sup_n \sum_m z_{mn}< \infty\Biggr\},\quad \text{where } z_{mn}:= {1\over m} \sum^m_{i= 1} |x_{\sigma^i(n)}|^{p_i}. \] Then \(\text{ces}^\sigma(p_i)\subset \text{ces}^{\sigma\sigma}(p_i)\) and \(l_p\subset \text{ces}^\sigma(p)\) for \(p_i= p> 1\) \((i= 1,2,\dots)\). It is shown that \(\text{ces}^\sigma(p_i)\) is a complete linear metric space paranormed by \(g(x)= \sup_n(\sum_m z_{mn})^{1/\max\{1, \sup_i p_i\}}\).
scientific article; zbMATH DE number 897333