Theorems · Definition · linear algebra

dualTensorHomEquivOfBasis

{ι : Type u_1} →
  {R : Type u_2} →
    {M : Type u_3} →
      {N : Type u_4} →
        [inst : CommSemiring R] →
          [inst_1 : AddCommMonoid M] →
            [inst_2 : AddCommMonoid N] →
              [inst_3 : Module R M] →
                [inst_4 : Module R N] →
                  Module.Basis ι R M →
                    [DecidableEq ι] → [Fintype ι] → TensorProduct R (Module.Dual R M) N ≃ₗ[R] M →ₗ[R] N

If M is free, the natural linear map $M^* ⊗ N → Hom(M, N)$ is an equivalence. This function provides this equivalence in return for a basis of M.

Defined in

Mathlib.LinearAlgebra.Contraction

Cited by

9 results in Mathlib

Foundations

Depth 101 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringAddCommMonoidAddCommMonoidModuleModuleDecidableEqFintype