Stone–Weierstrass 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)| < ε, or equivalently, the supremum norm || f  − p|| < ε.

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

The set C[a, b] of continuous real-valued functions on [a, b], together with the supremum norm || f || = sup_{a ≤ x ≤ b} | f (x)|, is a Banach algebra, (that is, an associative algebra and a Banach space such that || fg|| ≤ || f ||·||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.

Slightly more general is the following theorem, where we consider the algebra ${\displaystyle C(X,\mathbb {C} )}$ of complex-valued continuous functions on the compact space ${\displaystyle X}$, again with the topology of uniform convergence. This is a C*-algebra with the *-operation given by pointwise complex conjugation.

Stone–Weierstrass Theorem (complex numbers). Let ${\displaystyle X}$ be a compact Hausdorff space and let ${\displaystyle S}$ be a separating subset of ${\displaystyle C(X,\mathbb {C} )}$. Then the complex unital *-algebra generated by ${\displaystyle S}$ is dense in ${\displaystyle C(X,\mathbb {C} )}$.

The complex unital *-algebra generated by ${\displaystyle S}$ consists of all those functions that can be obtained from the elements of ${\displaystyle S}$ by throwing in the constant function 1 and adding them, multiplying them, conjugating them, or multiplying them with complex scalars, and repeating finitely many times.

This theorem implies the real version, because if a net of complex-valued functions uniformly approximates a given function, ${\displaystyle f_{n}\to f}$, then the real parts of those functions uniformly approximate the real part of that function, ${\displaystyle \operatorname {Re} f_{n}\to \operatorname {Re} f}$, and because for real subsets, ${\displaystyle S\subset C(X,\mathbb {R} )\subset C(X,\mathbb {C} ),}$ taking the real parts of the generated complex unital (selfadjoint) algebra agrees with the generated real unital algebra generated.

As in the real case, an analog of this theorem is true for locally compact Hausdorff spaces.

Following John C.Holladay (1957) : consider the algebra C(X, H) of quaternion-valued continuous functions on the compact space X, again with the topology of uniform convergence. If a quaternion q is written in the form q = a + ib;+ jc + kd then the scalar part a is the real number (q − iqi − jqj − kqk)/4. Likewise being the scalar part of −qi, −qj and −qk : b,c and d are respectively the real numbers (−qi − iq + jqk − kqj)/4, (−qj − iqk − jq + kqi)/4 and (−qk + iqj − jqk − kq)/4. Then we may state :

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

The space of complex-valued continuous functions on a compact Hausdorff space ${\displaystyle X}$ i.e. ${\displaystyle C(X,\mathbb {C} )}$ is the canonical example of a unital commutative C*-algebra ${\displaystyle {\mathfrak {A}}}$. The space X may be viewed as the space of pure states on ${\displaystyle {\mathfrak {A}}}$, with the weak-* topology. Following the above cue, a non-commutative extension of the Stone–Weierstrass theorem, which has remain unsolved, is as follows:

Conjecture. If a unital C*-algebra ${\displaystyle {\mathfrak {A}}}$ has a C*-subalgebra ${\displaystyle {\mathfrak {B}}}$ which separates the pure states of ${\displaystyle {\mathfrak {A}}}$, then ${\displaystyle {\mathfrak {A}}={\mathfrak {B}}}$.

In 1960, Jim Glimm proved a weaker version of the above conjecture.

Stone–Weierstrass theorem (C*-algebras).[2] If a unital C*-algebra ${\displaystyle {\mathfrak {A}}}$ has a C*-subalgebra ${\displaystyle {\mathfrak {B}}}$ which separates the pure state space (i.e. the weak-* closure of the pure states) of ${\displaystyle {\mathfrak {A}}}$, then ${\displaystyle {\mathfrak {A}}={\mathfrak {B}}}$.

Let X be a compact Hausdorff space. Stone's original proof of the theorem used the idea of lattices in C(X, R). A subset L of C(X, R) is called a lattice if for any two elements  f, g ∈ L, the functions max{ f, g}, min{ f, g} also belong to L. The lattice version of the Stone–Weierstrass theorem states:

Stone–Weierstrass Theorem (lattices). Suppose X is a compact Hausdorff space with at least two points and L is a lattice in C(X, R) with the property that for any two distinct elements x and y of X and any two real numbers a and b there exists an element  f  ∈ L with  f (x) = a and  f (y) = b. Then L is dense in C(X, R).

The above versions of Stone–Weierstrass can be proven from this version once one realizes that the lattice property can also be formulated using the absolute value | f | which in turn can be approximated by polynomials in  f . A variant of the theorem applies to linear subspaces of C(X, R) closed under max (Hewitt & Stromberg 1965, Theorem 7.29):

Stone–Weierstrass Theorem. Suppose X is a compact Hausdorff space and B is a family of functions in C(X, R) such that

1. B separates points.
2. B contains the constant function 1.
3. If  f  ∈ B then af  ∈ B for all constants a ∈ R.
4. If  f,  g ∈ B, then  f  + g, max{ f, g} ∈ B.

Then B is dense in C(X, R).

More precise information is available:

Suppose X is a compact Hausdorff space with at least two points and L is a lattice in C(X, R). The function φ ∈ C(X, R) belongs to the closure of L if and only if for each pair of distinct points x and y in X and for each ε > 0 there exists some  f  ∈ L for which | f (x) − φ(x)| < ε and | f (y) − φ(y)| < ε.

Another generalization of the Stone–Weierstrass theorem is due to Errett Bishop. Bishop's theorem is as follows (Bishop 1961):

Let A be a closed subalgebra of the complex Banach algebra C(X, C) of continuous complex-valued functions on a compact Hausdorff space X, using the supremum norm. For S ⊂ X we write A_S = {g|_S : g ∈ A}. Suppose that  f  ∈ C(X, C) has the following property:

 f |_S ∈ A_S for every maximal set S ⊂ X such that all real functions of A_S are constant.

Then  f  ∈ A.

Glicksberg (1962) gives a short proof of Bishop's theorem using the Krein–Milman theorem in an essential way, as well as the Hahn–Banach theorem : the process of Louis de Branges (1959). See also Rudin (1973, §5.7).

Nachbin's theorem gives an analog for Stone–Weierstrass theorem for algebras of complex valued smooth functions on a smooth manifold (Nachbin 1949). Nachbin's theorem is as follows (Llavona 1986):

Let A be a subalgebra of the algebra C^∞(M) of smooth functions on a finite dimensional smooth manifold M. Suppose that A separates the points of M and also separates the tangent vectors of M: for each point m ∈ M and tangent vector v at the tangent space at m, there is a f ∈ A such that df(x)(v) ≠ 0. Then A is dense in C^∞(M).

• Müntz–Szász theorem.
• Bernstein polynomial.
• Runge's phenomenon shows that finding a polynomial P such that  f (x) = P(x) for some finely spaced x = x_n is a bad way to attempt to find a polynomial approximating  f  uniformly. A better approach, explained e.g. in (Rudin 1976), p. 160, eq. (51) ff., is to construct polynomials P uniformly approximating  f  by taking the convolution of  f  with a family of suitably chosen polynomial kernels.
• Mergelyan's theorem, concerning polynomial approximations of complex functions.

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