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A new class of sequences related to the $l_{p}$ spaces defined by sequences of Orlicz functions. (English) Zbl 1236.46006
A new sequence space $m(M,\phi,q,\Lambda)$ is defined using an Orlicz function, seminorms and $\lambda$-sequences in order to generalize the space $m(\phi)$, introduced and studied by W.L.C. Sargent in 1960. Later on, many authors defined several other spaces in comparison with the space $m(\phi)$. First, the author investigates the linearity, examines solidity and monotonicity and gives some inclusion results involving the spaces $m(M,\phi,q,\Lambda)$. Then the relation between the space $m(M,\phi,q,\Lambda)$ and the space of $S^0_{\theta}(\phi,\Lambda)$-statistically convergent sequences is studied. In the last part, the relation between the space $m(M,\phi,q,\Lambda)$ and the space $m^c_{\theta}(M,\phi,q,\Lambda)$ of Cesàro convergence type sequences is given. Reviewer’s remark: Certain properties like solid space, monotonicity of the space $m(M,\phi,q,\Lambda)$ are investigated without any specific aim and almost all results are generalized versions of other results already studied by others.

46A45Sequence spaces
40A05Convergence and divergence of series and sequences
40G05Cesàro, Euler, Nörlund and Hausdorff methods
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