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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 M (p)={(x k ): k=1 (M(|x k |/ρ)) p k < for some ρ>0} and prove that for 1p k <, M (p) is a complete paranormed space. Also, it is shown that if 0<p k q k and (q k /p k ) is bounded, then W(M,q)W(M,p) where

W(M,p)=(x k ):1 n k=1 n M |x k -| ρ p k asnforsomeρand>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).

46A45Sequence spaces