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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
Citations:
Zbl 0796.46007
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References:
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