Sequence spaces defined by Orlicz functions. (English) Zbl 0802.46020

Let \(M\) be an Orlicz function and let \((p_ k)\) be a bounded sequence of positive real numbers. The authors define the sequence space \(\ell_ M(p)= \{(x_ k)\): \(\sum_{k=1}^ \infty (M (| x_ k|/\rho))^{p_ k} <\infty\) for some \(\rho> 0\}\) and prove that for \(1\leq p_ k< \infty\), \(\ell_ M(p)\) is a complete paranormed space. Also, it is shown that if \(0<p_ k \leq q_ k\) and \((q_ k/p_ k)\) is bounded, then \(W(M,q) \subseteq W(M,p)\) where \[ W(M,p)= \left\{ (x_ k):\;{1\over n} \sum_{k=1}^ n \Biggl( M \biggl( {{| x_ k- \ell|} \over \rho} \biggr) \Biggr)^{p_ k} \to \text{ as } n\to\infty \right\} \text{ for some } \rho \text{ and } \ell>0. \] Some similar results appear, and there are no new techniques. (In the proof of (i) of Theorem 6, the summation should be from 1 to \(n\)).


46A45 Sequence spaces (including Köthe sequence spaces)