📂Banach Space

Approximation, Best Approximation

Definition¹

Let $(X, d)$ be a metric space. For a subset $U \subset X$, a mapping $X \to U$ is called an approximation (method).

The best approximation for $f \in X$ is defined as follows: $$u^{\ast} = \argmin_{u \in U} d(f, u)$$

Explanation

The term “approximation” refers to something close, thus “approximation to $f$ (an approximation of $f$)” means something close to $f$. Mathematically, the concepts of being far or close are defined through a distance function $d$. Hence, the concepts of approximation and best approximation are defined on a metric space.

Related to this, possible mathematical inquiries include: (1) Does it exist? (existence) (2) If it exists, is it unique? (uniqueness) (3) How can it be found? (algorithm).