Summary of Stone–Weierstraß

I spent far too much time yesterday wondering why the lecturer gave a rather stronger statement of Stone–Weierstraß than that contained in any of the course books. In the end, it turned out to be a rather silly corollary he was rolling in to the statement of the theorem. Because the proof is beautiful, I feel I should educate you and show it.

Theorem (Stone–Weierstraß) Let X be compact, Hausdorff, and A ⊂ C ℂ X a sub-algebra of the continuous function space (that is, linear, also closed under pointwise-multiplication), which separates the points of C ℂ X , and finally is closed under conjugation (so ∀ f ∈ A , f ¯ : x ↦ f ⁡ x ¯ ∈ A ). Then either A ¯ is dense in C ℂ X (in the case ∀ x ∈ X , ∃ f ∈ A such that f ⁡ x ≠ 0 ) or ∃ x such that A ¯ is dense in f ∈ C ⁡ X : f ⁡ x = 0 (which is unique because of separation).

Remarks. This is the strongest statement I could come up with, combining a few corollaries in there for good measure. To explain the essential idea, what is going on is that we are using the gratuitously beautiful behaviour of the reals (Taylor series, and so on) to show that a hugely general class of function sequences converges and covers the whole space. Take the real version of the theorem first. The main objects in the proof are subalgebras and their closures, that is, inside the whole space of continuous functions on a well-behaved (specifically compact Hausdorff) space, we take some subset which is closed. Because we can combine continuous functions by pointwise addition and multiplication to get more continuous functions, the subspaces can be chosen to be subalgebras, whole arithmetical spaces of their own. Stone–Weierstraß is the assertion that once our subalgebra is big enough, we can approximate the entire function space (technically: we want the subalgebra to separate the points of X, that is given each pair of points in X we have a function that takes different values; this is just like saying we need enough functions to be able to tell apart every point in X; then what the theorem shows is that the subalgebra is dense, or equivalently its closure is the whole function space; in any case, for any real continuous function on X we can make a sequence of functions in the subalgebra which converges to it).

The complex case is just a refinement of the real case. The details are genuinely obvious to fill in, since with the added condition A is closed under conjugation we see that its real and imaginary parts are in the space; because the complex functions separate the space, just the real ones will do as well because at least one of the real and imaginary parts will differ. We then invoke the hard work done on the real case to get the complex case for free.

Pf. There are a lot of routine details to check. The main argument is simple: we want to show that the the subalgebra is closed under taking modulus, and then use that to stitch together an arbitrarily good approximation. The detail that the subalgebra is closed is a tad technical, but we can invoke a few lovely properties of real analysis and a clever trick with Taylor series to get there (or alternatively invoke some high-powered stuff). This clearly gives us the max and min functions. The last bit of the statement of the theorem is a silly detail about needing a non-zero function; when everything is sensible we can take a separating function for each pair of points and scale it. We are trying to build an approximation to g say. Fix x in X, and for every other point in X build a function which is close to g at those two points. We invoke compactness to pick a finite number of those and use min to bound them from above. Compactness again on the x lets us pick a finite number and use max to bound from below. ∎

I am skating over too many of the details to get much of the idea across, but I want to avoid too much notation. If you have to pick up a book to find out more, then good. I am saving the best until last though.

Application. This is a key theorem in a number of fields because with no work it instantly shows that polynomials uniformly approximate everything. In one fell swoop, numerical analysis, Fourier series, whole worlds of real-world approximations, suddenly become justified. That is a real theorem at work.