---
tag: 7N4988HN3P
kind: concept
name: Best approximation
msc: [41A50, 41A65]
sources:
  - url: https://en.wikipedia.org/wiki/Approximation_theory
---

## Summary

A best approximation of a point $x$ from a set $A$ in a normed space is a point of $A$ nearest to $x$.
It is the optimal answer to an approximation problem.
Existence, uniqueness and characterization of best approximations are the first questions of approximation theory.

## Explanations

### L1 human

Think of a point in the plane and a line: the best approximation is the foot of the perpendicular, the point of the line nearest to the given point.
In approximation theory the "points" are functions and the set is, for example, all polynomials of degree at most $n$.
For $f(x) = x^2$ on $[0, 1]$ and the uniform distance, the best straight line is $x - 1/8$, which is never further than $1/8$ from the parabola.
A best approximation may fail to exist, for instance when the set has a hole where the nearest point should be.