Overview
A Banach Space is a vector space equipped with a norm. Approximation in a Banach space is accomplished by using the norm to define a distance function. (see below) The best approximation to an element in the space, is then the element with the smallest distance to the target element.
Distance
Let {% K %} be a subset of a normed space {% X %}. Then the function , {% d(x,K) %}, called the distance from {% x \in X %} to {% K %} is defined to be
{% d(x,K) := inf_{y \in K} ||x-y|| %}
Best Approximation
The best approximation of an element {% x %} by an element {% y \in K %} is an element {% y %} such that
{% ||x-y|| = d(x,K) %}
(see deutsch chap 2)