Theorems · Definition · linear algebra

TensorProduct.dualDistribEquivOfBasis

{R : Type u_7} →
  {M : Type u_8} →
    {N : Type u_9} →
      {ι : Type u_10} →
        {κ : Type u_11} →
          [DecidableEq ι] →
            [DecidableEq κ] →
              [Fintype ι] →
                [Fintype κ] →
                  [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 →
                              Module.Basis κ R N →
                                TensorProduct R (Module.Dual R M) (Module.Dual R N) ≃ₗ[R]
                                  Module.Dual R (TensorProduct R M N)

A linear equivalence between Dual M ⊗ Dual N and Dual (M ⊗ N) given bases for M and N. It sends f ⊗ g to the composition of TensorProduct.map f g with the natural isomorphism R ⊗ R ≃ R.

Defined in

Mathlib.LinearAlgebra.Contraction

Cited by

2 results in Mathlib

Foundations

Depth 102 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

DecidableEqDecidableEqFintypeFintypeCommSemiringAddCommMonoidAddCommMonoidModuleModule