Geodesic
Deviation of elements of a~Banach space from a~system of subspaces
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function spaces and related problems of analysis, Tome 284 (2014), pp. 212-215.

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We prove that if $X$ is a real Banach space, $Y_1\subset Y_2\subset\dots$ is a sequence of strictly embedded closed linear subspaces of $X$, and $d_1\ge d_2\ge\dots$ is a nonincreasing sequence converging to zero, then there exists an element $x\in X$ such that the distance $\rho(x,Y_n)$ from $x$ to $Y_n$ satisfies the inequalities $d_n\le\rho(x,Y_n)\le8d_n$ for $n=1,2,\dots$. 
@article{TM_2014_284_a13,
     author = {S. V. Konyagin},
     title = {Deviation of elements of {a~Banach} space from a~system of subspaces},
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S. V. Konyagin. Deviation of elements of a~Banach space from a~system of subspaces. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function spaces and related problems of analysis, Tome 284 (2014), pp. 212-215. http://geodesic.mathdoc.fr/item/TM_2014_284_a13/

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