View source for Element of best approximation

An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035260/e0352601.png" /> in a given set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035260/e0352602.png" /> that is a [[Best approximation|best approximation]] to a given element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035260/e0352603.png" /> in a metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035260/e0352604.png" />, i.e. is such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035260/e0352605.png" /></td> </tr></table>

This is a generalization of the classical concept of a [[Polynomial of best approximation|polynomial of best approximation]]. The main questions concerning elements of best approximation are: their existence and uniqueness, their characteristic properties (see [[Chebyshev theorem|Chebyshev theorem]]), the properties of the operator that associates with each element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035260/e0352606.png" /> the set of elements of best approximation (see [[Metric projection|Metric projection]]; [[Approximately-compact set|Approximately-compact set]]), and numerical methods for the construction of elements of best approximation.