Dual vector spaces

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

Main definitions

• Duals and transposes:
  • Module.Dual R M defines the dual space of the R-module M, as M →ₗ[R] R.
  • Module.dualPairing R M is the canonical pairing between Dual R M and M.
  • Module.Dual.eval R M : M →ₗ[R] Dual R (Dual R) is the canonical map to the double dual.
  • Module.Dual.transpose is the linear map from M →ₗ[R] M' to Dual R M' →ₗ[R] Dual R M.
  • LinearMap.dualMap is Module.Dual.transpose of a given linear map, for dot notation.
  • LinearEquiv.dualMap is for the dual of an equivalence.
• Bases:
  • Basis.toDual produces the map M →ₗ[R] Dual R M associated to a basis for an R-module M.
  • Basis.toDual_equiv is the equivalence M ≃ₗ[R] Dual R M associated to a finite basis.
  • Basis.dualBasis is a basis for Dual R M given a finite basis for M.
  • Module.dual_bases e ε is the proposition that the families e of vectors and ε of dual vectors have the characteristic properties of a basis and a dual.
• Submodules:
  • Submodule.dualRestrict W is the transpose Dual R M →ₗ[R] Dual R W of the inclusion map.
  • Submodule.dualAnnihilator W is the kernel of W.dualRestrict. That is, it is the submodule of dual R M whose elements all annihilate W.
  • Submodule.dualRestrict_comap W' is the dual annihilator of W' : Submodule R (Dual R M), pulled back along Module.Dual.eval R M.
  • Submodule.dualCopairing W is the canonical pairing between W.dualAnnihilator and M ⧸ W. It is nondegenerate for vector spaces (subspace.dualCopairing_nondegenerate).
  • Submodule.dualPairing W is the canonical pairing between Dual R M ⧸ W.dualAnnihilator and W. It is nondegenerate for vector spaces (Subspace.dualPairing_nondegenerate).
• Vector spaces:
  • Subspace.dualLift W is an arbitrary section (using choice) of Submodule.dualRestrict W.

Main results

• Bases:
  • Module.dualBasis.basis and Module.dualBasis.coe_basis: if e and ε form a dual pair, then e is a basis.
  • Module.dualBasis.coe_dualBasis: if e and ε form a dual pair, then ε is a basis.
• Annihilators:
  • Module.dualAnnihilator_gc R M is the antitone Galois correspondence between Submodule.dualAnnihilator and Submodule.dualConnihilator.
  • LinearMap.ker_dual_map_eq_dualAnnihilator_range says that f.dual_map.ker = f.range.dualAnnihilator
  • LinearMap.range_dual_map_eq_dualAnnihilator_ker_of_subtype_range_surjective says that f.dual_map.range = f.ker.dualAnnihilator; this is specialized to vector spaces in LinearMap.range_dual_map_eq_dualAnnihilator_ker.
  • Submodule.dualQuotEquivDualAnnihilator is the equivalence Dual R (M ⧸ W) ≃ₗ[R] W.dualAnnihilator
  • Submodule.quotDualCoannihilatorToDual is the nondegenerate pairing M ⧸ W.dualCoannihilator →ₗ[R] Dual R W. It is an perfect pairing when R is a field and W is finite-dimensional.
• Vector spaces:
  • Subspace.dualAnnihilator_dualConnihilator_eq says that the double dual annihilator, pulled back ground Module.Dual.eval, is the original submodule.
  • Subspace.dualAnnihilator_gci says that module.dualAnnihilator_gc R M is an antitone Galois coinsertion.
  • Subspace.quotAnnihilatorEquiv is the equivalence Dual K V ⧸ W.dualAnnihilator ≃ₗ[K] Dual K W.
  • LinearMap.dualPairing_nondegenerate says that Module.dualPairing is nondegenerate.
  • Subspace.is_compl_dualAnnihilator says that the dual annihilator carries complementary subspaces to complementary subspaces.
• Finite-dimensional vector spaces:
  • Module.evalEquiv is the equivalence V ≃ₗ[K] Dual K (Dual K V)
  • Module.mapEvalEquiv is the order isomorphism between subspaces of V and subspaces of Dual K (Dual K V).
  • Subspace.orderIsoFiniteCodimDim is the antitone order isomorphism between finite-codimensional subspaces of V and finite-dimensional subspaces of Dual K V.
  • Subspace.orderIsoFiniteDimensional is the antitone order isomorphism between subspaces of a finite-dimensional vector space V and subspaces of its dual.
  • Subspace.quotDualEquivAnnihilator W is the equivalence (Dual K V ⧸ W.dualLift.range) ≃ₗ[K] W.dualAnnihilator, where W.dualLift.range is a copy of Dual K W inside Dual K V.
  • Subspace.quotEquivAnnihilator W is the equivalence (V ⧸ W) ≃ₗ[K] W.dualAnnihilator
  • Subspace.dualQuotDistrib W is an equivalence Dual K (V₁ ⧸ W) ≃ₗ[K] Dual K V₁ ⧸ W.dualLift.range from an arbitrary choice of splitting of V₁.

@[reducible, inline]

abbrev 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.

Equations

• Module.Dual R M = (M →ₗ[R] R)

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

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

Equations

• Module.dualPairing R M = LinearMap.id

@[simp]

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

instance Module.Dual.instInhabited (R : Type uR) (M : Type uM) [CommSemiring R] [AddCommMonoid M] [Module R M] : Inhabited (Dual R M)

Equations

• Module.Dual.instInhabited R M = { default := 0 }

def Module.Dual.eval (R : Type uR) (M : Type uM) [CommSemiring R] [AddCommMonoid M] [Module R M] : M →ₗ[R] Dual R (Dual R M)

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

Equations

• Module.Dual.eval R M = LinearMap.id.flip

@[simp]

theorem Module.Dual.eval_apply (R : Type uR) (M : Type uM) [CommSemiring R] [AddCommMonoid M] [Module R M] (v : M) (a : Dual R M) : ((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'] : (M →ₗ[R] M') →ₗ[R] Dual R M' →ₗ[R] Dual 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.

Equations

• Module.Dual.transpose = (LinearMap.llcomp R M M' R).flip

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 : Dual R M') : (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') : transpose (u ∘ₗ v) = transpose v ∘ₗ transpose u

def Module.dualProdDualEquivDual (R : Type uR) (M : Type uM) [CommSemiring R] [AddCommMonoid M] [Module R M] (M' : Type uM') [AddCommMonoid M'] [Module R M'] : (Dual R M × Dual R M') ≃ₗ[R] Dual R (M × M')

Taking duals distributes over products.

Equations

• Module.dualProdDualEquivDual R M M' = LinearMap.coprodEquiv R

@[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') : ((dualProdDualEquivDual R M M') f) a = f.1 a.1 + f.2 a.2

@[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) : (dualProdDualEquivDual R M M').symm f = (f ∘ₗ LinearMap.inl R M M', f ∘ₗ LinearMap.inr R M M')

@[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 : Dual R M) (ψ : Dual R M') : (dualProdDualEquivDual R M M') (φ, ψ) = LinearMap.coprod φ ψ

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) : Module.Dual R M₂ →ₗ[R] Module.Dual R M₁

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.

Equations

• f.dualMap = Module.Dual.transpose f

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) : f.dualMap = lcomp R R f

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) : f.dualMap = 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) : f.dualMap 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) : (f.dualMap g) x = g (f x)

@[simp]

theorem LinearMap.dualMap_id {R : Type u} [CommSemiring R] {M₁ : Type v} [AddCommMonoid M₁] [Module R M₁] : id.dualMap = 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) : f.dualMap ∘ₗ g.dualMap = (g ∘ₗ f).dualMap

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) : Function.Injective ⇑f.dualMap

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) : Module.Dual R M₂ ≃ₗ[R] Module.Dual R M₁

The Linear_equiv version of LinearMap.dualMap.

Equations

• f.dualMap = { toLinearMap := (↑f).dualMap, invFun := ⇑(↑f.symm).dualMap, left_inv := ⋯, right_inv := ⋯ }

@[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) : (f.dualMap g) x = g (f x)

@[simp]

theorem LinearEquiv.dualMap_refl {R : Type u} [CommSemiring R] {M₁ : Type v} [AddCommMonoid M₁] [Module R M₁] : (refl R M₁).dualMap = refl R (Module.Dual R M₁)

@[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} : f.dualMap.symm = f.symm.dualMap

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) : g.dualMap ≪≫ₗ f.dualMap = (f ≪≫ₗ g).dualMap

theorem Module.Dual.eval_naturality {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) : f.dualMap.dualMap ∘ₗ eval R M₁ = eval R M₂ ∘ₗ f

@[simp]

theorem Dual.apply_one_mul_eq {R : Type u} [CommSemiring R] (f : Module.Dual R R) (r : R) : f 1 * r = f r

@[simp]

theorem LinearMap.range_dualMap_dual_eq_span_singleton {R : Type u} [CommSemiring R] {M₁ : Type v} [AddCommMonoid M₁] [Module R M₁] (f : Module.Dual R M₁) : range (dualMap f) = Submodule.span R {f}

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) : M →ₗ[R] Module.Dual 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.

Equations

• b.toDual = (b.constr ℕ) fun (v : ι) => (b.constr ℕ) fun (w : ι) => if w = v then 1 else 0

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 j : ι) : (b.toDual (b i)) (b j) = if i = j then 1 else 0

@[simp]

theorem Basis.toDual_linearCombination_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) : (b.toDual ((Finsupp.linearCombination R ⇑b) f)) (b i) = f i

@[deprecated Basis.toDual_linearCombination_left (since := "2024-08-29")]

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) : (b.toDual ((Finsupp.linearCombination R ⇑b) f)) (b i) = f i

Alias of Basis.toDual_linearCombination_left.

@[simp]

theorem Basis.toDual_linearCombination_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) : (b.toDual (b i)) ((Finsupp.linearCombination R ⇑b) f) = f i

@[deprecated Basis.toDual_linearCombination_right (since := "2024-08-29")]

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) : (b.toDual (b i)) ((Finsupp.linearCombination R ⇑b) f) = f i

Alias of Basis.toDual_linearCombination_right.

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) : (b.toDual 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) : (b.toDual (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) : b.toDual (b i) = b.coord 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) : M →ₗ[R] R

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

Equations

• b.toDualFlip m = b.toDual.flip m

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 m₂ : M) : (b.toDualFlip m₁) m₂ = (b.toDual 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) : (b.toDual 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) : (b.toDual m) (b i) = b.equivFun 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) : Function.Injective ⇑b.toDual

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 : b.toDual 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) : LinearMap.ker b.toDual = ⊥

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] : LinearMap.range b.toDual = ⊤

@[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) : ∑ x : ι, f (b x) • b.coord 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] : M ≃ₗ[R] Module.Dual R M

A vector space is linearly equivalent to its dual space.

Equations

• b.toDualEquiv = LinearEquiv.ofBijective b.toDual ⋯

@[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) : b.toDualEquiv m = b.toDual m

theorem Basis.linearEquiv_dual_iff_finiteDimensional {K : Type uK} {V : Type uV} [Field K] [AddCommGroup V] [Module K V] : Nonempty (V ≃ₗ[K] Module.Dual K V) ↔ FiniteDimensional K V

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.

Equations

• b.dualBasis = b.map b.toDualEquiv

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 j : ι) : (b.dualBasis i) (b j) = if j = i then 1 else 0

theorem Basis.linearCombination_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.linearCombination R ⇑b.dualBasis) f) (b i) = f i

@[deprecated Basis.linearCombination_dualBasis (since := "2024-08-29")]

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.linearCombination R ⇑b.dualBasis) f) (b i) = f i

Alias of Basis.linearCombination_dualBasis.

@[simp]

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) : (b.dualBasis.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) : (b.dualBasis 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] : ⇑b.dualBasis = b.coord

@[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] : b.dualBasis.toDual ∘ₗ b.toDual = Module.Dual.eval R M

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) : b.dualBasis.equivFun 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) : LinearMap.ker (Module.Dual.eval 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) : LinearMap.range (Module.Dual.eval 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] : Module.Free R (Module.Dual R M)

instance Basis.dual_projective {R : Type uR} {M : Type uM} [CommRing R] [AddCommGroup M] [Module R M] [Module.Finite R M] [Module.Projective R M] : Module.Projective R (Module.Dual R M)

instance Basis.dual_finite {R : Type uR} {M : Type uM} [CommRing R] [AddCommGroup M] [Module R M] [Module.Finite R M] [Module.Projective R M] : Module.Finite R (Module.Dual R M)

@[simp]

theorem Basis.linearCombination_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.linearCombination R b.coord) f) (b i) = f i

simp normal form version of linearCombination_dualBasis

@[deprecated Basis.linearCombination_coord (since := "2024-08-29")]

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.linearCombination R b.coord) f) (b i) = f i

Alias of Basis.linearCombination_coord.

---

simp normal form version of linearCombination_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) : Cardinal.lift.{uK, uV} (Module.rank K V) = Module.rank K (Module.Dual K V)

theorem Module.eval_ker (K : Type uK) (V : Type uV) [CommRing K] [AddCommGroup V] [Module K V] [Projective K V] : LinearMap.ker (Dual.eval K V) = ⊥

theorem Module.map_eval_injective (K : Type uK) (V : Type uV) [CommRing K] [AddCommGroup V] [Module K V] [Projective K V] : Function.Injective (Submodule.map (Dual.eval K V))

theorem Module.comap_eval_surjective (K : Type uK) (V : Type uV) [CommRing K] [AddCommGroup V] [Module K V] [Projective K V] : Function.Surjective (Submodule.comap (Dual.eval K V))

theorem Module.eval_apply_eq_zero_iff (K : Type uK) {V : Type uV} [CommRing K] [AddCommGroup V] [Module K V] [Projective K V] (v : V) : (Dual.eval K V) v = 0 ↔ v = 0

theorem Module.eval_apply_injective (K : Type uK) {V : Type uV} [CommRing K] [AddCommGroup V] [Module K V] [Projective K V] : Function.Injective ⇑(Dual.eval K V)

theorem Module.forall_dual_apply_eq_zero_iff (K : Type uK) {V : Type uV} [CommRing K] [AddCommGroup V] [Module K V] [Projective K V] (v : V) : (∀ (φ : Dual K V), φ v = 0) ↔ v = 0

@[simp]

theorem Module.subsingleton_dual_iff (K : Type uK) {V : Type uV} [CommRing K] [AddCommGroup V] [Module K V] [Projective K V] : Subsingleton (Dual K V) ↔ Subsingleton V

@[simp]

theorem Module.nontrivial_dual_iff (K : Type uK) {V : Type uV} [CommRing K] [AddCommGroup V] [Module K V] [Projective K V] : Nontrivial (Dual K V) ↔ Nontrivial V

instance Module.instNontrivialDual (K : Type uK) {V : Type uV} [CommRing K] [AddCommGroup V] [Module K V] [Projective K V] [Nontrivial V] : Nontrivial (Dual K V)

theorem Module.finite_dual_iff (K : Type uK) {V : Type uV} [CommRing K] [AddCommGroup V] [Module K V] [Free K V] : Module.Finite K (Dual K V) ↔ Module.Finite K V

theorem Module.dual_rank_eq {K : Type uK} {V : Type uV} [CommRing K] [AddCommGroup V] [Module K V] [Free K V] [Module.Finite K V] : Cardinal.lift.{uK, uV} (Module.rank K V) = Module.rank K (Dual K V)

class Module.IsReflexive (R : Type u_1) (M : Type u_2) [CommRing R] [AddCommGroup M] [Module R M] : Prop

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

Any finitely-generated projective module (and thus any finite-dimensional vector space) is reflexive. See Module.instIsReflexiveOfFiniteOfProjective.

• bijective_dual_eval' : Function.Bijective ⇑(Dual.eval R M)

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

theorem Module.bijective_dual_eval (R : Type u_1) (M : Type u_2) [CommRing R] [AddCommGroup M] [Module R M] [IsReflexive R M] : Function.Bijective ⇑(Dual.eval R M)

@[instance 900]

instance Module.IsReflexive.of_finite_of_free (R : Type u_1) (M : Type u_2) [CommRing R] [AddCommGroup M] [Module R M] [Module.Finite R M] [Free R M] : IsReflexive R M

See also Module.instFiniteDimensionalOfIsReflexive for the converse over a field.

theorem Module.erange_coe (R : Type u_1) (M : Type u_2) [CommRing R] [AddCommGroup M] [Module R M] [IsReflexive R M] : LinearMap.range (Dual.eval R M) = ⊤

def Module.evalEquiv (R : Type u_1) (M : Type u_2) [CommRing R] [AddCommGroup M] [Module R M] [IsReflexive R M] : M ≃ₗ[R] Dual R (Dual R M)

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

Equations

• Module.evalEquiv R M = LinearEquiv.ofBijective (Module.Dual.eval R M) ⋯

@[simp]

theorem Module.evalEquiv_toLinearMap (R : Type u_1) (M : Type u_2) [CommRing R] [AddCommGroup M] [Module R M] [IsReflexive R M] : ↑(evalEquiv R M) = Dual.eval R M

@[simp]

theorem Module.evalEquiv_apply (R : Type u_1) (M : Type u_2) [CommRing R] [AddCommGroup M] [Module R M] [IsReflexive R M] (m : M) : (evalEquiv R M) m = (Dual.eval R M) m

@[simp]

theorem Module.apply_evalEquiv_symm_apply (R : Type u_1) (M : Type u_2) [CommRing R] [AddCommGroup M] [Module R M] [IsReflexive R M] (f : Dual R M) (g : Dual R (Dual R M)) : f ((evalEquiv R M).symm g) = g f

@[simp]

theorem Module.symm_dualMap_evalEquiv (R : Type u_1) (M : Type u_2) [CommRing R] [AddCommGroup M] [Module R M] [IsReflexive R M] : ↑(evalEquiv R M).symm.dualMap = Dual.eval R (Dual R M)

@[simp]

theorem Module.Dual.eval_comp_comp_evalEquiv_eq (R : Type u_1) (M : Type u_2) [CommRing R] [AddCommGroup M] [Module R M] [IsReflexive R M] {M' : Type u_4} [AddCommGroup M'] [Module R M'] {f : M →ₗ[R] M'} : eval R M' ∘ₗ f ∘ₗ ↑(evalEquiv R M).symm = f.dualMap.dualMap

theorem Module.dualMap_dualMap_eq_iff_of_injective (R : Type u_1) (M : Type u_2) [CommRing R] [AddCommGroup M] [Module R M] [IsReflexive R M] {M' : Type u_4} [AddCommGroup M'] [Module R M'] {f g : M →ₗ[R] M'} (h : Function.Injective ⇑(Dual.eval R M')) : f.dualMap.dualMap = g.dualMap.dualMap ↔ f = g

@[simp]

theorem Module.dualMap_dualMap_eq_iff (R : Type u_1) (M : Type u_2) [CommRing R] [AddCommGroup M] [Module R M] [IsReflexive R M] {M' : Type u_4} [AddCommGroup M'] [Module R M'] [IsReflexive R M'] {f g : M →ₗ[R] M'} : f.dualMap.dualMap = g.dualMap.dualMap ↔ f = g

instance Module.Dual.instIsReflecive (R : Type u_1) (M : Type u_2) [CommRing R] [AddCommGroup M] [Module R M] [IsReflexive R M] : IsReflexive R (Dual R M)

The dual of a reflexive module is reflexive.

theorem Module.IsReflexive.of_split {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommRing R] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [IsReflexive R M] (i : N →ₗ[R] M) (s : M →ₗ[R] N) (H : s ∘ₗ i = LinearMap.id) : IsReflexive R N

A direct summand of a reflexive module is reflexive.

@[instance 900]

instance Module.instIsReflexiveOfFiniteOfProjective (R : Type u_1) (N : Type u_3) [CommRing R] [AddCommGroup N] [Module R N] [Module.Finite R N] [Projective R N] : IsReflexive R N

def Module.mapEvalEquiv (R : Type u_1) (M : Type u_2) [CommRing R] [AddCommGroup M] [Module R M] [IsReflexive R M] : Submodule R M ≃o Submodule R (Dual R (Dual R M))

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

Equations

• Module.mapEvalEquiv R M = Submodule.orderIsoMapComap (Module.evalEquiv R M)

@[simp]

theorem Module.mapEvalEquiv_apply (R : Type u_1) (M : Type u_2) [CommRing R] [AddCommGroup M] [Module R M] [IsReflexive R M] (W : Submodule R M) : (mapEvalEquiv R M) W = Submodule.map (Dual.eval R M) W

@[simp]

theorem Module.mapEvalEquiv_symm_apply (R : Type u_1) (M : Type u_2) [CommRing R] [AddCommGroup M] [Module R M] [IsReflexive R M] (W'' : Submodule R (Dual R (Dual R M))) : (mapEvalEquiv R M).symm W'' = Submodule.comap (Dual.eval R M) W''

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] : Module.IsReflexive R (M × N)

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] [IsReflexive R M] (e : M ≃ₗ[R] N) : IsReflexive R N

instance MulOpposite.instModuleIsReflexive (R : Type u_1) (M : Type u_2) [CommRing R] [AddCommGroup M] [Module R M] [Module.IsReflexive R M] : Module.IsReflexive R Mᵐᵒᵖ

instance ULift.instModuleIsReflexive (R : Type u_1) (M : Type u_2) [CommRing R] [AddCommGroup M] [Module R M] [Module.IsReflexive R M] : Module.IsReflexive R (ULift.{w, u_2} M)

instance Module.instFiniteDimensionalOfIsReflexive (K : Type u_4) (V : Type u_5) [Field K] [AddCommGroup V] [Module K V] [IsReflexive K V] : FiniteDimensional K V

@[instance 100]

instance Module.instNoZeroSMulDivisorsOfIsDomain (R : Type u_1) (M : Type u_2) [CommRing R] [AddCommGroup M] [Module R M] [IsReflexive R M] [IsDomain R] : NoZeroSMulDivisors R M

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 : x ∉ p) (hp' : Module.Free R (M ⧸ p)) : ∃ (f : Module.Dual R M), f x ≠ 0 ∧ 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 ∧ map f p = ⊥

theorem Submodule.span_eq_top_of_ne_zero {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] [Module.IsReflexive R M] {s : Set (M →ₗ[R] R)} [Module.Free R ((M →ₗ[R] R) ⧸ span R s)] (h : ∀ (z : M), z ≠ 0 → ∃ f ∈ s, f z ≠ 0) : span R s = ⊤

Consider a reflexive module and a set s of linear forms. If for any z ≠ 0 there exists f ∈ s such that f z ≠ 0, then s spans the whole dual space.

theorem FiniteDimensional.mem_span_of_iInf_ker_le_ker {ι : Type u_3} {𝕜 : Type u_4} {E : Type u_5} [Field 𝕜] [AddCommGroup E] [Module 𝕜 E] [FiniteDimensional 𝕜 E] {L : ι → E →ₗ[𝕜] 𝕜} {K : E →ₗ[𝕜] 𝕜} (h : ⨅ (i : ι), LinearMap.ker (L i) ≤ LinearMap.ker K) : K ∈ Submodule.span 𝕜 (Set.range L)

theorem mem_span_of_iInf_ker_le_ker {ι : Type u_3} {𝕜 : Type u_4} {E : Type u_5} [Field 𝕜] [AddCommGroup E] [Module 𝕜 E] [Finite ι] {L : ι → E →ₗ[𝕜] 𝕜} {K : E →ₗ[𝕜] 𝕜} (h : ⨅ (i : ι), LinearMap.ker (L i) ≤ LinearMap.ker K) : K ∈ Submodule.span 𝕜 (Set.range L)

Given some linear forms $L_1,...,L_n,K$ over a vector space $E$, if $\bigcap _{i=1}^n\ker (L_i)\subseteq \ker (K)$, then $K$ is in the space generated by $L_1,...,L_n$.

def evalUseFiniteInstance : Lean.Elab.Tactic.TacticM Unit

Try using Set.toFinite to dispatch a Set.Finite goal.

Equations

• One or more equations did not get rendered due to their size.

def tacticUse_finite_instance : Lean.ParserDescr

Equations

• tacticUse_finite_instance = Lean.ParserDescr.node `tacticUse_finite_instance 1024 (Lean.ParserDescr.nonReservedSymbol "use_finite_instance" false)

structure Module.DualBases {R : Type u_1} {M : Type u_2} {ι : Type u_3} [CommSemiring R] [AddCommMonoid M] [Module R M] (e : ι → M) (ε : ι → Dual R M) : Prop

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

• eval_same (i : ι) : (ε i) (e i) = 1
• eval_of_ne : Pairwise fun (i j : ι) => (ε i) (e j) = 0
• total {m : M} : (∀ (i : ι), (ε i) m = 0) → m = 0
• finite (m : M) : {i : ι | (ε i) m ≠ 0}.Finite

def Module.DualBases.coeffs {R : Type u_1} {M : Type u_2} {ι : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] {e : ι → M} {ε : ι → Dual R M} (h : DualBases e ε) (m : M) : ι →₀ R

The coefficients of v on the basis e

Equations

• h.coeffs m = { support := ⋯.toFinset, toFun := fun (i : ι) => (ε i) m, mem_support_toFun := ⋯ }

@[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 : ι → M} {ε : ι → Dual R M} (h : DualBases e ε) (m : M) (i : ι) : (h.coeffs 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 : ι → M) (l : ι →₀ R) : M

linear combinations of elements of e. This is a convenient abbreviation for Finsupp.linearCombination R e l

Equations

• Module.DualBases.lc e l = l.sum fun (i : ι) (a : R) => a • e i

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 : ι → M) (l : ι →₀ R) : lc e l = (Finsupp.linearCombination R e) l

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 : ι → M} {ε : ι → Dual R M} (h : DualBases e ε) (l : ι →₀ R) (i : ι) : (ε i) (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 : ι → M} {ε : ι → Dual R M} (h : DualBases e ε) (l : ι →₀ R) : h.coeffs (lc e l) = l

@[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 : ι → M} {ε : ι → Dual R M} (h : DualBases e ε) (m : M) : lc e (h.coeffs m) = m

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

def Module.DualBases.basis {R : Type u_1} {M : Type u_2} {ι : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] {e : ι → M} {ε : ι → Dual R M} (h : DualBases e ε) : Basis ι R M

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

Equations

• h.basis = { repr := { toFun := h.coeffs, map_add' := ⋯, map_smul' := ⋯, invFun := Module.DualBases.lc e, left_inv := ⋯, right_inv := ⋯ } }

@[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 : ι → M} {ε : ι → Dual R M} (h : DualBases e ε) (m : M) : h.basis.repr m = h.coeffs m

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 : ι → M} {ε : ι → Dual R M} (h : DualBases e ε) (l : ι →₀ R) : h.basis.repr.symm l = lc e l

@[simp]

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 : ι → M} {ε : ι → Dual R M} (h : DualBases e ε) : ⇑h.basis = e

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 : ι → M} {ε : ι → Dual R M} (h : DualBases e ε) {H : Set ι} {x : M} (hmem : x ∈ Submodule.span R (e '' H)) (i : ι) : (ε i) x ≠ 0 → i ∈ 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 : ι → M} {ε : ι → Dual R M} (h : DualBases e ε) [DecidableEq ι] [Finite ι] : ⇑h.basis.dualBasis = ε

def Submodule.dualRestrict {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] (W : Submodule R M) : Module.Dual R M →ₗ[R] Module.Dual R ↥W

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.

Equations

• W.dualRestrict = LinearMap.domRestrict' W

theorem Submodule.dualRestrict_def {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] (W : Submodule R M) : W.dualRestrict = W.subtype.dualMap

@[simp]

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) : (W.dualRestrict φ) x = φ ↑x

def Submodule.dualAnnihilator {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] (W : Submodule R M) : Submodule R (Module.Dual R M)

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

Equations

• W.dualAnnihilator = LinearMap.ker W.dualRestrict

@[simp]

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 ∈ W.dualAnnihilator ↔ ∀ w ∈ W, φ w = 0

theorem Submodule.dualRestrict_ker_eq_dualAnnihilator {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] (W : Submodule R M) : LinearMap.ker W.dualRestrict = W.dualAnnihilator

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

def Submodule.dualCoannihilator {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] (Φ : Submodule R (Module.Dual R M)) : Submodule R M

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

Equations

• Φ.dualCoannihilator = Submodule.comap (Module.Dual.eval R M) Φ.dualAnnihilator

@[simp]

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 ∈ Φ.dualCoannihilator ↔ ∀ φ ∈ Φ, φ x = 0

theorem Submodule.comap_dualAnnihilator {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] (Φ : Submodule R (Module.Dual R M)) : comap (Module.Dual.eval R M) Φ.dualAnnihilator = Φ.dualCoannihilator

theorem Submodule.map_dualCoannihilator_le {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] (Φ : Submodule R (Module.Dual R M)) : map (Module.Dual.eval R M) Φ.dualCoannihilator ≤ Φ.dualAnnihilator

theorem Submodule.dualAnnihilator_gc (R : Type u) (M : Type v) [CommSemiring R] [AddCommMonoid M] [Module R M] : GaloisConnection (⇑OrderDual.toDual ∘ dualAnnihilator) (dualCoannihilator ∘ ⇑OrderDual.ofDual)

theorem Submodule.le_dualAnnihilator_iff_le_dualCoannihilator {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] {U : Submodule R (Module.Dual R M)} {V : Submodule R M} : U ≤ V.dualAnnihilator ↔ V ≤ U.dualCoannihilator

@[simp]

theorem Submodule.dualAnnihilator_bot {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] : ⊥.dualAnnihilator = ⊤

@[simp]

theorem Submodule.dualAnnihilator_top {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] : ⊤.dualAnnihilator = ⊥

@[simp]

theorem Submodule.dualCoannihilator_bot {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] : ⊥.dualCoannihilator = ⊤

theorem Submodule.dualAnnihilator_anti {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] {U V : Submodule R M} (hUV : U ≤ V) : V.dualAnnihilator ≤ U.dualAnnihilator

theorem Submodule.dualCoannihilator_anti {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] {U V : Submodule R (Module.Dual R M)} (hUV : U ≤ V) : V.dualCoannihilator ≤ U.dualCoannihilator

theorem Submodule.le_dualAnnihilator_dualCoannihilator {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] (U : Submodule R M) : U ≤ U.dualAnnihilator.dualCoannihilator

theorem Submodule.le_dualCoannihilator_dualAnnihilator {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] (U : Submodule R (Module.Dual R M)) : U ≤ U.dualCoannihilator.dualAnnihilator

theorem Submodule.dualAnnihilator_dualCoannihilator_dualAnnihilator {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] (U : Submodule R M) : U.dualAnnihilator.dualCoannihilator.dualAnnihilator = U.dualAnnihilator

theorem Submodule.dualCoannihilator_dualAnnihilator_dualCoannihilator {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] (U : Submodule R (Module.Dual R M)) : U.dualCoannihilator.dualAnnihilator.dualCoannihilator = U.dualCoannihilator

theorem Submodule.dualAnnihilator_sup_eq {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] (U V : Submodule R M) : (U ⊔ V).dualAnnihilator = U.dualAnnihilator ⊓ V.dualAnnihilator

theorem Submodule.dualCoannihilator_sup_eq {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] (U V : Submodule R (Module.Dual R M)) : (U ⊔ V).dualCoannihilator = U.dualCoannihilator ⊓ V.dualCoannihilator

theorem Submodule.dualAnnihilator_iSup_eq {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] {ι : Sort u_1} (U : ι → Submodule R M) : (⨆ (i : ι), U i).dualAnnihilator = ⨅ (i : ι), (U i).dualAnnihilator

theorem Submodule.dualCoannihilator_iSup_eq {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] {ι : Sort u_1} (U : ι → Submodule R (Module.Dual R M)) : (⨆ (i : ι), U i).dualCoannihilator = ⨅ (i : ι), (U i).dualCoannihilator

theorem Submodule.sup_dualAnnihilator_le_inf {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] (U V : Submodule R M) : U.dualAnnihilator ⊔ V.dualAnnihilator ≤ (U ⊓ V).dualAnnihilator

See also Subspace.dualAnnihilator_inf_eq for vector subspaces.

theorem Submodule.iSup_dualAnnihilator_le_iInf {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] {ι : Sort u_1} (U : ι → Submodule R M) : ⨆ (i : ι), (U i).dualAnnihilator ≤ (⨅ (i : ι), U i).dualAnnihilator

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

@[simp]

theorem Submodule.coe_dualAnnihilator_span {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] (s : Set M) : ↑(span R s).dualAnnihilator = {f : Module.Dual R M | s ⊆ ↑(LinearMap.ker f)}

@[simp]

theorem Submodule.coe_dualCoannihilator_span {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] (s : Set (Module.Dual R M)) : ↑(span R s).dualCoannihilator = {x : M | ∀ f ∈ s, f x = 0}

@[simp]

theorem Subspace.dualCoannihilator_top {K : Type u} {V : Type v} [Field K] [AddCommGroup V] [Module K V] (W : Subspace K V) : Submodule.dualCoannihilator ⊤ = ⊥

@[simp]

theorem Subspace.dualAnnihilator_dualCoannihilator_eq {K : Type u} {V : Type v} [Field K] [AddCommGroup V] [Module K V] {W : Subspace K V} : (Submodule.dualAnnihilator W).dualCoannihilator = W

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) : (∀ φ ∈ Submodule.dualAnnihilator W, φ v = 0) ↔ v ∈ W

theorem Subspace.comap_dualAnnihilator_dualAnnihilator {K : Type u} {V : Type v} [Field K] [AddCommGroup V] [Module K V] (W : Subspace K V) : Submodule.comap (Module.Dual.eval K V) (Submodule.dualAnnihilator W).dualAnnihilator = W

theorem Subspace.map_le_dualAnnihilator_dualAnnihilator {K : Type u} {V : Type v} [Field K] [AddCommGroup V] [Module K V] (W : Subspace K V) : Submodule.map (Module.Dual.eval K V) W ≤ (Submodule.dualAnnihilator W).dualAnnihilator

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.

Equations

• One or more equations did not get rendered due to their size.

theorem Subspace.dualAnnihilator_le_dualAnnihilator_iff {K : Type u} {V : Type v} [Field K] [AddCommGroup V] [Module K V] {W W' : Subspace K V} : Submodule.dualAnnihilator W ≤ Submodule.dualAnnihilator W' ↔ W' ≤ W

theorem Subspace.dualAnnihilator_inj {K : Type u} {V : Type v} [Field K] [AddCommGroup V] [Module K V] {W W' : Subspace K V} : Submodule.dualAnnihilator W = Submodule.dualAnnihilator W' ↔ W = W'

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

Equations

• W.dualLift = (Classical.choose ⋯).dualMap

@[simp]

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) : (W.dualLift 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) : (W.dualLift φ) w = φ ⟨w, hw⟩

@[simp]

theorem Subspace.dualRestrict_comp_dualLift {K : Type u} {V : Type v} [Field K] [AddCommGroup V] [Module K V] (W : Subspace K V) : Submodule.dualRestrict W ∘ₗ W.dualLift = 1

theorem Subspace.dualRestrict_leftInverse {K : Type u} {V : Type v} [Field K] [AddCommGroup V] [Module K V] (W : Subspace K V) : Function.LeftInverse ⇑(Submodule.dualRestrict W) ⇑W.dualLift

theorem Subspace.dualLift_rightInverse {K : Type u} {V : Type v} [Field K] [AddCommGroup V] [Module K V] (W : Subspace K V) : Function.RightInverse ⇑W.dualLift ⇑(Submodule.dualRestrict W)

theorem Subspace.dualRestrict_surjective {K : Type u} {V : Type v} [Field K] [AddCommGroup V] [Module K V] {W : Subspace K V} : Function.Surjective ⇑(Submodule.dualRestrict W)

theorem Subspace.dualLift_injective {K : Type u} {V : Type v} [Field K] [AddCommGroup V] [Module K V] {W : Subspace K V} : Function.Injective ⇑W.dualLift

noncomputable def Subspace.quotAnnihilatorEquiv {K : Type u} {V : Type v} [Field K] [AddCommGroup V] [Module K V] (W : Subspace K V) : (Module.Dual K V ⧸ Submodule.dualAnnihilator W) ≃ₗ[K] Module.Dual K ↥W

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

Equations

• W.quotAnnihilatorEquiv = ((LinearMap.ker (Submodule.dualRestrict W)).quotEquivOfEq (Submodule.dualAnnihilator W) ⋯).symm.trans ((Submodule.dualRestrict W).quotKerEquivOfSurjective ⋯)

@[simp]

theorem Subspace.quotAnnihilatorEquiv_apply {K : Type u} {V : Type v} [Field K] [AddCommGroup V] [Module K V] (W : Subspace K V) (φ : Module.Dual K V) : W.quotAnnihilatorEquiv (Submodule.Quotient.mk φ) = (Submodule.dualRestrict W) φ

noncomputable def Subspace.dualEquivDual {K : Type u} {V : Type v} [Field K] [AddCommGroup V] [Module K V] (W : Subspace K V) : Module.Dual K ↥W ≃ₗ[K] ↥(LinearMap.range W.dualLift)

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

Equations

• W.dualEquivDual = LinearEquiv.ofInjective W.dualLift ⋯

theorem Subspace.dualEquivDual_def {K : Type u} {V : Type v} [Field K] [AddCommGroup V] [Module K V] (W : Subspace K V) : ↑W.dualEquivDual = W.dualLift.rangeRestrict

@[simp]

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) : W.dualEquivDual φ = ⟨W.dualLift φ, ⋯⟩

instance Subspace.instModuleDualFiniteDimensional {K : Type u} {V : Type v} [Field K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] : FiniteDimensional K (Module.Dual K V)

@[simp]

theorem Subspace.dual_finrank_eq {K : Type u} {V : Type v} [Field K] [AddCommGroup V] [Module K V] : Module.finrank K (Module.Dual K V) = Module.finrank K V

theorem Subspace.dualAnnihilator_dualAnnihilator_eq {K : Type u} {V : Type v} [Field K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] (W : Subspace K V) : (Submodule.dualAnnihilator W).dualAnnihilator = (Module.mapEvalEquiv K V) W

noncomputable def Subspace.quotDualEquivAnnihilator {K : Type u} {V : Type v} [Field K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] (W : Subspace K V) : (Module.Dual K V ⧸ LinearMap.range W.dualLift) ≃ₗ[K] ↥(Submodule.dualAnnihilator W)

The quotient by the dual is isomorphic to its dual annihilator.

Equations

• W.quotDualEquivAnnihilator = FiniteDimensional.LinearEquiv.quotEquivOfQuotEquiv (W.quotAnnihilatorEquiv.trans W.dualEquivDual)

noncomputable def Subspace.quotEquivAnnihilator {K : Type u} {V : Type v} [Field K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] (W : Subspace K V) : (V ⧸ W) ≃ₗ[K] ↥(Submodule.dualAnnihilator W)

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

Equations

• One or more equations did not get rendered due to their size.

theorem Subspace.finrank_add_finrank_dualAnnihilator_eq {K : Type u} {V : Type v} [Field K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] (W : Subspace K V) : Module.finrank K ↥W + Module.finrank K ↥(Submodule.dualAnnihilator W) = Module.finrank K V

@[simp]

theorem Subspace.finrank_dualCoannihilator_eq {K : Type u} {V : Type v} [Field K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] {Φ : Subspace K (Module.Dual K V)} : Module.finrank K ↥(Submodule.dualCoannihilator Φ) = Module.finrank K ↥(Submodule.dualAnnihilator Φ)

theorem Subspace.finrank_add_finrank_dualCoannihilator_eq {K : Type u} {V : Type v} [Field K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] (W : Subspace K (Module.Dual K V)) : Module.finrank K ↥W + Module.finrank K ↥(Submodule.dualCoannihilator W) = Module.finrank K V

theorem LinearMap.ker_dualMap_eq_dualAnnihilator_range {R : Type uR} [CommSemiring R] {M₁ : Type uM₁} {M₂ : Type uM₂} [AddCommMonoid M₁] [Module R M₁] [AddCommMonoid M₂] [Module R M₂] (f : M₁ →ₗ[R] M₂) : ker f.dualMap = (range f).dualAnnihilator

theorem LinearMap.range_dualMap_le_dualAnnihilator_ker {R : Type uR} [CommSemiring R] {M₁ : Type uM₁} {M₂ : Type uM₂} [AddCommMonoid M₁] [Module R M₁] [AddCommMonoid M₂] [Module R M₂] (f : M₁ →ₗ[R] M₂) : range f.dualMap ≤ (ker f).dualAnnihilator

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

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

See Subspace.dualCopairing_nondegenerate.

Equations

• W.dualCopairing = (W.liftQ ((Module.dualPairing R M).domRestrict W.dualAnnihilator).flip ⋯).flip

instance Submodule.instFunLikeSubtypeDualMemDualAnnihilator {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] (W : Submodule R M) : FunLike (↥W.dualAnnihilator) M R

Equations

• W.instFunLikeSubtypeDualMemDualAnnihilator = { coe := fun (φ : ↥W.dualAnnihilator) => ⇑↑φ, coe_injective' := ⋯ }

@[simp]

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

def Submodule.dualPairing {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [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.

Equations

• W.dualPairing = W.dualAnnihilator.liftQ W.dualRestrict ⋯

@[simp]

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) : (W.dualPairing (Quotient.mk φ)) x = φ ↑x

theorem Submodule.range_dualMap_mkQ_eq {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] (W : Submodule R M) : LinearMap.range W.mkQ.dualMap = W.dualAnnihilator

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

def Submodule.dualQuotEquivDualAnnihilator {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] (W : Submodule R M) : Module.Dual R (M ⧸ W) ≃ₗ[R] ↥W.dualAnnihilator

Equivalence $(M/W{)}^{\ast }\cong \mathrm{ann}(W)$. 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.

Equations

• W.dualQuotEquivDualAnnihilator = LinearEquiv.ofLinear (LinearMap.codRestrict W.dualAnnihilator W.mkQ.dualMap ⋯) W.dualCopairing ⋯ ⋯

@[simp]

theorem Submodule.dualQuotEquivDualAnnihilator_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 ⧸ W)) (x : M) : (W.dualQuotEquivDualAnnihilator φ) x = φ (Quotient.mk x)

theorem Submodule.dualCopairing_eq {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] (W : Submodule R M) : W.dualCopairing = ↑W.dualQuotEquivDualAnnihilator.symm

@[simp]

theorem Submodule.dualQuotEquivDualAnnihilator_symm_apply_mk {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] (W : Submodule R M) (φ : ↥W.dualAnnihilator) (x : M) : (W.dualQuotEquivDualAnnihilator.symm φ) (Quotient.mk x) = φ x

theorem Submodule.finite_dualAnnihilator_iff {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] {W : Submodule R M} [Module.Free R (M ⧸ W)] : Module.Finite R ↥W.dualAnnihilator ↔ Module.Finite R (M ⧸ W)

def Submodule.quotDualCoannihilatorToDual {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] (W : Submodule R (Module.Dual R M)) : M ⧸ W.dualCoannihilator →ₗ[R] Module.Dual R ↥W

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.

Equations

• W.quotDualCoannihilatorToDual = W.dualCoannihilator.liftQ W.subtype.flip ⋯

@[simp]

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) : (W.quotDualCoannihilatorToDual (Quotient.mk m)) w = ↑w m

theorem Submodule.quotDualCoannihilatorToDual_injective {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] (W : Submodule R (Module.Dual R M)) : Function.Injective ⇑W.quotDualCoannihilatorToDual

theorem Submodule.flip_quotDualCoannihilatorToDual_injective {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] (W : Submodule R (Module.Dual R M)) : Function.Injective ⇑W.quotDualCoannihilatorToDual.flip

theorem Submodule.quotDualCoannihilatorToDual_nondegenerate {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] (W : Submodule R (Module.Dual R M)) : W.quotDualCoannihilatorToDual.Nondegenerate

theorem LinearMap.range_dualMap_eq_dualAnnihilator_ker_of_surjective {R : Type u_1} {M : Type u_2} {M' : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup M'] [Module R M'] (f : M →ₗ[R] M') (hf : Function.Surjective ⇑f) : range f.dualMap = (ker f).dualAnnihilator

theorem LinearMap.range_dualMap_eq_dualAnnihilator_ker_of_subtype_range_surjective {R : Type u_1} {M : Type u_2} {M' : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup M'] [Module R M'] (f : M →ₗ[R] M') (hf : Function.Surjective ⇑(range f).subtype.dualMap) : range f.dualMap = (ker f).dualAnnihilator

theorem LinearMap.ker_dualMap_eq_dualCoannihilator_range {R : Type u_1} {M : Type u_2} {M' : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup M'] [Module R M'] (f : M →ₗ[R] M') : ker f.dualMap = (range (Module.Dual.eval R M' ∘ₗ f)).dualCoannihilator

@[simp]

theorem LinearMap.dualCoannihilator_range_eq_ker_flip {R : Type u_1} {M : Type u_2} {M' : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup M'] [Module R M'] (B : M →ₗ[R] M' →ₗ[R] R) : (range B).dualCoannihilator = ker B.flip

theorem Module.Dual.range_eq_top_of_ne_zero {K : Type uK} [Field K] {V₁ : Type uV₁} [AddCommGroup V₁] [Module K V₁] {f : Dual K V₁} (hf : f ≠ 0) : LinearMap.range f = ⊤

theorem Module.Dual.finrank_ker_add_one_of_ne_zero {K : Type uK} [Field K] {V₁ : Type uV₁} [AddCommGroup V₁] [Module K V₁] {f : Dual K V₁} (hf : f ≠ 0) [FiniteDimensional K V₁] : finrank K ↥(LinearMap.ker f) + 1 = finrank K V₁

theorem Module.Dual.isCompl_ker_of_disjoint_of_ne_bot {K : Type uK} [Field K] {V₁ : Type uV₁} [AddCommGroup V₁] [Module K V₁] {f : Dual K V₁} (hf : f ≠ 0) [FiniteDimensional K V₁] {p : Submodule K V₁} (hpf : Disjoint (LinearMap.ker f) p) (hp : p ≠ ⊥) : IsCompl (LinearMap.ker f) p

theorem Module.Dual.eq_of_ker_eq_of_apply_eq {K : Type uK} [Field K] {V₁ : Type uV₁} [AddCommGroup V₁] [Module K V₁] [FiniteDimensional K V₁] {f g : Dual K V₁} (x : V₁) (h : LinearMap.ker f = LinearMap.ker g) (h' : f x = g x) (hx : f x ≠ 0) : f = g

theorem LinearMap.dualPairing_nondegenerate {K : Type uK} [Field K] {V₁ : Type uV₁} [AddCommGroup V₁] [Module K V₁] : (Module.dualPairing K V₁).Nondegenerate

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) : Function.Surjective ⇑f.dualMap

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) : range f.dualMap = (ker f).dualAnnihilator

@[simp]

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} : Function.Surjective ⇑f.dualMap ↔ Function.Injective ⇑f

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) : Submodule.dualPairing W = ↑W.quotAnnihilatorEquiv

theorem Subspace.dualPairing_nondegenerate {K : Type uK} [Field K] {V₁ : Type uV₁} [AddCommGroup V₁] [Module K V₁] (W : Subspace K V₁) : (Submodule.dualPairing W).Nondegenerate

theorem Subspace.dualCopairing_nondegenerate {K : Type uK} [Field K] {V₁ : Type uV₁} [AddCommGroup V₁] [Module K V₁] (W : Subspace K V₁) : (Submodule.dualCopairing W).Nondegenerate

theorem Subspace.dualAnnihilator_inf_eq {K : Type uK} [Field K] {V₁ : Type uV₁} [AddCommGroup V₁] [Module K V₁] (W W' : Subspace K V₁) : Submodule.dualAnnihilator (W ⊓ W') = Submodule.dualAnnihilator W ⊔ Submodule.dualAnnihilator W'

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 : ι → Subspace 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 W' : Subspace K V₁} (h : IsCompl W W') : IsCompl (Submodule.dualAnnihilator W) (Submodule.dualAnnihilator 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) : Module.Dual K (V₁ ⧸ W) ≃ₗ[K] Module.Dual K V₁ ⧸ LinearMap.range W.dualLift

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.

Equations

• W.dualQuotDistrib = (Submodule.dualQuotEquivDualAnnihilator W).trans W.quotDualEquivAnnihilator.symm

@[simp]

theorem LinearMap.finrank_range_dualMap_eq_finrank_range {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) : Module.finrank K ↥(range f.dualMap) = Module.finrank K ↥(range f)

@[simp]

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} : Function.Injective ⇑f.dualMap ↔ Function.Surjective ⇑f

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

@[simp]

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} : Function.Bijective ⇑f.dualMap ↔ Function.Bijective ⇑f

f.dualMap is bijective if and only if f is

@[simp]

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] : (ker B).dualAnnihilator = range B.flip

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] : Function.Injective ⇑B.flip ↔ Function.Surjective ⇑B

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] : Function.Injective ⇑B.flip ↔ Function.Surjective ⇑B

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] : Function.Surjective ⇑B.flip ↔ Function.Injective ⇑B

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] : Function.Surjective ⇑B.flip ↔ Function.Injective ⇑B

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] : Function.Bijective ⇑B.flip ↔ Function.Bijective ⇑B

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] : Function.Bijective ⇑B.flip ↔ Function.Bijective ⇑B

theorem Subspace.quotDualCoannihilatorToDual_bijective {K : Type u_1} {V : Type u_2} [Field K] [AddCommGroup V] [Module K V] (W : Subspace K (Module.Dual K V)) [FiniteDimensional K ↥W] : Function.Bijective ⇑(Submodule.quotDualCoannihilatorToDual W)

theorem Subspace.flip_quotDualCoannihilatorToDual_bijective {K : Type u_1} {V : Type u_2} [Field K] [AddCommGroup V] [Module K V] (W : Subspace K (Module.Dual K V)) [FiniteDimensional K ↥W] : Function.Bijective ⇑(Submodule.quotDualCoannihilatorToDual W).flip

theorem Subspace.dualCoannihilator_dualAnnihilator_eq {K : Type u_1} {V : Type u_2} [Field K] [AddCommGroup V] [Module K V] {W : Subspace K (Module.Dual K V)} [FiniteDimensional K ↥W] : (Submodule.dualCoannihilator W).dualAnnihilator = W

theorem Subspace.finiteDimensional_quot_dualCoannihilator_iff {K : Type u_1} {V : Type u_2} [Field K] [AddCommGroup V] [Module K V] {W : Submodule K (Module.Dual K V)} : FiniteDimensional K (V ⧸ W.dualCoannihilator) ↔ FiniteDimensional K ↥W

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.

Equations

• One or more equations did not get rendered due to their size.

def Subspace.orderIsoFiniteDimensional {K : Type u_1} {V : Type u_2} [Field K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] : Subspace K V ≃o (Subspace K (Module.Dual K V))ᵒᵈ

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.

Equations

• One or more equations did not get rendered due to their size.

theorem Subspace.dualAnnihilator_dualAnnihilator_eq_map {K : Type u_1} {V : Type u_2} [Field K] [AddCommGroup V] [Module K V] (W : Subspace K V) [FiniteDimensional K ↥W] : (Submodule.dualAnnihilator W).dualAnnihilator = Submodule.map (Module.Dual.eval K V) W

theorem Subspace.map_dualCoannihilator {K : Type u_1} {V : Type u_2} [Field K] [AddCommGroup V] [Module K V] (W : Subspace K (Module.Dual K V)) [FiniteDimensional K V] : Submodule.map (Module.Dual.eval K V) (Submodule.dualCoannihilator W) = Submodule.dualAnnihilator W

def TensorProduct.dualDistrib (R : Type u_1) (M : Type u_3) (N : Type u_4) [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] : TensorProduct R (Module.Dual R M) (Module.Dual R N) →ₗ[R] Module.Dual R (TensorProduct R M N)

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.

Equations

• TensorProduct.dualDistrib R M N = (↑(TensorProduct.lid R R)).compRight ∘ₗ TensorProduct.homTensorHomMap R M N R R

@[simp]

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) : ((dualDistrib R M N) (f ⊗ₜ[R] g)) (m ⊗ₜ[R] n) = f m * g n

def TensorProduct.AlgebraTensorModule.dualDistrib (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] : TensorProduct R (Module.Dual A M) (Module.Dual R N) →ₗ[A] Module.Dual A (TensorProduct R M N)

Heterobasic version of TensorProduct.dualDistrib

Equations

• TensorProduct.AlgebraTensorModule.dualDistrib R A M N = (Algebra.TensorProduct.rid R A A).toLinearMap.compRight ∘ₗ TensorProduct.AlgebraTensorModule.homTensorHomMap R A A M N A R

@[simp]

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) : ((dualDistrib R A M N) (f ⊗ₜ[R] g)) (m ⊗ₜ[R] n) = g n • f m

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) : Module.Dual R (TensorProduct R M N) →ₗ[R] TensorProduct R (Module.Dual R M) (Module.Dual R N)

An inverse to TensorProduct.dualDistrib given bases.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

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)) : (dualDistribInvOfBasis b c) f = ∑ i : ι, ∑ j : κ, f (b i ⊗ₜ[R] c j) • b.dualBasis i ⊗ₜ[R] c.dualBasis 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) : dualDistrib R M N ∘ₗ dualDistribInvOfBasis b c = LinearMap.id

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) : dualDistribInvOfBasis b c ∘ₗ dualDistrib R M N = LinearMap.id

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) : 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.

Equations

• TensorProduct.dualDistribEquivOfBasis b c = LinearEquiv.ofLinear (TensorProduct.dualDistrib R M N) (TensorProduct.dualDistribInvOfBasis b c) ⋯ ⋯

@[simp]

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)) : (dualDistribEquivOfBasis b c).symm a = ∑ x : ι, ∑ x_1 : κ, a (b x ⊗ₜ[R] c x_1) • b.coord x ⊗ₜ[R] c.coord x_1

@[simp]

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) : ((dualDistribEquivOfBasis b c) a✝) a✝¹ = (TensorProduct.lid R R) (((homTensorHomMap R M N R R) a✝) a✝¹)

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] : 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) 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.

Equations

• TensorProduct.dualDistribEquiv R M N = TensorProduct.dualDistribEquivOfBasis (Module.Free.chooseBasis R M) (Module.Free.chooseBasis R N)