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Banach spaces with a supershrinking basis. (English) Zbl 0939.46011
Let \(X\) be a Banach space with a normalized and shrinking basis \((e_n)\) and with associated functionals \((f_n)\). Let us suppose that \(X\) has no infinite-dimensional reflexive subspaces. There are proved various necessary and sufficient conditions in order that \((e_n)\) be supershrinking. Also, it is proved that a non-reflexive Banach space \(X\) with a normalized supershrinking basis \((e_n)\) and associated functionals \((f_n)\), without subspaces associated to \(c_0\), contains order-one quasireflexive subspaces.

46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
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