Stone-Weierstrass Theorem

From Local Patches to a Uniform Approximation

The previous tutorial developed the ingredients behind the Stone-Weierstrass argument. A unital algebra that separates points can interpolate a continuous target at two chosen points, and its uniform closure is closed under maxima and minima. Compactness then gives a one-sided patch at a fixed point. The remaining task is to combine patches based at different points so that the resulting function is close to the target everywhere.

Throughout, let \(K\) be a nonempty compact subset of \(\mathbb{R}\), and let \(A\) be a unital subalgebra of \(C(K)\) that separates points. Write \(\overline{A}\) for its closure in the supremum norm. We use the One-Sided Patching at a Fixed Point Theorem from the previous tutorial: for \(f\in C(K)\), \(x\in K\), and \(\delta>0\), there is an \(h_x\in\overline{A}\) such that \(h_x(x)=f(x)\) and \(h_x(z)<f(z)+\delta\) for every \(z\in K\).

Proof. Fix \(f\in C(K)\) and \(\varepsilon>0\), and put \(\delta=\varepsilon/3\). For each \(x\in K\), apply the one-sided patching theorem to obtain \(h_x\in\overline{A}\) with \(h_x(x)=f(x)\) and \(h_x(z)<f(z)+\delta\) for every \(z\in K\).

The function \(h_x-f\) is continuous and has value zero at \(x\). Therefore the set

is open relative to \(K\) and contains \(x\). The sets \(U_x\), as \(x\) varies over \(K\), form an open cover. Compactness supplies a finite subcover \(U_{x_1},\ldots,U_{x_m}\). Define

Each \(h_{x_i}\) belongs to \(\overline{A}\), and the uniform closure is closed under finite maxima by the Absolute Values in the Uniform Closure Theorem from the previous tutorial. Hence \(g\in\overline{A}\). For any \(z\in K\), the upper bound for each patch gives

Because the sets \(U_{x_i}\) cover \(K\), there is at least one index \(i\) for which \(z\in U_{x_i}\). For that index, \(h_{x_i}(z)>f(z)-\delta\), so

Together these inequalities show \(|g(z)-f(z)|<\delta\) at every \(z\in K\), and therefore \(\|g-f\|_\infty\leq\delta\). Since \(g\in\overline{A}\), choose \(a\in A\) with \(\|a-g\|_\infty<\delta\). The triangle inequality now yields

Thus every continuous \(f\) can be approximated uniformly to any prescribed positive accuracy by an element of \(A\). This is exactly density of \(A\) in \(C(K)\). \(\square\)

Polynomial Restrictions on Compact Sets

The theorem immediately gives a useful conclusion beyond approximation on an interval. If \(K\) is any nonempty compact subset of the real line, a polynomial can be restricted to \(K\), even when \(K\) has gaps or consists of isolated points. These restrictions still have enough algebraic flexibility to approximate every continuous function on \(K\).

Proof. Let \(A\) be the set of restrictions to \(K\) of real polynomials. Sums, scalar multiples, and products of polynomial restrictions are again polynomial restrictions, so \(A\) is a subalgebra of \(C(K)\). Constant polynomials show that \(A\) is unital. If \(x,y\in K\) and \(x\neq y\), the polynomial \(p(t)=t\) satisfies \(p(x)=x\neq y=p(y)\); thus \(A\) separates points. The Real Stone-Weierstrass Theorem applies and gives the claimed density. \(\square\)

Other Algebras Can Be Dense Too

Polynomials are a familiar example, but the theorem is about algebraic structure rather than a particular formula. A smaller-looking collection of functions can still be dense if it contains constants and can distinguish every pair of points. The next example uses polynomials in the square of the coordinate function.

Why Point Separation Matters

The hypotheses are sufficient conditions, not decoration. In particular, if two different points always receive the same value from every member of an algebra, then no uniform limit of its members can distinguish those points either. A continuous target that does distinguish them cannot be uniformly approximated arbitrarily well. The next example quantifies this obstruction.

What the Theorem Does—and Does Not—Promise

The proof separates the roles of the assumptions. Point separation enables two-point interpolation. The algebra operations and the Weierstrass Approximation Theorem supply maxima and minima in the uniform closure. Continuity turns equality at a point into a useful local inequality, and compactness reduces the resulting neighborhoods to a finite collection. The finite maximum combines the lower bounds from those neighborhoods without losing the upper bound that each patch already satisfies everywhere.

The conclusion is qualitative: for each target and each positive error, at least one algebra element achieves that error. It does not give an explicit approximant, a degree bound, or a rate of convergence. Those questions require additional information about the target and the algebra. Also, the result here concerns real-valued continuous functions and real subalgebras; hypotheses for approximation by complex-valued algebras require separate care.