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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Bounds for nearly best approximations


Author: Rudolf Wegmann
Journal: Proc. Amer. Math. Soc. 52 (1975), 252-256
MSC: Primary 41A65
DOI: https://doi.org/10.1090/S0002-9939-1975-0442563-1
MathSciNet review: 0442563
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Abstract: Let $X$ be a uniformly convex space and $\psi$ be the inverse function of the modulus of convexity $\delta ( \cdot )$. Assume here that $\psi$ is a concave function. Let $V$ be a linear subspace of $X$ and let $f$ in $X$ be such that $||f|| = 1 = \min \{ ||f - v||:v\epsilon V\}$. Then for $0 < \delta \leqslant 1$ and for $v$ in $V$ with $||f - v|| \leqslant 1 + \delta$, it follows that $||v|| \leqslant K \cdot \psi (\delta )$. Let $T$ be a compact Hausdorff-space and $V$ a finite-dimensional subspace of $C(T,X)$. When $V$ has the interpolation property $({P_m})$ with $V = m \cdot \dim X$, then the same type of estimate as above holds.


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Article copyright: © Copyright 1975 American Mathematical Society