Stone-Weierstrass Theorem

The Weierstrass approximation theorem states that every continuous function defined on a closed interval [a, b] can be uniformly approximated as closely as desired by a polynomial function. Because polynomials are among the simplest functions, and because computers can directly evaluate polynomials, this theorem has both practical and theoretical relevance, especially in polynomial interpolation. The original version of this result was established by Karl Weierstrass in 1885 using the Weierstrass transform.

Marshall H. Stone considerably generalized the theorem and simplified the proof. His result is known as the Stone–Weierstrass theorem. The Stone–Weierstrass theorem generalizes the Weierstrass approximation theorem in two directions: instead of the real interval [a, b], an arbitrary compact Hausdorff space X is considered, and instead of the algebra of polynomial functions, a variety of other families of continuous functions on $X$ are shown to suffice. The Stone–Weierstrass theorem is a vital result in the study of the algebra of continuous functions on a compact Hausdorff space.

Further, there is a generalization of the Stone–Weierstrass theorem to noncompact Tychonoff spaces, namely, any continuous function on a Tychonoff space is approximated uniformly on compact sets by algebras of the type appearing in the Stone–Weierstrass theorem and described below.

A different generalization of Weierstrass' original theorem is Mergelyan's theorem, which generalizes it to functions defined on certain subsets of the complex plane.

Weierstrass approximation theorem

The statement of the approximation theorem as originally discovered by Weierstrass is as follows:

Weierstrass Approximation Theorem. Suppose  f  is a continuous real-valued function defined on the real interval [a, b]. For every ε > 0, there exists a polynomial p such that for all x in [a, b], we have $|f(x)-p(x)|<\varepsilon$, or equivalently, the supremum norm $\|f-p\|<\varepsilon$ .

A constructive proof of this theorem using Bernstein polynomials is outlined on that page.

Applications

As a consequence of the Weierstrass approximation theorem, one can show that the space C[a, b] is separable: the polynomial functions are dense, and each polynomial function can be uniformly approximated by one with rational coefficients; there are only countably many polynomials with rational coefficients. Since C[a, b] is metrizable and separable it follows that C[a, b] has cardinality at most 2^ℵ₀. (Remark: This cardinality result also follows from the fact that a continuous function on the reals is uniquely determined by its restriction to the rationals.)

Stone–Weierstrass theorem, real version

The set C[a, b] of continuous real-valued functions on [a, b], together with the supremum norm $\|f\|={\text{sup}}_{a\leq x\leq b}|f(x)|$, is a Banach algebra, (that is, an associative algebra and a Banach space such that $\|fg\|\leq \|f\|\cdot \|g\|$ for all  f, g). The set of all polynomial functions forms a subalgebra of C[a, b] (that is, a vector subspace of C[a, b] that is closed under multiplication of functions), and the content of the Weierstrass approximation theorem is that this subalgebra is dense in C[a, b].

Stone starts with an arbitrary compact Hausdorff space X and considers the algebra C(X, R) of real-valued continuous functions on X, with the topology of uniform convergence. He wants to find subalgebras of C(X, R) which are dense. It turns out that the crucial property that a subalgebra must satisfy is that it separates points: a set A of functions defined on X is said to separate points if, for every two different points x and y in X there exists a function p in A with p(x) ≠ p(y). Now we may state:

Stone–Weierstrass Theorem (real numbers). Suppose X is a compact Hausdorff space and A is a subalgebra of C(X, R) which contains a non-zero constant function. Then A is dense in C(X, R) if and only if it separates points.

This implies Weierstrass' original statement since the polynomials on [a, b] form a subalgebra of C[a, b] which contains the constants and separates points.

Locally compact version

A version of the Stone–Weierstrass theorem is also true when X is only locally compact. Let C₀(X, R) be the space of real-valued continuous functions on X which vanish at infinity; that is, a continuous function  f  is in C₀(X, R) if, for every ε > 0, there exists a compact set K ⊂ X such that  | f |  < ε on X \ K. Again, C₀(X, R) is a Banach algebra with the supremum norm. A subalgebra A of C₀(X, R) is said to vanish nowhere if not all of the elements of A simultaneously vanish at a point; that is, for every x in X, there is some  f  in A such that  f (x) ≠ 0. The theorem generalizes as follows:

Stone–Weierstrass Theorem (locally compact spaces). Suppose X is a locally compact Hausdorff space and A is a subalgebra of C₀(X, R). Then A is dense in C₀(X, R) (given the topology of uniform convergence) if and only if it separates points and vanishes nowhere.

This version clearly implies the previous version in the case when X is compact, since in that case $C_{0}(X,\mathbb {R} )=C(X,\mathbb {R} )$. There are also more general versions of the Stone–Weierstrass that weaken the assumption of local compactness.

Applications

The Stone–Weierstrass theorem can be used to prove the following two statements which go beyond Weierstrass's result.

• If  f  is a continuous real-valued function defined on the set [a, b] × [c, d] and ε > 0, then there exists a polynomial function p in two variables such that $|f(x,y)-p(x,y)|<\varepsilon$ for all x in [a, b] and y in [c, d].
• If X and Y are two compact Hausdorff spaces and f : X × Y → R is a continuous function, then for every ε > 0 there exist n > 0 and continuous functions  f₁, ...,  f_n  on X and continuous functions g₁, ..., g_n on Y such that $\|f-\Sigma f_{i}g_{i}\|<\varepsilon$.

The theorem has many other applications to analysis, including:

• Fourier series: The set of linear combinations of functions $e^{n}x=e^{2\pi inx},n\in \mathbb {Z}$ is dense in C([0, 1]/{0, 1}), where we identify the endpoints of the interval [0, 1] to obtain a circle. An important consequence of this is that the e_n are an orthonormal basis of the space L²([0, 1]) of square-integrable functions on [0, 1].

Licensing

Content obtained and/or adapted from:

• Stone-Weierstrass theorem, Wikipedia under a CC BY-SA license