nLab
Stone-Weierstrass theorem

Idea

The Stone–Weierstrass theorem says given a compact Hausdorff space $X$, one can uniformly approximate continuous functions $f:X\to ℝ$ by elements of any subalgebra that has enough elements to distinguish points. It is a far-reaching generalization of a classical theorem of Weierstrass, that real-valued continuous functions on a closed interval are uniformly approximable by polynomial functions.

Precise statement

Let $X$ be a compact Hausdorff topological space; for a constructive version take $X$ to be a compact regular locale (see compactum). Recall that the algebra $C(X)$ of real-valued continuous functions $f:X\to ℝ$ is a commutative (real) Banach algebra with unit, under pointwise-defined addition and multiplication, and where the norm is the sup-norm

$$\parallel f\parallel :={\sup }_{x\in X}\mid f(x)\mid$$\|f\| := sup_{x \in X} |f(x)|

A subalgebra of $C(X)$ is a vector subspace $A\subseteq C(X)$ that is closed under the unit and algebra multiplication operations on $C(X)$. A Banach subalgebra is a subalgebra $A\subseteq C(X)$ which is closed as a subspace of the metric space $C(X)$ under the sup-norm metric. We say that $A\subseteq C(X)$ separates points if, given distinct points $x,y\in X$, there exists $f\in A$ such that $f(x)\ne f(y)$.

Theorem (Stone–Weierstrass)

A subalgebra inclusion $A\subseteq C(X)$ is dense if and only if it separates points. Equivalently, a Banach subalgebra inclusion $A\subseteq C(X)$ is the identity if and only if it separates points.

Outline of proof

• The first step is the classical Weierstrass approximation, that polynomial functions $p(x):[a,b]\to ℝ$ are dense in $C([a,b])$; without loss of generality, take $a=-\frac{1}{4}$ and $b=\frac{1}{4}$. We use the method of constructing polynomials which approximate the identity of the convolution product (the Dirac distribution). Explicitly, consider the normalizations $K_n=s_n/\parallel s_n{\parallel }_1$ where $s_n(x)=(1-x^2{)}^n$ over $[-1,1]$ and is compactly supported on $[-1,1]$, so that $K_n$ approximates the Dirac distribution concentrated at 0. Given $f:[-\frac{1}{4},\frac{1}{4}]\to ℝ$, extend to a function compactly supported on $[-\frac{1}{2},\frac{1}{2}]$. The convolution product

  $$(K_n*f)(x)={\int }_{-1}^1{K}_n(x-y)f(y)dy$$(K_n * f)(x) = \int_{-1}^1 K_n(x-y) f(y) d y

  is polynomial (since by differentiating under the integral enough times, we eventually kill the convolving polynomial factor $K_n$), and one may verify that $(K_n*f)$ converges to $f$ in sup norm (i.e., uniformly) as $n\to \mathrm{\infty }$, and in particular when restricted over the interval $[-\frac{1}{4},\frac{1}{4}]$.

Now suppose given a Banach subalgebra $A\subseteq C(X)$.

• Given $f\in A\subseteq C(X)$, the values of $f$ are contained in some interval $[a,b]$. If polynomials $p_n$ converge to the absolute value function $\mid \cdot \mid :[a,b]\to ℝ$ in sup norm, then $p_n(f)\in A$ converges to $\mid f\mid \in C(X)$ in sup norm. Since $A$ is closed in $C(X)$ with respect to the sup-norm, it follows that $\mid f\mid \in A$.

• Next, $A$ is partially ordered by $f\le g$ if $f(x)\le g(x)$ for all $x\in X$, and we claim the poset $A$ is closed under binary meets and binary joins. For,

  $$f\vee g=\frac{1}{2}(\mid f-g\mid +(f+g))$$f \vee g = \frac1{2}(|f - g| + (f + g))

  and $f\wedge g=-((-f)\vee (-g))$, and we showed in the last step that $A$ is closed under the operation $f\mapsto \mid f\mid$.

Finally, suppose the Banach subalgebra $A$ separates points. Given $g\in C(X)$ and $\epsilon >0$, the last step is to show there exists $f\in A$ such that $\parallel f-g\parallel \le \epsilon$.

• Lemma

  Given $x\in X$, there exists $f_x\in A$ such that $f_x(x)=g(x)$ and $g(y)\le f_x(y)+\epsilon$ for all $y\in X$.

  Proof

  For each $y\in X$, $y\ne x$ we can choose $s\in A$ such that $s(x)\ne s(y)$, since $A$ separates points. Thus, given any $x,y$, there exist $s\in A$ and scalars $a$ and $b$ such that

  $$a+b\cdot s(x)=g(x),a+b\cdot s(y)=g(y)$$a + b \cdot s(x) = g(x), \qquad a + b \cdot s(y) = g(y)

  Denote $a\cdot 1+bs$ by $f_{x,y}$ (to indicate dependence on $x$ and $y$). For each $y$, choose a neighborhood $V_y$ so that $g(z)\le f_{x,y}(z)+\epsilon$ for all $z\in V_y$. Finitely many such neighborhoods $V_{y_1},\dots ,V_{y_n}$ cover $X$; let $f_x$ be the join of $f_{x,y_1},\dots ,f_{x,y_n}$. Then $g(y)\le f_x(y)+\epsilon$ for all $y\in X$.

• Given a choice of $f_x$ for each $x\in X$, as in the preceding lemma, we may choose a neighborhood $V_x$ such that $f_x(z)-\epsilon \le g(z)$ for all $z\in V_x$. Finitely many such neighborhoods $V_{x_1},\dots ,V_{x_n}$ cover $X$; let $f$ be the meet of $f_{x_1},\dots ,f_{x_n}$. Then

  $$f(y)-\epsilon \le g(y)\le f(y)+\epsilon$$f(y) - \varepsilon \leq g(y) \leq f(y) + \varepsilon

  for all $y\in X$, as was to be shown.

Variations

There is a complex-valued version of Stone–Weierstrass. Let $C(X,ℂ)$ denote the commutative $C^{*}$-algebra of complex-valued functions $f:X\to ℂ$, where the star operation is pointwise-defined conjugation. A $C^{*}$-subalgebra is a subalgebra $A\subseteq C(X,ℂ)$ which is closed under the star operation.

Theorem

A $C^{*}$-subalgebra $A\subseteq C(X,ℂ)$ is dense if and only if it separates points.

There is also a locally compact version. Let $X$ be a locally compact Hausdorff space and let $C_0(X)$ be the space of (say real-valued) functions $f$ which “vanish at infinity”: for every $\epsilon >0$ there exists a compact set $K\subseteq X$ such that $\mid f(x)\mid <\epsilon$ for all $x$ outside $K$. ($C_0(X)$ is no longer a Banach space, but it is locally convex and complete in its uniformity, and a Fréchet space if $X$ is second countable.) Under pointwise multiplication, $C_0(X)$ is a commutative algebra without unit. As before, we have a notion of subalgebra $A\subseteq C_0(X)$.

Theorem

$A\subseteq C_0(X)$ is dense if and only if it separates points and for no $x\in X$ is it true that every $f\in A$ vanishes at $x$.

References

• B. Banaschewski, C. J. Mulvey, A constructive proof of the Stone-Weierstrass theorem, J. Pure Appl. Algebra 116 (1997), pp. 25–40, doi