%0 Journal Article %T Constructing an Element of a Banach Space with Given Deviation from its Nested Subspaces %J Khayyam Journal of Mathematics %I Department of Pure Mathematics, Ferdowsi University of Mashhad (in cooperation with the Center of Excellence in Analysis on Algebraic Structures) %Z 2423-4788 %A Aksoy, Asuman %A Peng, Qidi %D 2018 %\ 01/01/2018 %V 4 %N 1 %P 59-76 %! Constructing an Element of a Banach Space with Given Deviation from its Nested Subspaces %K Best approximation %K Bernstein's lethargy theorem %K Banach space %K Hahn-Banach theorem %R 10.22034/kjm.2018.55158 %X This paper contains two improvements on a theorem of S. N. Bernstein for Banach spaces. We show that if $X$ is an arbitrary infinite-dimensional Banach space, $\{Y_n\}$ is a sequence of strictly nested subspaces of $ X$ and if $\{d_n\}$ is a non-increasing sequence of non-negative numbers tending to 0, then for any $c\in(0,1]$ we can find $x_{c} \in X$, such that the distance $\rho(x_{c}, Y_n)$ from $x_{c}$ to $Y_n$ satisfies$$c d_n \leq \rho(x_{c},Y_n) \leq 4c d_n,~\mbox{for all $n\in\mathbb N$}.$$We prove the above inequality by first improving Borodin (2006)'s result for Banach spaces by weakening his condition on the sequence $\{d_n\}$. The weakened condition on $d_n$ requires refinement of Borodin's construction to extract an element in $X$, whose distances from the nested subspaces are precisely the given values $d_n$. %U https://www.kjm-math.org/article_55158_6967a156928a4b5003d50eae0fedc911.pdf