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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.

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: DOI
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