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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 $$n$$).

##### MSC:
 46A45 Sequence spaces (including Köthe sequence spaces)
##### Keywords:
Orlicz function; sequence space; complete paranormed space