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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.

##### MSC:
 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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