Error of best approximation

Summary

The error of best approximation of $x$ from a set $A$ is the distance from $x$ to $A$, the infimum of $\norm{x - a}$ over $a \in A$. For a function $f$ and the polynomials of degree at most $n$ it is written $E_n(f)$. It measures how well $f$ can be approximated at all, whether or not a best approximation exists. Its decay as $n \to \infty$ reflects the smoothness of $f$.