Stone Weierstrass, An Introduction
Introduction
The continuous functions from a space S, usually a compact space, into the reals,
denoted c(S),
forms a ring, an R vector space, a partially ordered lattice, and a complete metric space.
In the next few sections we will analyze c(S),
and prove the Stone
(biography)
Weierstrass
(biography)
theorem.
This is used to approximate arbitrary functions in c(S) using a specific set of functions,
rather like spanning a vector space with a basis.
In a direct application, sines and cosines combine to build any periodic wave form,
in a process known as fourier analysis.
Any continuous wave can be approximated, uniformly, with arbitrary precision,
using these basic trig functions.
In other words, a basis in c(S) spans all of c(S).