Geodesic
Deviation of elements of a Banach space from a system of subspaces
Informatics and Automation, Function spaces and related problems of analysis, Tome 284 (2014), pp. 212-215 
S. V. Konyagin. Deviation of elements of a Banach space from a system of subspaces. Informatics and Automation, Function spaces and related problems of analysis, Tome 284 (2014), pp. 212-215. http://geodesic.mathdoc.fr/item/TRSPY_2014_284_a13/
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     author = {S. V. Konyagin},
     title = {Deviation of elements of {a~Banach} space from a~system of subspaces},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2014_284_a13/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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 $n=1,2,\dots$.

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