Constructing an Element of a Banach Space with Given Deviation from its Nested Subspaces

@article{Aksoy2016ConstructingAE,
  title={Constructing an Element of a Banach Space with Given Deviation from its Nested Subspaces},
  author={Asuman G{\"u}ven Aksoy and Qidi Peng},
  journal={arXiv: Functional Analysis},
  year={2016}
}
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… 
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