Commit 2021-05-01 20:07 ea379b09

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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 of C(X, ℝ), if f ∈ A, then abs f ∈ A.topological_closure. This follows from the Weierstrass approximation theorem on [-∥f∥, ∥f∥] by approximating abs uniformly thereon by polynomials.
  • This ensures that A.topological_closure is actually a sublattice: if it contains f and g, then it contains the pointwise supremum f ⊔ g and the pointwise infimum f ⊓ g.
  • Any nonempty sublattice L of C(X, ℝ) which separates points is dense, by a nice argument approximating a given f above and below using separating functions. For each x y : X, we pick a function g x y ∈ L so g x y x = f x and g x y y = f y. By continuity these functions remain close to f on small patches around x and y. We use compactness to identify a certain finitely indexed infimum of finitely indexed supremums which is then close to f everywhere, obtaining the desired approximation.
  • Finally we put these pieces together. L = A.topological_closure is a nonempty sublattice which separates points since A 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.

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