Documentation

Mathlib.LinearAlgebra.Dual

Dual vector spaces #

The dual space of an $R$-module $M$ is the $R$-module of $R$-linear maps $M \to R$.

Main definitions #

Main results #

@[reducible]
def Module.Dual (R : Type uR) (M : Type uM) [CommSemiring R] [AddCommMonoid M] [Module R M] :
Type (max uM uR)

The dual space of an R-module M is the R-module of linear maps M → R.

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    def Module.dualPairing (R : Type u_1) (M : Type u_2) [CommSemiring R] [AddCommMonoid M] [Module R M] :

    The canonical pairing of a vector space and its algebraic dual.

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      @[simp]
      theorem Module.dualPairing_apply (R : Type uR) (M : Type uM) [CommSemiring R] [AddCommMonoid M] [Module R M] (v : Module.Dual R M) (x : M) :
      ((Module.dualPairing R M) v) x = v x
      Equations

      Maps a module M to the dual of the dual of M. See Module.erange_coe and Module.evalEquiv.

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        @[simp]
        theorem Module.Dual.eval_apply (R : Type uR) (M : Type uM) [CommSemiring R] [AddCommMonoid M] [Module R M] (v : M) (a : Module.Dual R M) :
        ((Module.Dual.eval R M) v) a = a v
        def Module.Dual.transpose {R : Type uR} {M : Type uM} [CommSemiring R] [AddCommMonoid M] [Module R M] {M' : Type uM'} [AddCommMonoid M'] [Module R M'] :

        The transposition of linear maps, as a linear map from M →ₗ[R] M' to Dual R M' →ₗ[R] Dual R M.

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          theorem Module.Dual.transpose_apply {R : Type uR} {M : Type uM} [CommSemiring R] [AddCommMonoid M] [Module R M] {M' : Type uM'} [AddCommMonoid M'] [Module R M'] (u : M →ₗ[R] M') (l : Module.Dual R M') :
          (Module.Dual.transpose u) l = l ∘ₗ u
          theorem Module.Dual.transpose_comp {R : Type uR} {M : Type uM} [CommSemiring R] [AddCommMonoid M] [Module R M] {M' : Type uM'} [AddCommMonoid M'] [Module R M'] {M'' : Type uM''} [AddCommMonoid M''] [Module R M''] (u : M' →ₗ[R] M'') (v : M →ₗ[R] M') :
          Module.Dual.transpose (u ∘ₗ v) = Module.Dual.transpose v ∘ₗ Module.Dual.transpose u
          @[simp]
          theorem Module.dualProdDualEquivDual_symm_apply (R : Type uR) (M : Type uM) [CommSemiring R] [AddCommMonoid M] [Module R M] (M' : Type uM') [AddCommMonoid M'] [Module R M'] (f : M × M' →ₗ[R] R) :
          (LinearEquiv.symm (Module.dualProdDualEquivDual R M M')) f = (f ∘ₗ LinearMap.inl R M M', f ∘ₗ LinearMap.inr R M M')
          @[simp]
          theorem Module.dualProdDualEquivDual_apply_apply (R : Type uR) (M : Type uM) [CommSemiring R] [AddCommMonoid M] [Module R M] (M' : Type uM') [AddCommMonoid M'] [Module R M'] (f : (M →ₗ[R] R) × (M' →ₗ[R] R)) (a : M × M') :
          ((Module.dualProdDualEquivDual R M M') f) a = f.1 a.1 + f.2 a.2
          def Module.dualProdDualEquivDual (R : Type uR) (M : Type uM) [CommSemiring R] [AddCommMonoid M] [Module R M] (M' : Type uM') [AddCommMonoid M'] [Module R M'] :

          Taking duals distributes over products.

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            @[simp]
            theorem Module.dualProdDualEquivDual_apply (R : Type uR) (M : Type uM) [CommSemiring R] [AddCommMonoid M] [Module R M] (M' : Type uM') [AddCommMonoid M'] [Module R M'] (φ : Module.Dual R M) (ψ : Module.Dual R M') :
            def LinearMap.dualMap {R : Type u} [CommSemiring R] {M₁ : Type v} {M₂ : Type v'} [AddCommMonoid M₁] [Module R M₁] [AddCommMonoid M₂] [Module R M₂] (f : M₁ →ₗ[R] M₂) :

            Given a linear map f : M₁ →ₗ[R] M₂, f.dualMap is the linear map between the dual of M₂ and M₁ such that it maps the functional φ to φ ∘ f.

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              theorem LinearMap.dualMap_eq_lcomp {R : Type u} [CommSemiring R] {M₁ : Type v} {M₂ : Type v'} [AddCommMonoid M₁] [Module R M₁] [AddCommMonoid M₂] [Module R M₂] (f : M₁ →ₗ[R] M₂) :
              theorem LinearMap.dualMap_def {R : Type u} [CommSemiring R] {M₁ : Type v} {M₂ : Type v'} [AddCommMonoid M₁] [Module R M₁] [AddCommMonoid M₂] [Module R M₂] (f : M₁ →ₗ[R] M₂) :
              LinearMap.dualMap f = Module.Dual.transpose f
              theorem LinearMap.dualMap_apply' {R : Type u} [CommSemiring R] {M₁ : Type v} {M₂ : Type v'} [AddCommMonoid M₁] [Module R M₁] [AddCommMonoid M₂] [Module R M₂] (f : M₁ →ₗ[R] M₂) (g : Module.Dual R M₂) :
              (LinearMap.dualMap f) g = g ∘ₗ f
              @[simp]
              theorem LinearMap.dualMap_apply {R : Type u} [CommSemiring R] {M₁ : Type v} {M₂ : Type v'} [AddCommMonoid M₁] [Module R M₁] [AddCommMonoid M₂] [Module R M₂] (f : M₁ →ₗ[R] M₂) (g : Module.Dual R M₂) (x : M₁) :
              ((LinearMap.dualMap f) g) x = g (f x)
              @[simp]
              theorem LinearMap.dualMap_id {R : Type u} [CommSemiring R] {M₁ : Type v} [AddCommMonoid M₁] [Module R M₁] :
              LinearMap.dualMap LinearMap.id = LinearMap.id
              theorem LinearMap.dualMap_comp_dualMap {R : Type u} [CommSemiring R] {M₁ : Type v} {M₂ : Type v'} [AddCommMonoid M₁] [Module R M₁] [AddCommMonoid M₂] [Module R M₂] {M₃ : Type u_1} [AddCommGroup M₃] [Module R M₃] (f : M₁ →ₗ[R] M₂) (g : M₂ →ₗ[R] M₃) :
              theorem LinearMap.dualMap_injective_of_surjective {R : Type u} [CommSemiring R] {M₁ : Type v} {M₂ : Type v'} [AddCommMonoid M₁] [Module R M₁] [AddCommMonoid M₂] [Module R M₂] {f : M₁ →ₗ[R] M₂} (hf : Function.Surjective f) :

              If a linear map is surjective, then its dual is injective.

              def LinearEquiv.dualMap {R : Type u} [CommSemiring R] {M₁ : Type v} {M₂ : Type v'} [AddCommMonoid M₁] [Module R M₁] [AddCommMonoid M₂] [Module R M₂] (f : M₁ ≃ₗ[R] M₂) :

              The Linear_equiv version of LinearMap.dualMap.

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                @[simp]
                theorem LinearEquiv.dualMap_apply {R : Type u} [CommSemiring R] {M₁ : Type v} {M₂ : Type v'} [AddCommMonoid M₁] [Module R M₁] [AddCommMonoid M₂] [Module R M₂] (f : M₁ ≃ₗ[R] M₂) (g : Module.Dual R M₂) (x : M₁) :
                ((LinearEquiv.dualMap f) g) x = g (f x)
                @[simp]
                theorem LinearEquiv.dualMap_symm {R : Type u} [CommSemiring R] {M₁ : Type v} {M₂ : Type v'} [AddCommMonoid M₁] [Module R M₁] [AddCommMonoid M₂] [Module R M₂] {f : M₁ ≃ₗ[R] M₂} :
                theorem LinearEquiv.dualMap_trans {R : Type u} [CommSemiring R] {M₁ : Type v} {M₂ : Type v'} [AddCommMonoid M₁] [Module R M₁] [AddCommMonoid M₂] [Module R M₂] {M₃ : Type u_1} [AddCommGroup M₃] [Module R M₃] (f : M₁ ≃ₗ[R] M₂) (g : M₂ ≃ₗ[R] M₃) :
                @[simp]
                theorem Dual.apply_one_mul_eq {R : Type u} [CommSemiring R] (f : Module.Dual R R) (r : R) :
                f 1 * r = f r
                def Basis.toDual {R : Type uR} {M : Type uM} {ι : Type uι} [CommSemiring R] [AddCommMonoid M] [Module R M] [DecidableEq ι] (b : Basis ι R M) :

                The linear map from a vector space equipped with basis to its dual vector space, taking basis elements to corresponding dual basis elements.

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                  theorem Basis.toDual_apply {R : Type uR} {M : Type uM} {ι : Type uι} [CommSemiring R] [AddCommMonoid M] [Module R M] [DecidableEq ι] (b : Basis ι R M) (i : ι) (j : ι) :
                  ((Basis.toDual b) (b i)) (b j) = if i = j then 1 else 0
                  @[simp]
                  theorem Basis.toDual_total_left {R : Type uR} {M : Type uM} {ι : Type uι} [CommSemiring R] [AddCommMonoid M] [Module R M] [DecidableEq ι] (b : Basis ι R M) (f : ι →₀ R) (i : ι) :
                  ((Basis.toDual b) ((Finsupp.total ι M R b) f)) (b i) = f i
                  @[simp]
                  theorem Basis.toDual_total_right {R : Type uR} {M : Type uM} {ι : Type uι} [CommSemiring R] [AddCommMonoid M] [Module R M] [DecidableEq ι] (b : Basis ι R M) (f : ι →₀ R) (i : ι) :
                  ((Basis.toDual b) (b i)) ((Finsupp.total ι M R b) f) = f i
                  theorem Basis.toDual_apply_left {R : Type uR} {M : Type uM} {ι : Type uι} [CommSemiring R] [AddCommMonoid M] [Module R M] [DecidableEq ι] (b : Basis ι R M) (m : M) (i : ι) :
                  ((Basis.toDual b) m) (b i) = (b.repr m) i
                  theorem Basis.toDual_apply_right {R : Type uR} {M : Type uM} {ι : Type uι} [CommSemiring R] [AddCommMonoid M] [Module R M] [DecidableEq ι] (b : Basis ι R M) (i : ι) (m : M) :
                  ((Basis.toDual b) (b i)) m = (b.repr m) i
                  theorem Basis.coe_toDual_self {R : Type uR} {M : Type uM} {ι : Type uι} [CommSemiring R] [AddCommMonoid M] [Module R M] [DecidableEq ι] (b : Basis ι R M) (i : ι) :
                  (Basis.toDual b) (b i) = Basis.coord b i
                  def Basis.toDualFlip {R : Type uR} {M : Type uM} {ι : Type uι} [CommSemiring R] [AddCommMonoid M] [Module R M] [DecidableEq ι] (b : Basis ι R M) (m : M) :

                  h.toDual_flip v is the linear map sending w to h.toDual w v.

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                    theorem Basis.toDualFlip_apply {R : Type uR} {M : Type uM} {ι : Type uι} [CommSemiring R] [AddCommMonoid M] [Module R M] [DecidableEq ι] (b : Basis ι R M) (m₁ : M) (m₂ : M) :
                    (Basis.toDualFlip b m₁) m₂ = ((Basis.toDual b) m₂) m₁
                    theorem Basis.toDual_eq_repr {R : Type uR} {M : Type uM} {ι : Type uι} [CommSemiring R] [AddCommMonoid M] [Module R M] [DecidableEq ι] (b : Basis ι R M) (m : M) (i : ι) :
                    ((Basis.toDual b) m) (b i) = (b.repr m) i
                    theorem Basis.toDual_eq_equivFun {R : Type uR} {M : Type uM} {ι : Type uι} [CommSemiring R] [AddCommMonoid M] [Module R M] [DecidableEq ι] (b : Basis ι R M) [Finite ι] (m : M) (i : ι) :
                    ((Basis.toDual b) m) (b i) = (Basis.equivFun b) m i
                    theorem Basis.toDual_injective {R : Type uR} {M : Type uM} {ι : Type uι} [CommSemiring R] [AddCommMonoid M] [Module R M] [DecidableEq ι] (b : Basis ι R M) :
                    theorem Basis.toDual_inj {R : Type uR} {M : Type uM} {ι : Type uι} [CommSemiring R] [AddCommMonoid M] [Module R M] [DecidableEq ι] (b : Basis ι R M) (m : M) (a : (Basis.toDual b) m = 0) :
                    m = 0
                    theorem Basis.toDual_ker {R : Type uR} {M : Type uM} {ι : Type uι} [CommSemiring R] [AddCommMonoid M] [Module R M] [DecidableEq ι] (b : Basis ι R M) :
                    theorem Basis.toDual_range {R : Type uR} {M : Type uM} {ι : Type uι} [CommSemiring R] [AddCommMonoid M] [Module R M] [DecidableEq ι] (b : Basis ι R M) [Finite ι] :
                    @[simp]
                    theorem Basis.sum_dual_apply_smul_coord {R : Type uR} {M : Type uM} {ι : Type uι} [CommSemiring R] [AddCommMonoid M] [Module R M] [Fintype ι] (b : Basis ι R M) (f : Module.Dual R M) :
                    (Finset.sum Finset.univ fun (x : ι) => f (b x) Basis.coord b x) = f
                    def Basis.toDualEquiv {R : Type uR} {M : Type uM} {ι : Type uι} [CommRing R] [AddCommGroup M] [Module R M] [DecidableEq ι] (b : Basis ι R M) [Finite ι] :

                    A vector space is linearly equivalent to its dual space.

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                      @[simp]
                      theorem Basis.toDualEquiv_apply {R : Type uR} {M : Type uM} {ι : Type uι} [CommRing R] [AddCommGroup M] [Module R M] [DecidableEq ι] (b : Basis ι R M) [Finite ι] (m : M) :

                      A vector space over a field is isomorphic to its dual if and only if it is finite-dimensional: a consequence of the Erdős-Kaplansky theorem.

                      def Basis.dualBasis {R : Type uR} {M : Type uM} {ι : Type uι} [CommRing R] [AddCommGroup M] [Module R M] [DecidableEq ι] (b : Basis ι R M) [Finite ι] :
                      Basis ι R (Module.Dual R M)

                      Maps a basis for V to a basis for the dual space.

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                        theorem Basis.dualBasis_apply_self {R : Type uR} {M : Type uM} {ι : Type uι} [CommRing R] [AddCommGroup M] [Module R M] [DecidableEq ι] (b : Basis ι R M) [Finite ι] (i : ι) (j : ι) :
                        ((Basis.dualBasis b) i) (b j) = if j = i then 1 else 0
                        theorem Basis.total_dualBasis {R : Type uR} {M : Type uM} {ι : Type uι} [CommRing R] [AddCommGroup M] [Module R M] [DecidableEq ι] (b : Basis ι R M) [Finite ι] (f : ι →₀ R) (i : ι) :
                        ((Finsupp.total ι (Module.Dual R M) R (Basis.dualBasis b)) f) (b i) = f i
                        theorem Basis.dualBasis_repr {R : Type uR} {M : Type uM} {ι : Type uι} [CommRing R] [AddCommGroup M] [Module R M] [DecidableEq ι] (b : Basis ι R M) [Finite ι] (l : Module.Dual R M) (i : ι) :
                        ((Basis.dualBasis b).repr l) i = l (b i)
                        theorem Basis.dualBasis_apply {R : Type uR} {M : Type uM} {ι : Type uι} [CommRing R] [AddCommGroup M] [Module R M] [DecidableEq ι] (b : Basis ι R M) [Finite ι] (i : ι) (m : M) :
                        ((Basis.dualBasis b) i) m = (b.repr m) i
                        @[simp]
                        theorem Basis.coe_dualBasis {R : Type uR} {M : Type uM} {ι : Type uι} [CommRing R] [AddCommGroup M] [Module R M] [DecidableEq ι] (b : Basis ι R M) [Finite ι] :
                        @[simp]
                        theorem Basis.toDual_toDual {R : Type uR} {M : Type uM} {ι : Type uι} [CommRing R] [AddCommGroup M] [Module R M] [DecidableEq ι] (b : Basis ι R M) [Finite ι] :
                        theorem Basis.dualBasis_equivFun {R : Type uR} {M : Type uM} {ι : Type uι} [CommRing R] [AddCommGroup M] [Module R M] [DecidableEq ι] (b : Basis ι R M) [Finite ι] (l : Module.Dual R M) (i : ι) :
                        (Basis.equivFun (Basis.dualBasis b)) l i = l (b i)
                        theorem Basis.eval_ker {R : Type uR} {M : Type uM} [CommRing R] [AddCommGroup M] [Module R M] {ι : Type u_1} (b : Basis ι R M) :
                        theorem Basis.eval_range {R : Type uR} {M : Type uM} [CommRing R] [AddCommGroup M] [Module R M] {ι : Type u_1} [Finite ι] (b : Basis ι R M) :
                        instance Basis.dual_free {R : Type uR} {M : Type uM} [CommRing R] [AddCommGroup M] [Module R M] [Module.Finite R M] [Module.Free R M] :
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                        • =
                        instance Basis.dual_finite {R : Type uR} {M : Type uM} [CommRing R] [AddCommGroup M] [Module R M] [Module.Finite R M] [Module.Free R M] :
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                        • =
                        @[simp]
                        theorem Basis.total_coord {R : Type uR} {M : Type uM} {ι : Type uι} [CommRing R] [AddCommGroup M] [Module R M] [Finite ι] (b : Basis ι R M) (f : ι →₀ R) (i : ι) :
                        ((Finsupp.total ι (Module.Dual R M) R (Basis.coord b)) f) (b i) = f i

                        simp normal form version of total_dualBasis

                        theorem Basis.dual_rank_eq {K : Type uK} {V : Type uV} {ι : Type uι} [CommRing K] [AddCommGroup V] [Module K V] [Finite ι] (b : Basis ι K V) :
                        theorem Module.eval_apply_eq_zero_iff (K : Type uK) {V : Type uV} [CommRing K] [AddCommGroup V] [Module K V] [Module.Free K V] (v : V) :
                        (Module.Dual.eval K V) v = 0 v = 0
                        theorem Module.forall_dual_apply_eq_zero_iff (K : Type uK) {V : Type uV} [CommRing K] [AddCommGroup V] [Module K V] [Module.Free K V] (v : V) :
                        (∀ (φ : Module.Dual K V), φ v = 0) v = 0
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                        Equations
                        • =
                        class Module.IsReflexive (R : Type u_1) (M : Type u_2) [CommRing R] [AddCommGroup M] [Module R M] :

                        A reflexive module is one for which the natural map to its double dual is a bijection.

                        Any finitely-generated free module (and thus any finite-dimensional vector space) is reflexive. See Module.IsReflexive.of_finite_of_free.

                        Instances
                          Equations
                          • =

                          The bijection between a reflexive module and its double dual, bundled as a LinearEquiv.

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                            @[simp]
                            theorem Module.evalEquiv_apply (R : Type u_1) (M : Type u_2) [CommRing R] [AddCommGroup M] [Module R M] [Module.IsReflexive R M] (m : M) :
                            @[simp]
                            theorem Module.apply_evalEquiv_symm_apply (R : Type u_1) (M : Type u_2) [CommRing R] [AddCommGroup M] [Module R M] [Module.IsReflexive R M] (f : Module.Dual R M) (g : Module.Dual R (Module.Dual R M)) :

                            The dual of a reflexive module is reflexive.

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                            • =

                            The isomorphism Module.evalEquiv induces an order isomorphism on subspaces.

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                              @[simp]
                              instance Prod.instModuleIsReflexive (R : Type u_1) (M : Type u_2) (N : Type u_3) [CommRing R] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [Module.IsReflexive R M] [Module.IsReflexive R N] :
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                              theorem Module.equiv {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommRing R] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [Module.IsReflexive R M] (e : M ≃ₗ[R] N) :
                              Equations
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                              theorem Submodule.exists_dual_map_eq_bot_of_nmem {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] {p : Submodule R M} {x : M} (hx : xp) (hp' : Module.Free R (M p)) :
                              ∃ (f : Module.Dual R M), f x 0 Submodule.map f p =
                              theorem Submodule.exists_dual_map_eq_bot_of_lt_top {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] {p : Submodule R M} (hp : p < ) (hp' : Module.Free R (M p)) :
                              ∃ (f : Module.Dual R M), f 0 Submodule.map f p =

                              Try using Set.to_finite to dispatch a Set.finite goal.

                              Equations
                              • One or more equations did not get rendered due to their size.
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                                structure Module.DualBases {R : Type u_1} {M : Type u_2} {ι : Type u_3} [CommSemiring R] [AddCommMonoid M] [Module R M] [DecidableEq ι] (e : ιM) (ε : ιModule.Dual R M) :

                                e and ε have characteristic properties of a basis and its dual

                                • eval : ∀ (i j : ι), (ε i) (e j) = if i = j then 1 else 0
                                • total : ∀ {m : M}, (∀ (i : ι), (ε i) m = 0)m = 0
                                • finite : ∀ (m : M), Set.Finite {i : ι | (ε i) m 0}
                                Instances For
                                  def Module.DualBases.coeffs {R : Type u_1} {M : Type u_2} {ι : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] {e : ιM} {ε : ιModule.Dual R M} [DecidableEq ι] (h : Module.DualBases e ε) (m : M) :
                                  ι →₀ R

                                  The coefficients of v on the basis e

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                                    @[simp]
                                    theorem Module.DualBases.coeffs_apply {R : Type u_1} {M : Type u_2} {ι : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] {e : ιM} {ε : ιModule.Dual R M} [DecidableEq ι] (h : Module.DualBases e ε) (m : M) (i : ι) :
                                    (Module.DualBases.coeffs h m) i = (ε i) m
                                    def Module.DualBases.lc {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] {ι : Type u_4} (e : ιM) (l : ι →₀ R) :
                                    M

                                    linear combinations of elements of e. This is a convenient abbreviation for Finsupp.total _ M R e l

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                                      theorem Module.DualBases.lc_def {R : Type u_1} {M : Type u_2} {ι : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] (e : ιM) (l : ι →₀ R) :
                                      theorem Module.DualBases.dual_lc {R : Type u_1} {M : Type u_2} {ι : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] {e : ιM} {ε : ιModule.Dual R M} [DecidableEq ι] (h : Module.DualBases e ε) (l : ι →₀ R) (i : ι) :
                                      (ε i) (Module.DualBases.lc e l) = l i
                                      @[simp]
                                      theorem Module.DualBases.coeffs_lc {R : Type u_1} {M : Type u_2} {ι : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] {e : ιM} {ε : ιModule.Dual R M} [DecidableEq ι] (h : Module.DualBases e ε) (l : ι →₀ R) :
                                      @[simp]
                                      theorem Module.DualBases.lc_coeffs {R : Type u_1} {M : Type u_2} {ι : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] {e : ιM} {ε : ιModule.Dual R M} [DecidableEq ι] (h : Module.DualBases e ε) (m : M) :

                                      For any m : M n, \sum_{p ∈ Q n} (ε p m) • e p = m

                                      @[simp]
                                      theorem Module.DualBases.basis_repr_apply {R : Type u_1} {M : Type u_2} {ι : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] {e : ιM} {ε : ιModule.Dual R M} [DecidableEq ι] (h : Module.DualBases e ε) (m : M) :
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                                      theorem Module.DualBases.basis_repr_symm_apply {R : Type u_1} {M : Type u_2} {ι : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] {e : ιM} {ε : ιModule.Dual R M} [DecidableEq ι] (h : Module.DualBases e ε) (l : ι →₀ R) :
                                      def Module.DualBases.basis {R : Type u_1} {M : Type u_2} {ι : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] {e : ιM} {ε : ιModule.Dual R M} [DecidableEq ι] (h : Module.DualBases e ε) :
                                      Basis ι R M

                                      (h : DualBases e ε).basis shows the family of vectors e forms a basis.

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                                        theorem Module.DualBases.coe_basis {R : Type u_1} {M : Type u_2} {ι : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] {e : ιM} {ε : ιModule.Dual R M} [DecidableEq ι] (h : Module.DualBases e ε) :
                                        theorem Module.DualBases.mem_of_mem_span {R : Type u_1} {M : Type u_2} {ι : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] {e : ιM} {ε : ιModule.Dual R M} [DecidableEq ι] (h : Module.DualBases e ε) {H : Set ι} {x : M} (hmem : x Submodule.span R (e '' H)) (i : ι) :
                                        (ε i) x 0i H
                                        theorem Module.DualBases.coe_dualBasis {R : Type u_1} {M : Type u_2} {ι : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] {e : ιM} {ε : ιModule.Dual R M} [DecidableEq ι] (h : Module.DualBases e ε) [Finite ι] :

                                        The dualRestrict of a submodule W of M is the linear map from the dual of M to the dual of W such that the domain of each linear map is restricted to W.

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                                          theorem Submodule.dualRestrict_apply {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] (W : Submodule R M) (φ : Module.Dual R M) (x : W) :
                                          ((Submodule.dualRestrict W) φ) x = φ x

                                          The dualAnnihilator of a submodule W is the set of linear maps φ such that φ w = 0 for all w ∈ W.

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                                            theorem Submodule.mem_dualAnnihilator {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] {W : Submodule R M} (φ : Module.Dual R M) :
                                            φ Submodule.dualAnnihilator W wW, φ w = 0

                                            That $\operatorname{ker}(\iota^* : V^* \to W^*) = \operatorname{ann}(W)$. This is the definition of the dual annihilator of the submodule $W$.

                                            The dualAnnihilator of a submodule of the dual space pulled back along the evaluation map Module.Dual.eval.

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                                              theorem Submodule.mem_dualCoannihilator {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] {Φ : Submodule R (Module.Dual R M)} (x : M) :
                                              x Submodule.dualCoannihilator Φ φΦ, φ x = 0
                                              theorem Submodule.dualAnnihilator_gc (R : Type u) (M : Type v) [CommSemiring R] [AddCommMonoid M] [Module R M] :
                                              GaloisConnection (OrderDual.toDual Submodule.dualAnnihilator) (Submodule.dualCoannihilator OrderDual.ofDual)
                                              theorem Submodule.dualAnnihilator_iSup_eq {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] {ι : Type u_1} (U : ιSubmodule R M) :
                                              Submodule.dualAnnihilator (⨆ (i : ι), U i) = ⨅ (i : ι), Submodule.dualAnnihilator (U i)
                                              theorem Submodule.dualCoannihilator_iSup_eq {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] {ι : Type u_1} (U : ιSubmodule R (Module.Dual R M)) :
                                              Submodule.dualCoannihilator (⨆ (i : ι), U i) = ⨅ (i : ι), Submodule.dualCoannihilator (U i)
                                              theorem Submodule.iSup_dualAnnihilator_le_iInf {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] {ι : Type u_1} (U : ιSubmodule R M) :
                                              ⨆ (i : ι), Submodule.dualAnnihilator (U i) Submodule.dualAnnihilator (⨅ (i : ι), U i)

                                              See also Subspace.dualAnnihilator_iInf_eq for vector subspaces when ι is finite.

                                              theorem Subspace.forall_mem_dualAnnihilator_apply_eq_zero_iff {K : Type u} {V : Type v} [Field K] [AddCommGroup V] [Module K V] (W : Subspace K V) (v : V) :
                                              (φSubmodule.dualAnnihilator W, φ v = 0) v W
                                              def Subspace.dualAnnihilatorGci (K : Type u_1) (V : Type u_2) [Field K] [AddCommGroup V] [Module K V] :
                                              GaloisCoinsertion (OrderDual.toDual Submodule.dualAnnihilator) (Submodule.dualCoannihilator OrderDual.ofDual)

                                              Submodule.dualAnnihilator and Submodule.dualCoannihilator form a Galois coinsertion.

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                                                noncomputable def Subspace.dualLift {K : Type u} {V : Type v} [Field K] [AddCommGroup V] [Module K V] (W : Subspace K V) :

                                                Given a subspace W of V and an element of its dual φ, dualLift W φ is an arbitrary extension of φ to an element of the dual of V. That is, dualLift W φ sends w ∈ W to φ x and x in a chosen complement of W to 0.

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                                                  theorem Subspace.dualLift_of_subtype {K : Type u} {V : Type v} [Field K] [AddCommGroup V] [Module K V] {W : Subspace K V} {φ : Module.Dual K W} (w : W) :
                                                  ((Subspace.dualLift W) φ) w = φ w
                                                  theorem Subspace.dualLift_of_mem {K : Type u} {V : Type v} [Field K] [AddCommGroup V] [Module K V] {W : Subspace K V} {φ : Module.Dual K W} {w : V} (hw : w W) :
                                                  ((Subspace.dualLift W) φ) w = φ { val := w, property := hw }
                                                  noncomputable def Subspace.quotAnnihilatorEquiv {K : Type u} {V : Type v} [Field K] [AddCommGroup V] [Module K V] (W : Subspace K V) :

                                                  The quotient by the dualAnnihilator of a subspace is isomorphic to the dual of that subspace.

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                                                    noncomputable def Subspace.dualEquivDual {K : Type u} {V : Type v} [Field K] [AddCommGroup V] [Module K V] (W : Subspace K V) :

                                                    The natural isomorphism from the dual of a subspace W to W.dualLift.range.

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                                                      theorem Subspace.dualEquivDual_apply {K : Type u} {V : Type v} [Field K] [AddCommGroup V] [Module K V] {W : Subspace K V} (φ : Module.Dual K W) :
                                                      (Subspace.dualEquivDual W) φ = { val := (Subspace.dualLift W) φ, property := }
                                                      noncomputable def Subspace.quotEquivAnnihilator {K : Type u} {V : Type v} [Field K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] (W : Subspace K V) :

                                                      The quotient by a subspace is isomorphic to its dual annihilator.

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                                                        def Submodule.dualCopairing {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] (W : Submodule R M) :

                                                        Given a submodule, corestrict to the pairing on M ⧸ W by simultaneously restricting to W.dualAnnihilator.

                                                        See Subspace.dualCopairing_nondegenerate.

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                                                          theorem Submodule.dualCopairing_apply {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] {W : Submodule R M} (φ : (Submodule.dualAnnihilator W)) (x : M) :

                                                          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.

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                                                            theorem Submodule.dualPairing_apply {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] {W : Submodule R M} (φ : Module.Dual R M) (x : W) :

                                                            That $\operatorname{im}(q^* : (V/W)^* \to V^*) = \operatorname{ann}(W)$.

                                                            Equivalence $(M/W)^* \cong \operatorname{ann}(W)$. That is, there is a one-to-one correspondence between the dual of M ⧸ W and those elements of the dual of M that vanish on W.

                                                            The inverse of this is Submodule.dualCopairing.

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                                                              The pairing between a submodule W of a dual module Dual R M and the quotient of M by the coannihilator of W, which is always nondegenerate.

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                                                                theorem Submodule.quotDualCoannihilatorToDual_apply {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] (W : Submodule R (Module.Dual R M)) (m : M) (w : W) :
                                                                theorem Module.Dual.range_eq_top_of_ne_zero {K : Type uK} [Field K] {V₁ : Type uV₁} [AddCommGroup V₁] [Module K V₁] {f : Module.Dual K V₁} (hf : f 0) :
                                                                theorem Module.Dual.finrank_ker_add_one_of_ne_zero {K : Type uK} [Field K] {V₁ : Type uV₁} [AddCommGroup V₁] [Module K V₁] [FiniteDimensional K V₁] {f : Module.Dual K V₁} (hf : f 0) :
                                                                theorem Module.Dual.isCompl_ker_of_disjoint_of_ne_bot {K : Type uK} [Field K] {V₁ : Type uV₁} [AddCommGroup V₁] [Module K V₁] [FiniteDimensional K V₁] {f : Module.Dual K V₁} (hf : f 0) {p : Submodule K V₁} (hpf : Disjoint (LinearMap.ker f) p) (hp : p ) :
                                                                theorem LinearMap.dualMap_surjective_of_injective {K : Type uK} [Field K] {V₁ : Type uV₁} {V₂ : Type uV₂} [AddCommGroup V₁] [Module K V₁] [AddCommGroup V₂] [Module K V₂] {f : V₁ →ₗ[K] V₂} (hf : Function.Injective f) :
                                                                theorem LinearMap.range_dualMap_eq_dualAnnihilator_ker {K : Type uK} [Field K] {V₁ : Type uV₁} {V₂ : Type uV₂} [AddCommGroup V₁] [Module K V₁] [AddCommGroup V₂] [Module K V₂] (f : V₁ →ₗ[K] V₂) :
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                                                                theorem LinearMap.dualMap_surjective_iff {K : Type uK} [Field K] {V₁ : Type uV₁} {V₂ : Type uV₂} [AddCommGroup V₁] [Module K V₁] [AddCommGroup V₂] [Module K V₂] {f : V₁ →ₗ[K] V₂} :

                                                                For vector spaces, f.dualMap is surjective if and only if f is injective

                                                                theorem Subspace.dualPairing_eq {K : Type uK} [Field K] {V₁ : Type uV₁} [AddCommGroup V₁] [Module K V₁] (W : Subspace K V₁) :
                                                                theorem Subspace.dualAnnihilator_iInf_eq {K : Type uK} [Field K] {V₁ : Type uV₁} [AddCommGroup V₁] [Module K V₁] {ι : Type u_1} [Finite ι] (W : ιSubspace K V₁) :
                                                                Submodule.dualAnnihilator (⨅ (i : ι), W i) = ⨆ (i : ι), Submodule.dualAnnihilator (W i)
                                                                theorem Subspace.isCompl_dualAnnihilator {K : Type uK} [Field K] {V₁ : Type uV₁} [AddCommGroup V₁] [Module K V₁] {W : Subspace K V₁} {W' : Subspace K V₁} (h : IsCompl W W') :

                                                                For vector spaces, dual annihilators carry direct sum decompositions to direct sum decompositions.

                                                                def Subspace.dualQuotDistrib {K : Type uK} [Field K] {V₁ : Type uV₁} [AddCommGroup V₁] [Module K V₁] [FiniteDimensional K V₁] (W : Subspace K V₁) :

                                                                For finite-dimensional vector spaces, one can distribute duals over quotients by identifying W.dualLift.range with W. Note that this depends on a choice of splitting of V₁.

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                                                                  theorem LinearMap.dualMap_injective_iff {K : Type uK} [Field K] {V₁ : Type uV₁} {V₂ : Type uV₂} [AddCommGroup V₁] [Module K V₁] [AddCommGroup V₂] [Module K V₂] {f : V₁ →ₗ[K] V₂} :

                                                                  f.dualMap is injective if and only if f is surjective

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                                                                  theorem LinearMap.dualMap_bijective_iff {K : Type uK} [Field K] {V₁ : Type uV₁} {V₂ : Type uV₂} [AddCommGroup V₁] [Module K V₁] [AddCommGroup V₂] [Module K V₂] {f : V₁ →ₗ[K] V₂} :

                                                                  f.dualMap is bijective if and only if f is

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                                                                  theorem LinearMap.dualAnnihilator_ker_eq_range_flip {K : Type uK} [Field K] {V₁ : Type uV₁} {V₂ : Type uV₂} [AddCommGroup V₁] [Module K V₁] [AddCommGroup V₂] [Module K V₂] {B : V₁ →ₗ[K] V₂ →ₗ[K] K} [Module.IsReflexive K V₂] :
                                                                  theorem LinearMap.flip_injective_iff₁ {K : Type uK} [Field K] {V₁ : Type uV₁} {V₂ : Type uV₂} [AddCommGroup V₁] [Module K V₁] [AddCommGroup V₂] [Module K V₂] {B : V₁ →ₗ[K] V₂ →ₗ[K] K} [FiniteDimensional K V₁] :
                                                                  theorem LinearMap.flip_injective_iff₂ {K : Type uK} [Field K] {V₁ : Type uV₁} {V₂ : Type uV₂} [AddCommGroup V₁] [Module K V₁] [AddCommGroup V₂] [Module K V₂] {B : V₁ →ₗ[K] V₂ →ₗ[K] K} [FiniteDimensional K V₂] :
                                                                  theorem LinearMap.flip_surjective_iff₁ {K : Type uK} [Field K] {V₁ : Type uV₁} {V₂ : Type uV₂} [AddCommGroup V₁] [Module K V₁] [AddCommGroup V₂] [Module K V₂] {B : V₁ →ₗ[K] V₂ →ₗ[K] K} [FiniteDimensional K V₁] :
                                                                  theorem LinearMap.flip_surjective_iff₂ {K : Type uK} [Field K] {V₁ : Type uV₁} {V₂ : Type uV₂} [AddCommGroup V₁] [Module K V₁] [AddCommGroup V₂] [Module K V₂] {B : V₁ →ₗ[K] V₂ →ₗ[K] K} [FiniteDimensional K V₂] :
                                                                  theorem LinearMap.flip_bijective_iff₁ {K : Type uK} [Field K] {V₁ : Type uV₁} {V₂ : Type uV₂} [AddCommGroup V₁] [Module K V₁] [AddCommGroup V₂] [Module K V₂] {B : V₁ →ₗ[K] V₂ →ₗ[K] K} [FiniteDimensional K V₁] :
                                                                  theorem LinearMap.flip_bijective_iff₂ {K : Type uK} [Field K] {V₁ : Type uV₁} {V₂ : Type uV₂} [AddCommGroup V₁] [Module K V₁] [AddCommGroup V₂] [Module K V₂] {B : V₁ →ₗ[K] V₂ →ₗ[K] K} [FiniteDimensional K V₂] :
                                                                  def Subspace.orderIsoFiniteCodimDim {K : Type u_1} {V : Type u_2} [Field K] [AddCommGroup V] [Module K V] :
                                                                  { W : Subspace K V // FiniteDimensional K (V W) } ≃o { W : Subspace K (Module.Dual K V) // FiniteDimensional K W }ᵒᵈ

                                                                  For any vector space, dualAnnihilator and dualCoannihilator gives an antitone order isomorphism between the finite-codimensional subspaces in the vector space and the finite-dimensional subspaces in its dual.

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                                                                    For any finite-dimensional vector space, dualAnnihilator and dualCoannihilator give an antitone order isomorphism between the subspaces in the vector space and the subspaces in its dual.

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                                                                      The canonical linear map from Dual M ⊗ Dual N to Dual (M ⊗ N), sending f ⊗ g to the composition of TensorProduct.map f g with the natural isomorphism R ⊗ R ≃ R.

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                                                                        theorem TensorProduct.dualDistrib_apply {R : Type u_1} {M : Type u_3} {N : Type u_4} [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (f : Module.Dual R M) (g : Module.Dual R N) (m : M) (n : N) :
                                                                        ((TensorProduct.dualDistrib R M N) (f ⊗ₜ[R] g)) (m ⊗ₜ[R] n) = f m * g n

                                                                        Heterobasic version of TensorProduct.dualDistrib

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                                                                          theorem TensorProduct.AlgebraTensorModule.dualDistrib_apply {R : Type u_1} (A : Type u_2) {M : Type u_3} {N : Type u_4} [CommSemiring R] [CommSemiring A] [Algebra R A] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module A M] [Module R N] [IsScalarTower R A M] (f : Module.Dual A M) (g : Module.Dual R N) (m : M) (n : N) :
                                                                          noncomputable def TensorProduct.dualDistribInvOfBasis {R : Type u_1} {M : Type u_3} {N : Type u_4} {ι : Type u_5} {κ : Type u_6} [DecidableEq ι] [DecidableEq κ] [Fintype ι] [Fintype κ] [CommRing R] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] (b : Basis ι R M) (c : Basis κ R N) :

                                                                          An inverse to TensorProduct.dualDistrib given bases.

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                                                                            theorem TensorProduct.dualDistribInvOfBasis_apply {R : Type u_1} {M : Type u_3} {N : Type u_4} {ι : Type u_5} {κ : Type u_6} [DecidableEq ι] [DecidableEq κ] [Fintype ι] [Fintype κ] [CommRing R] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] (b : Basis ι R M) (c : Basis κ R N) (f : Module.Dual R (TensorProduct R M N)) :
                                                                            (TensorProduct.dualDistribInvOfBasis b c) f = Finset.sum Finset.univ fun (i : ι) => Finset.sum Finset.univ fun (j : κ) => f (b i ⊗ₜ[R] c j) (Basis.dualBasis b) i ⊗ₜ[R] (Basis.dualBasis c) j
                                                                            theorem TensorProduct.dualDistrib_dualDistribInvOfBasis_left_inverse {R : Type u_1} {M : Type u_3} {N : Type u_4} {ι : Type u_5} {κ : Type u_6} [DecidableEq ι] [DecidableEq κ] [Fintype ι] [Fintype κ] [CommRing R] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] (b : Basis ι R M) (c : Basis κ R N) :
                                                                            theorem TensorProduct.dualDistrib_dualDistribInvOfBasis_right_inverse {R : Type u_1} {M : Type u_3} {N : Type u_4} {ι : Type u_5} {κ : Type u_6} [DecidableEq ι] [DecidableEq κ] [Fintype ι] [Fintype κ] [CommRing R] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] (b : Basis ι R M) (c : Basis κ R N) :
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                                                                            theorem TensorProduct.dualDistribEquivOfBasis_apply_apply {R : Type u_1} {M : Type u_3} {N : Type u_4} {ι : Type u_5} {κ : Type u_6} [DecidableEq ι] [DecidableEq κ] [Fintype ι] [Fintype κ] [CommRing R] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] (b : Basis ι R M) (c : Basis κ R N) :
                                                                            ∀ (a : TensorProduct R (Module.Dual R M) (Module.Dual R N)) (a_1 : TensorProduct R M N), ((TensorProduct.dualDistribEquivOfBasis b c) a) a_1 = (TensorProduct.lid R R) (((TensorProduct.homTensorHomMap R M N R R) a) a_1)
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                                                                            theorem TensorProduct.dualDistribEquivOfBasis_symm_apply {R : Type u_1} {M : Type u_3} {N : Type u_4} {ι : Type u_5} {κ : Type u_6} [DecidableEq ι] [DecidableEq κ] [Fintype ι] [Fintype κ] [CommRing R] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] (b : Basis ι R M) (c : Basis κ R N) (a : Module.Dual R (TensorProduct R M N)) :
                                                                            (LinearEquiv.symm (TensorProduct.dualDistribEquivOfBasis b c)) a = Finset.sum Finset.univ fun (x : ι) => Finset.sum Finset.univ fun (x_1 : κ) => a (b x ⊗ₜ[R] c x_1) Basis.coord b x ⊗ₜ[R] Basis.coord c x_1
                                                                            noncomputable def TensorProduct.dualDistribEquivOfBasis {R : Type u_1} {M : Type u_3} {N : Type u_4} {ι : Type u_5} {κ : Type u_6} [DecidableEq ι] [DecidableEq κ] [Fintype ι] [Fintype κ] [CommRing R] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] (b : Basis ι R M) (c : Basis κ R 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.

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                                                                              noncomputable def TensorProduct.dualDistribEquiv (R : Type u_1) (M : Type u_3) (N : Type u_4) [CommRing R] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [Module.Finite R M] [Module.Finite R N] [Module.Free R M] [Module.Free R N] :

                                                                              A linear equivalence between Dual M ⊗ Dual N and Dual (M ⊗ N) when M and N are finite free modules. It sends f ⊗ g to the composition of TensorProduct.map f g with the natural isomorphism R ⊗ R ≃ R.

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