Stone-Weierstrass Theorem

• categories: Real Analysis, Theorem

The Stone-Weierstrass Theorem generalizes the Weierstrass Approximation Theorem. It extends the result from polynomials to a more general class of functions and provides conditions under which continuous functions on a compact space can be uniformly approximated.

Statement of the Theorem

Let $X$ be a compact Hausdorff space, and let $\mathcal{A}$ be a subalgebra of $C(X)$ (the set of continuous real-valued functions on $X$). The Stone-Weierstrass Theorem states that if $\mathcal{A}$:

1. Contains a constant function,
2. Separates points on $X$ (i.e., for any two distinct points $x,y\in X$, there exists a function $f\in \mathcal{A}$ such that $f(x)\ne f(y)$),

then $\mathcal{A}$ is dense in $C(X)$ with respect to the uniform norm. This means that for any continuous function $f\in C(X)$ and any $ϵ>0$, there exists a function $g\in \mathcal{A}$ such that:

$$\underset{x\in X}{\sup }|f(x)-g(x)|<ϵ$$

Intuition

The Stone-Weierstrass Theorem tells us that if a subalgebra of continuous functions satisfies certain properties, then we can uniformly approximate any continuous function on a compact space using functions from this subalgebra.

• The Weierstrass Approximation Theorem is a special case where the space $X=[a,b]\subseteq \mathbb{R}$ and the subalgebra $\mathcal{A}$ is the set of polynomials. The Stone-Weierstrass Theorem applies to much more general spaces and function classes. In the Weierstrass case, the approximating functions are polynomials, whereas in the Stone-Weierstrass Theorem, the approximating functions can be elements of any subalgebra that satisfies the two conditions (constant function and point separation).

Example

Consider $X=[0,1]$ and the subalgebra $\mathcal{A}$ of continuous functions spanned by $\{1,\cos (\pi x),\cos (2\pi x),\cos (3\pi x),\dots \}$. This set satisfies the conditions of the Stone-Weierstrass Theorem, so any continuous function on $[0,1]$ can be uniformly approximated by a finite linear combination of cosine functions.

Applications

• Fourier Series: In many cases, trigonometric polynomials (sums of sines and cosines) are used to approximate functions, and the Stone-Weierstrass Theorem guarantees that this type of approximation is possible.

• Functional Analysis: The theorem is important in the study of Banach algebras and approximation theory, as it ensures that large classes of functions can be approximated by simpler or more structured ones.