Geodesic
On the existence of an element with given deviations from an expanding system of subspaces
Matematičeskie zametki, Tome 80 (2006) no. 5, pp. 657-667.

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The problem as to whether a Banach space contains an element with given deviations from an expanding system of strictly nested subspaces (which are not necessarily finite-dimensional) is solved under additional restrictions on the deviations or subspaces. 
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P. A. Borodin. On the existence of an element with given deviations from an expanding system of subspaces. Matematičeskie zametki, Tome 80 (2006) no. 5, pp. 657-667. http://geodesic.mathdoc.fr/item/MZM_2006_80_5_a1/

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