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Some paranormed Riesz sequence spaces of non-absolute type. (English) Zbl 1118.46009

Let \(t=( t_{k}) \) be a sequence of positive numbers and \(T_{n}:=t_{0}+\dots +t_{n}\) \(( n\in\mathbb{N}) \). For the weighted mean transformation \(y_{n}:=T_{n}^{-1}\sum_{k=0}^{n}t_{k}x_{k},\) the authors introduce the Riesz sequence spaces \(r_{\infty}^{t}( p) :=\{ ( x_{k}) \mid\sup_{n}| y_{n}| ^{p_{n}}<\infty\} \), \(r_{0}^{t}( p) :=\{ ( x_{k}) \mid\lim_{n}| y_{n}| ^{p_{n}}=0\} \) and \(r_{c}^{t}( p) :=r_{0}^{t}( p) \oplus \) span\(\{ ( 1,1,\dots ) \} \) (here \(p=( p_{n}) \) is a bounded sequence with \(p_{n}>0)\) and study some of their properties, such as completeness with respect to a paranorm and characterization of the continuous and the Köthe-Toeplitz duals. It is proved that these spaces are linearly isomorphic to the complete paranormed sequence spaces of Maddox, \(\ell^{\infty}( p) ,\) \(c_{0}( p) \) and \(c( p) \), respectively. Some matrix mappings related to the Riesz spaces are characterized.

MSC:

46A45 Sequence spaces (including Köthe sequence spaces)
46A35 Summability and bases in topological vector spaces
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