The Weierstrass approximation theorem for continuous functions on [a,b]

We've already proved the Weierstrass approximation theorem in the sense that we've shown that the Bernstein approximations to a continuous function on [0,1] converge uniformly.

Here we rephrase this more abstractly as polynomialFunctions_closure_eq_top' : (polynomialFunctions I).topologicalClosure = ⊤ and then, by precomposing with suitable affine functions, polynomialFunctions_closure_eq_top : (polynomialFunctions (Set.Icc a b)).topologicalClosure = ⊤

theorem polynomialFunctions_closure_eq_top' : Subalgebra.topologicalClosure (polynomialFunctions unitInterval) = ⊤

The special case of the Weierstrass approximation theorem for the interval [0,1]. This is just a matter of unravelling definitions and using the Bernstein approximations.

theorem polynomialFunctions_closure_eq_top (a : ℝ) (b : ℝ) : Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b)) = ⊤

The Weierstrass Approximation Theorem: polynomials functions on [a, b] ⊆ ℝ are dense in C([a,b],ℝ)

(While we could deduce this as an application of the Stone-Weierstrass theorem, our proof of that relies on the fact that abs is in the closure of polynomials on [-M, M], so we may as well get this done first.)

theorem continuousMap_mem_polynomialFunctions_closure (a : ℝ) (b : ℝ) (f : C(↑(Set.Icc a b), ℝ)) : f ∈ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b))

An alternative statement of Weierstrass' theorem.

Every real-valued continuous function on [a,b] is a uniform limit of polynomials.

theorem exists_polynomial_near_continuousMap (a : ℝ) (b : ℝ) (f : C(↑(Set.Icc a b), ℝ)) (ε : ℝ) (pos : 0 < ε) : ∃ (p : Polynomial ℝ), ‖Polynomial.toContinuousMapOn p (Set.Icc a b) - f‖ < ε

An alternative statement of Weierstrass' theorem, for those who like their epsilons.

Every real-valued continuous function on [a,b] is within any ε > 0 of some polynomial.

theorem exists_polynomial_near_of_continuousOn (a : ℝ) (b : ℝ) (f : ℝ → ℝ) (c : ContinuousOn f (Set.Icc a b)) (ε : ℝ) (pos : 0 < ε) : ∃ (p : Polynomial ℝ), ∀ x ∈ Set.Icc a b, |Polynomial.eval x p - f x| < ε

Another alternative statement of Weierstrass's theorem, for those who like epsilons, but not bundled continuous functions.

Every real-valued function ℝ → ℝ which is continuous on [a,b] can be approximated to within any ε > 0 on [a,b] by some polynomial.