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A class of Banach sequence spaces analogous to the space of Popov. (English) Zbl 1224.46017
Summary: J. N. Hagler and P. Azimi [Pac. J. Math. 122, 287–297 (1986; Zbl 0609.46012)] introduced a class of hereditarily \(l_{1}\) Banach spaces which do not possess the Schur property. Then, the first author extended these spaces to a class of hereditarily \(l_{p}\) Banach spaces for \(1\leq p<\infty \). Here, we use these spaces to introduce a new class of hereditarily \(l_{p}(c_{0})\) Banach spaces analogous to the space of Popov. In particular, for \(p=1\), the spaces are further examples of hereditarily \(l_{1}\) Banach spaces failing to have the Schur property.

MSC:
46B20 Geometry and structure of normed linear spaces
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Citations:
Zbl 0609.46012
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References:
[1] P. Azimi: A new class of Banach sequence spaces. Bull. of Iranian Math. Society 28 (2002), 57–68. · Zbl 1035.46006
[2] P. Azimi and J. Hagler: Examples of hereditarily 1 Banach spaces failing the Schur property. Pacific J. Math. 122 (1986), 287–297. · Zbl 0609.46012
[3] J. Bourgain: 1-subspace of Banach spaces. Lecture notes. Free University of Brussels.
[4] J. Lindenstrauss and L. Tzafriri: Classical Banach Spaces. Vol. I sequence Spaces, Springer Verlag, Berlin. · Zbl 0362.46013
[5] M. M. Popov: A hereditarily 1 subspace of L 1 without the Schur property. Proc. Amer. Math. Soc. 133 (2005), 2023–2028. · Zbl 1080.46007
[6] M. M. Popov: More examples of hereditarily p Banach spaces. Ukrainian Math. Bull. 2 (2005), 95–111.
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