nLab
Stone-Weierstrass theorem

Contents

Idea

The Stone–Weierstrass theorem says given a compact Hausdorff space XX, one can uniformly approximate continuous functions f:Xf: X \to \mathbb{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 XX be a compact Hausdorff topological space; for a constructive version take XX to be a compact regular locale (see compactum). Recall that the algebra C(X)C(X) of real-valued continuous functions f:Xf: X \to \mathbb{R} is a commutative (real) Banach algebra with unit, under pointwise-defined addition and multiplication, and where the norm is the sup-norm

f:=sup xX|f(x)|\|f\| := sup_{x \in X} |f(x)|

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

Theorem (Stone–Weierstrass)

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

Outline of proof

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

Finally, suppose the Banach subalgebra AA separates points. Given gC(X)g \in C(X) and ε>0\varepsilon \gt 0, the last step is to show there exists fAf \in A such that fgε\|f - g\| \leq \varepsilon.

Variations

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

Theorem

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

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

Theorem

AC 0(X)A \subseteq C_0(X) is dense if and only if it separates points and for no xXx \in X is it true that every fAf \in A vanishes at xx.

References

Named after Karl Weierstraß and Marshall Stone.

See

Discussion in constructive mathematics is in