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Banach spaces without a local basis structure. (English. Russian original) Zbl 0682.46005
Math. Notes 43, No. 2, 124-129 (1988); translation from Mat. Zametki 43, No. 2, 220-228 (1988).
The paper contains solutions of some questions on local basis structure (LBS) posed by S. J. Szarek [Banach spaces without bases which have the bounded approximation products, preprint (1985)]. (A separable Banach space is said to have the local basis structure iff it can be represented as a closure of an increasing sequence of finite dimensional subspaces having uniformly bounded basis constants.) The main results are:
1. There exists a Banach space with bounded approximation property (BAP) and with LBS but without basis.
2. There exists a Banach space with BAP but without LBS and without any non-trivial type.
3. For any \(q>2\) there exists a Banach space with BAP but without LBS and without cotype q.
Reviewer’s remark. The first and the second results are obtained also by S. J. Szarek [Acta Math. 159, No.1-2, 81-98 (1987; Zbl 0637.46013)].
Reviewer: I.V.Ostrovskii
MSC:
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
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References:
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[2] S. J. Szarek, ?The finite-dimensional basis problem with an appendix on nets of Grassmann manifolds,? Acta Math.,151, No. 3-4, 153-179 (1983). · Zbl 0554.46004 · doi:10.1007/BF02393205
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