span_fourier_closure_eq_top — Mathlib

English

The linear span of Fourier monomials f_n is dense in C(AddCircle(T), ℂ).

Русский

Линейный-span мономиал Фурье f_n плотно лежит в C(AddCircle(T), ℂ).

LaTeX

$$(span ℂ (range fourier)).topologicalClosure = ⊤$$

Lean4

/-- The linear span of the monomials `fourier n` is dense in `C(AddCircle T, ℂ)`. -/
theorem span_fourier_closure_eq_top : (span ℂ (range <| @fourier T)).topologicalClosure = ⊤ :=
  by
  rw [← fourierSubalgebra_coe]
  exact congr_arg (Subalgebra.toSubmodule <| StarSubalgebra.toSubalgebra ·) fourierSubalgebra_closure_eq_top

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