# 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}
}
• Published 15 May 2016
• Mathematics
• arXiv: Functional Analysis
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 $n\in\mathbb… 3 Citations • Mathematics • 2018 In this paper, we prove the equivalence of reflexive Banach spaces and those Banach spaces which satisfy the following form of Bernstein's Lethargy Theorem. Let$X$be an arbitrary In this paper, we examine the aptly-named "Lethargy Theorem" of Bernstein and survey its recent extensions. We show that one of these extensions shrinks the interval for best approximation by half In this paper, we examine the aptly-named “Lethargy Theorem” of Bernstein and survey its recent extensions. We show that one of these extensions shrinks the interval for best approximation by half ## References SHOWING 1-10 OF 27 REFERENCES • Mathematics • 2016 In this paper, we consider a condition on subspaces in order to improve bounds given in the Bernstein's Lethargy Theorem (BLT) for Banach spaces. Let$d_1 \geq d_2 \geq \dots d_n \geq \dots > 0\$ be
In a recent paper E. J. McShane [3]2 has given a theorem which is the common core of a variety of results about Baire sets, Baire functions, and convex sets in topological spaces including groups and
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Replacing the nested sequence of "finite" dimensional subspaces by the nested sequence of "closed" subspaces in the classical Bernstein lethargy theorem, we obtain a version of this theorem for the
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Shapiro’s lethargy theorem [48] states that if {An} is any non-trivial linear approximation scheme on a Banach space X ,t hen the sequences of errors of best approximation E(x,An )=i nfa∈An ∥x − an∥X
The problem as to whether a Banach space contains an element with given deviations from an expanding system of strictly nested subspaces (which are not necessarily finite-dimensional) is solved under
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Tyuriemskih's Lethargy Theorem is generalized to provide a useful tool for establishing when a sequence of (not necessarily) linear operators that converges point wise to the identity operator
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In this paper some inequalities for Dirichlet’s and Fejer’s kernels proved in [6] are refined and extended. Then we have obtained the conditions for L-convergence of the r-th derivatives of complex