On the linearity of some sets of sequences defined by \(L_{p}\)-functions and \(L_{1}\)-functions determining \(\ell_{1}\). (English) Zbl 1252.46004

Summary: We discuss the linearity of a sequence space \(\Lambda_{p}(f)\), and conditions such that \(\ell_{1} = \Lambda_{1}(f)\) holds are characterized in terms of the essential bounded variation of \(f\in L_{1}(\mathbb{R})\), i.e., \(\ell_{1} = \Lambda_{1}(f)\) if and only if \(f\in BV(\mathbb{R})\).


46A45 Sequence spaces (including Köthe sequence spaces)
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