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A hereditarily \(\ell_1\) subspace of \(L_1\) without the Schur property. (English) Zbl 1080.46007
The author constructs a subspace of \(L_1\) without the Schur property such that each closed infinite-dimensional subspace contains a copy of \(\ell_1\); the first examples of this kind were due to J. Bourgain [unpublished] and P. Azimi and J. Hagler [Pac. J. Math. 122, 287–297 (1986; Zbl 0609.46012)]. The virtue of the present example is that it is easily described; indeed, the author considers the subspace \(Z\) of \(\bigoplus_1 \ell_{p_n}\) for \(p_1 > p_2 > \dots >1\) spanned by the vectors \(z_i = \sum_n 2^{-n} e_{i,n}\) where \((e_{i,n})_i\) denotes the standard unit vector basis of \(\ell_{p_n}\). He shows that \(Z\) has the aforementioned properties; it embeds into \(L_1\) if \(p_1\leq2\).

MSC:
46B03 Isomorphic theory (including renorming) of Banach spaces
46B25 Classical Banach spaces in the general theory
Citations:
Zbl 0609.46012
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[1] Parviz Azimi and James N. Hagler, Examples of hereditarily \?\textonesuperior Banach spaces failing the Schur property, Pacific J. Math. 122 (1986), no. 2, 287 – 297. · Zbl 0609.46012
[2] J. Bourgain, \(\ell_1\)-subspaces of Banach spaces. Lecture notes. Free University of Brussels.
[3] J. Bourgain and H. P. Rosenthal, Martingales valued in certain subspaces of \?\textonesuperior , Israel J. Math. 37 (1980), no. 1-2, 54 – 75. · Zbl 0445.46015
[4] William B. Johnson and Joram Lindenstrauss, Basic concepts in the geometry of Banach spaces, Handbook of the geometry of Banach spaces, Vol. I, North-Holland, Amsterdam, 2001, pp. 1 – 84. · Zbl 1011.46009
[5] Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Springer-Verlag, Berlin-New York, 1977. Sequence spaces; Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 92. · Zbl 0362.46013
[6] Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Springer-Verlag, Berlin-New York, 1979. Function spaces. · Zbl 0403.46022
[7] H. P. Rosenthal, Convolution by a biased coin, Altgeld Book (Univ. of Illinois),II, 1975/76.
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