Theorems · Definition · linear algebra

Submodule.dualPairing

{R : Type u_1} →
  {M : Type u_2} →
    [inst : CommRing R] →
      [inst_1 : AddCommGroup M] →
        [inst_2 : Module R M] → (W : Submodule R M) → Module.Dual R M ⧸ W.dualAnnihilator →ₗ[R] ↥W →ₗ[R] R

Given a submodule, restrict to the pairing on W by simultaneously corestricting to Module.Dual R M ⧸ W.dualAnnihilator. This is Submodule.dualRestrict factored through the quotient by its kernel (which is W.dualAnnihilator by definition). See Subspace.dualPairing_nondegenerate.

Defined in

Mathlib.LinearAlgebra.Dual.Lemmas

Cited by

3 results in Mathlib

Foundations

Depth 85 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModule