# zbMATH — the first resource for mathematics

The unconditional basic sequence problem. (English) Zbl 0827.46008
The authors constructed a Banach space $$X$$ which contains no unconditional basic sequence, where a basic sequence is a basis of its closed linear span and it is unconditional if the basis is independent of its permutation. The counterexample resembles that of Th. Schlumprecht [Isr. J. Math. 76, 81-95 (1991; Zbl 0796.46007)] and is hereditarily indecomposable (H.I.), i.e. it does not contain a decomposable subspace. In particular, a real H.I. space is not isomorphic to its hyperplanes (Banach’s hyperplane problem). The development of the counterexample and the method of proof involved are of independent interest.

##### MSC:
 46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces 46B20 Geometry and structure of normed linear spaces 46B45 Banach sequence spaces
Zbl 0796.46007
Full Text:
##### References:
 [1] C. Bessaga and A. Pełczyński, A generalization of results of R. C. James concerning absolute bases in Banach spaces, Studia Math. 17 (1958), 165 – 174. · Zbl 0084.10001 [2] P. G. Casazza and T. Shura, Tsirelson’s space, Lecture Notes in Math., vol. 1363, Springer Verlag, New York, 1988. · Zbl 0709.46008 [3] N. Dunford and J. Schwartz, Linear operators, Vol. I, Interscience, New York, 1958. · Zbl 0084.10402 [4] Per Enflo, A counterexample to the approximation problem in Banach spaces, Acta Math. 130 (1973), 309 – 317. · Zbl 0267.46012 [5] W. T. Gowers, A solution to Banach’s hyperplane problem, preprint. · Zbl 0838.46011 [6] -, A solution to the Schroeder-Bernstein problem for Banach spaces, preprint. · Zbl 0863.46006 [7] Joram Lindenstrauss, Some aspects of the theory of Banach spaces, Advances in Math. 5 (1970), 159 – 180 (1970). · Zbl 0203.12002 [8] J. Lindenstrauss and A. Pełczyński, Contributions to the theory of the classical Banach spaces, J. Functional Analysis 8 (1971), 225 – 249. · Zbl 0224.46041 [9] Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces, Lecture Notes in Mathematics, Vol. 338, Springer-Verlag, Berlin-New York, 1973. · Zbl 0259.46011 [10] B. Maurey and H. P. Rosenthal, Normalized weakly null sequence with no unconditional subsequence, Studia Math. 61 (1977), no. 1, 77 – 98. · Zbl 0357.46025 [11] Edward Odell and Thomas Schlumprecht, The distortion problem, Acta Math. 173 (1994), no. 2, 259 – 281. · Zbl 0828.46005 [12] Thomas Schlumprecht, An arbitrarily distortable Banach space, Israel J. Math. 76 (1991), no. 1-2, 81 – 95. · Zbl 0796.46007 [13] -, A complementably minimal Banach space not containing $${c_0}$$ or $${\ell _p}$$, preprint. [14] B. S. Tsirelson, Not every Banach space contains $${\ell _p}$$ or $${c_0}$$, Funct. Anal. Appl. 8 (1974), 138-141. · Zbl 0296.46018
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.