range_dualMap_le_dualAnnihilator_ker

English

For any submodule U of Module.Dual(R,M), the dual coannihilator of the dual annihilator of U, then dual coannihilator again equals U.dualCoannihilator: U.dualCoannihilator.dualAnnihilator.dualCoannihilator = U.dualCoannihilator.

Русский

Для подмодуля U дуального пространства выполняется тройная дуализация через дуальный коаннигилятора: U.dualCoannihilator.dualAnnihilator.dualCoannihilator = U.dualCoannihilator.

LaTeX

$$$ U.dualCoannihilator.dualAnnihilator.dualCoannihilator = U.dualCoannihilator $$$

Lean4

theorem range_dualMap_le_dualAnnihilator_ker : LinearMap.range f.dualMap ≤ (ker f).dualAnnihilator :=
  by
  rintro _ ⟨ψ, rfl⟩
  simp_rw [Submodule.mem_dualAnnihilator, mem_ker]
  rintro x hx
  rw [dualMap_apply, hx, map_zero]

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