Element of best approximation

An element $u_0$ in a given set $F$ that is a best approximation to a given element $x$ in a metric space $X$, i.e. is such that $$\rho(u_0,x) = \inf \{ \rho(u,x) : x \in F \} \ .$$ This is a generalization of the classical concept of a polynomial of best approximation. The main questions concerning elements of best approximation are: their existence and uniqueness, their characteristic properties (see Chebyshev theorem), the properties of the operator that associates with each element $x \in X$ the set of elements of best approximation (see Metric projection; Approximately-compact set), and numerical methods for the construction of elements of best approximation.