# Deviation of elements of a Banach space from a system of subspaces

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## Abstract

We prove that if *X* is a real Banach space, *Y* _{1} ⊂ *Y* _{2} ⊂ ... is a sequence of strictly embedded closed linear subspaces of *X*, and *d* _{1} ≥ *d* _{2} ≥ ... is a nonincreasing sequence converging to zero, then there exists an element *x* ∈ *X* such that the distance *ρ*(*x*, *Y* _{ n }) from *x* to *Y* _{ n } satisfies the inequalities *d* _{ n } ≤ *ρ*(*x*, *Y* _{ n }) ≤ 8*d* _{ n } for *n* = 1, 2, ....

## Keywords

Hilbert Space Banach Space Positive Integer STEKLOV Institute Number Sequence
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