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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 \(f\in BV(\mathbb{R})\).

MSC:

46A45 Sequence spaces (including Köthe sequence spaces)
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References:

[1] A. Honda, Y. Okazaki and H. Sato, An \(L_{p}\)-function determines \(l_{p}\), Proc. Japan Acad. Ser. A Math. Sci. 84 (2008), no. 3, 39-41. · Zbl 1142.26314 · doi:10.3792/pjaa.84.39
[2] A. Honda, Y. Okazaki and H. Sato, A new sequence space defined by an \(L_{2}\)-function, in Banach and Function Spaces III (held at Kyushu Institute of Technology (KIT), Tobata Campus, Kitakyushu, JAPAN on September 14-17, 2009) , Proceedings of the Third International Symposium on Banach and Function Spaces 2009, Yokohama Publishers, Yokohama. (to appear).
[3] G. Leoni, A first course in Sobolev spaces , Graduate Studies in Mathematics, 105, Amer. Math. Soc., Providence, RI, 2009. · Zbl 1180.46001
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