Stone–Weierstrass Theorem

Stone–Weierstrass Theorem (real version): Let $K$ be a compact metric space and let $A\subseteq C(K,\mathbb{R})$ be a subalgebra (closed under addition, multiplication, and scalar multiplication). Assume:

• $A$ contains the constant functions, and
• $A$ separates points: for any distinct $x,y\in K$ there exists $f\in A$ such that $f(x)\neq f(y)$.

Then $A$ is dense in $C(K,\mathbb{R})$ with respect to the sup norm $\|\cdot\|_\infty$; i.e., for every $g\in C(K,\mathbb{R})$ and $\varepsilon>0$ there exists $f\in A$ with $\|g-f\|_\infty<\varepsilon.$ (For the complex version, one typically also assumes $A$ is closed under complex conjugation.)

Stone–Weierstrass generalizes the classical Weierstrass approximation theorem (polynomials) to many other function families and is a central density theorem in analysis.

Proof sketch: Using point separation, one shows $A$ can approximate continuous functions that interpolate prescribed values at finitely many points. One then builds “local bump-like” approximations and uses compactness to patch them into a uniform approximation. A key ingredient is showing the uniform closure of $A$ is stable under taking max/min (in the real case), which allows approximation of general continuous functions.