English
Same spectrum result in the appropriate setting, showing the robustness of the preimage-range relationship.
Русский
Та же связность спектра через предобраз диапазона в соответствующих условиях.
LaTeX
$$$\mathrm{spectrum}_{\mathbb{K}}(f) = (\mathrm{algebraMap})^{-1}(\mathrm{range}(f)).$$$
Lean4
/-- 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' : (polynomialFunctions I).topologicalClosure = ⊤ :=
by
apply top_unique
rintro f -
refine mem_closure_of_tendsto (bernsteinApproximation_uniform f) <| .of_forall fun n ↦ ?_
apply Subalgebra.sum_mem
rintro i -
rw [← SetLike.mem_coe, polynomialFunctions_coe]
use bernsteinPolynomial ℝ n i * .C (f (bernstein.z i))
ext
simp [bernstein]
