One of the central problems in approximation theory is to determine points that minimize distances (to a given point or subset). More precisely,

Problem - Let be a metric space and a subset. Given we want to know if there exists a point in that minimizes the distance to , i.e. if there exists such that

Definition - A point that satisfies the above conditions is called a best approximation of in .

In general, best approximations do not exist. Thus, the problem is usually to identify classes of spaces and where the existence of best approximations can be assured.

Example : When is compact, best approximations of a given point in always exist.

After one assures the existence of a best approximation, one can question about its uniqueness and how to calculate it explicitly.

Remark - There is no reason to restrict to metric spaces. The definition of best approximation can be given for pseudo-metric spaces, semimetric spaces or any other space where a suitable notion of "distance" can be given.