Commit 2021-05-01 20:07 ea379b09
View on Github →feat(topology/continuous_function): the Stone-Weierstrass theorem (#7305)
The Stone-Weierstrass theorem
If a subalgebra A
of C(X, ℝ)
, where X
is a compact Hausdorff space,
separates points, then it is dense.
We argue as follows.
- In any subalgebra
A
ofC(X, ℝ)
, iff ∈ A
, thenabs f ∈ A.topological_closure
. This follows from the Weierstrass approximation theorem on[-∥f∥, ∥f∥]
by approximatingabs
uniformly thereon by polynomials. - This ensures that
A.topological_closure
is actually a sublattice: if it containsf
andg
, then it contains the pointwise supremumf ⊔ g
and the pointwise infimumf ⊓ g
. - Any nonempty sublattice
L
ofC(X, ℝ)
which separates points is dense, by a nice argument approximating a givenf
above and below using separating functions. For eachx y : X
, we pick a functiong x y ∈ L
sog x y x = f x
andg x y y = f y
. By continuity these functions remain close tof
on small patches aroundx
andy
. We use compactness to identify a certain finitely indexed infimum of finitely indexed supremums which is then close tof
everywhere, obtaining the desired approximation. - Finally we put these pieces together.
L = A.topological_closure
is a nonempty sublattice which separates points sinceA
does, and so is dense (in fact equal to⊤
).
Future work
Prove the complex version for self-adjoint subalgebras A
, by separately approximating
the real and imaginary parts using the real subalgebra of real-valued functions in A
(which still separates points, by taking the norm-square of a separating function).
Extend to cover the case of subalgebras of the continuous functions vanishing at infinity,
on non-compact Hausdorff spaces.