# zbMATH — the first resource for mathematics

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infinite-dimensional subspace contains a copy of $$\ell_1$$; the first examples of this kind were due to J. Bourgain [unpublished] and P. Azimi and J. Hagler [Pac.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
##### Keywords:
Schur property; hereditarily-$$\ell_1$$ Banach space
Zbl 0609.46012
Full Text:
##### References:
 [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.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.