quotEquivAnnihilator — Mathlib

English

The quotient by the dual is isomorphic to the dual annihilator via a constructed equivalence.

Русский

Квантическое отношение: квотирование дуала эквивалентно дуальному аннигилятору через конструкцию эквивалентности.

LaTeX

$$quotDualEquivAnnihilator (W) : (Module.Dual K V ⧸ LinearMap.range W.dualLift) ≃ₗ[K] W.dualAnnihilator$$

Lean4

/-- The quotient by a subspace is isomorphic to its dual annihilator. -/
noncomputable def quotEquivAnnihilator (W : Subspace K V) : (V ⧸ W) ≃ₗ[K] W.dualAnnihilator :=
  let φ := (Basis.ofVectorSpace K W).toDualEquiv.trans W.dualEquivDual
  let ψ := LinearEquiv.quotEquivOfEquiv φ (Basis.ofVectorSpace K V).toDualEquiv
  ψ ≪≫ₗ W.quotDualEquivAnnihilator

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