Bounds for nearly best approximations

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.