ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContinuousLinearMap.apply_norm_eq_sqrt_inner_adjoint_left
theorem apply_norm_eq_sqrt_inner_adjoint_left (A : E →L[𝕜] F) (x : E) :
    ‖A x‖ = √(re ⟪(A† ∘L A) x, x⟫) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 E] [InnerProductSpace 𝕜 F] [CompleteSpace E] [CompleteSpace F] (A : E →L[𝕜] F) (x : E)
Goal: ‖A x‖ = √(RCLike.re ⟪((ContinuousLinearMap.adjoint A).comp A) x, x⟫_𝕜)
-- ContinuousLinearMap.apply_norm_sq_eq_inner_adjoint_right
theorem apply_norm_sq_eq_inner_adjoint_right (A : E →L[𝕜] F) (x : E) :
    ‖A x‖ ^ 2 = re ⟪x, (A† ∘L A) x⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 E] [InnerProductSpace 𝕜 F] [CompleteSpace E] [CompleteSpace F] (A : E →L[𝕜] F) (x : E)
Goal: ‖A x‖ ^ 2 = RCLike.re ⟪x, ((ContinuousLinearMap.adjoint A).comp A) x⟫_𝕜
-- ContinuousLinearMap.apply_norm_eq_sqrt_inner_adjoint_right
theorem apply_norm_eq_sqrt_inner_adjoint_right (A : E →L[𝕜] F) (x : E) :
    ‖A x‖ = √(re ⟪x, (A† ∘L A) x⟫) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 E] [InnerProductSpace 𝕜 F] [CompleteSpace E] [CompleteSpace F] (A : E →L[𝕜] F) (x : E)
Goal: ‖A x‖ = √(RCLike.re ⟪x, ((ContinuousLinearMap.adjoint A).comp A) x⟫_𝕜)
-- ContinuousLinearMap.eq_adjoint_iff
theorem eq_adjoint_iff (A : E →L[𝕜] F) (B : F →L[𝕜] E) : A = B† ↔ ∀ x y, ⟪A x, y⟫ = ⟪x, B y⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 E] [InnerProductSpace 𝕜 F] [CompleteSpace E] [CompleteSpace F] (A : E →L[𝕜] F) (B : F →L[𝕜] E)
Goal: A = ContinuousLinearMap.adjoint B ↔ ∀ (x : E) (y : F), ⟪A x, y⟫_𝕜 = ⟪x, B y⟫_𝕜
-- ContinuousLinearMap.adjoint_id
@[simp]
theorem adjoint_id :
    ContinuousLinearMap.adjoint (ContinuousLinearMap.id 𝕜 E) = ContinuousLinearMap.id 𝕜 E 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E]
Goal: ContinuousLinearMap.adjoint (ContinuousLinearMap.id 𝕜 E) = ContinuousLinearMap.id 𝕜 E
-- Submodule.adjoint_subtypeL
theorem _root_.Submodule.adjoint_subtypeL (U : Submodule 𝕜 E) [CompleteSpace U] :
    U.subtypeL† = orthogonalProjection U 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] (U : Submodule 𝕜 E) [CompleteSpace ↥U]
Goal: ContinuousLinearMap.adjoint U.subtypeL = orthogonalProjection U
-- Submodule.adjoint_orthogonalProjection
theorem _root_.Submodule.adjoint_orthogonalProjection (U : Submodule 𝕜 E) [CompleteSpace U] :
    (orthogonalProjection U : E →L[𝕜] U)† = U.subtypeL 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] (U : Submodule 𝕜 E) [CompleteSpace ↥U]
Goal: ContinuousLinearMap.adjoint (orthogonalProjection U) = U.subtypeL
-- ContinuousLinearMap.star_eq_adjoint
theorem star_eq_adjoint (A : E →L[𝕜] E) : star A = A† 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] (A : E →L[𝕜] E)
Goal: star A = ContinuousLinearMap.adjoint A
-- ContinuousLinearMap.isSelfAdjoint_iff'
theorem isSelfAdjoint_iff' {A : E →L[𝕜] E} : IsSelfAdjoint A ↔ ContinuousLinearMap.adjoint A = A 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] {A : E →L[𝕜] E}
Goal: IsSelfAdjoint A ↔ ContinuousLinearMap.adjoint A = A
-- ContinuousLinearMap.norm_adjoint_comp_self
theorem norm_adjoint_comp_self (A : E →L[𝕜] F) :
    ‖ContinuousLinearMap.adjoint A ∘L A‖ = ‖A‖ * ‖A‖ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 E] [InnerProductSpace 𝕜 F] [CompleteSpace E] [CompleteSpace F] (A : E →L[𝕜] F)
Goal: ‖(ContinuousLinearMap.adjoint A).comp A‖ = ‖A‖ * ‖A‖
-- ContinuousLinearMap.isAdjointPair_inner
theorem isAdjointPair_inner (A : E →L[𝕜] F) :
    LinearMap.IsAdjointPair (sesqFormOfInner : E →ₗ[𝕜] E →ₗ⋆[𝕜] 𝕜)
      (sesqFormOfInner : F →ₗ[𝕜] F →ₗ⋆[𝕜] 𝕜) A (A†) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 E] [InnerProductSpace 𝕜 F] [CompleteSpace E] [CompleteSpace F] (A : E →L[𝕜] F)
Goal: sesqFormOfInner.IsAdjointPair sesqFormOfInner ↑A ↑(ContinuousLinearMap.adjoint A)
-- IsSelfAdjoint.adjoint_eq
theorem adjoint_eq {A : E →L[𝕜] E} (hA : IsSelfAdjoint A) : ContinuousLinearMap.adjoint A = A 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] {A : E →L[𝕜] E} (hA : IsSelfAdjoint A)
Goal: ContinuousLinearMap.adjoint A = A
-- IsSelfAdjoint.isSymmetric
theorem isSymmetric {A : E →L[𝕜] E} (hA : IsSelfAdjoint A) : (A : E →ₗ[𝕜] E).IsSymmetric 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] {A : E →L[𝕜] E} (hA : IsSelfAdjoint A)
Goal: (↑A).IsSymmetric
-- IsSelfAdjoint.conj_adjoint
theorem conj_adjoint {T : E →L[𝕜] E} (hT : IsSelfAdjoint T) (S : E →L[𝕜] F) :
    IsSelfAdjoint (S ∘L T ∘L ContinuousLinearMap.adjoint S) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 E] [InnerProductSpace 𝕜 F] [CompleteSpace E] [CompleteSpace F] {T : E →L[𝕜] E} (hT : IsSelfAdjoint T) (S : E →L[𝕜] F)
Goal: IsSelfAdjoint (S.comp (T.comp (ContinuousLinearMap.adjoint S)))
-- IsSelfAdjoint.adjoint_conj
theorem adjoint_conj {T : E →L[𝕜] E} (hT : IsSelfAdjoint T) (S : F →L[𝕜] E) :
    IsSelfAdjoint (ContinuousLinearMap.adjoint S ∘L T ∘L S) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 E] [InnerProductSpace 𝕜 F] [CompleteSpace E] [CompleteSpace F] {T : E →L[𝕜] E} (hT : IsSelfAdjoint T) (S : F →L[𝕜] E)
Goal: IsSelfAdjoint ((ContinuousLinearMap.adjoint S).comp (T.comp S))
-- ContinuousLinearMap.isSelfAdjoint_iff_isSymmetric
theorem _root_.ContinuousLinearMap.isSelfAdjoint_iff_isSymmetric {A : E →L[𝕜] E} :
    IsSelfAdjoint A ↔ (A : E →ₗ[𝕜] E).IsSymmetric 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] {A : E →L[𝕜] E}
Goal: IsSelfAdjoint A ↔ (↑A).IsSymmetric
-- LinearMap.IsSymmetric.isSelfAdjoint
theorem _root_.LinearMap.IsSymmetric.isSelfAdjoint {A : E →L[𝕜] E}
    (hA : (A : E →ₗ[𝕜] E).IsSymmetric) : IsSelfAdjoint A 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] {A : E →L[𝕜] E} (hA : (↑A).IsSymmetric)
Goal: IsSelfAdjoint A
-- orthogonalProjection_isSelfAdjoint
theorem _root_.orthogonalProjection_isSelfAdjoint (U : Submodule 𝕜 E) [CompleteSpace U] :
    IsSelfAdjoint (U.subtypeL ∘L orthogonalProjection U) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] (U : Submodule 𝕜 E) [CompleteSpace ↥U]
Goal: IsSelfAdjoint (U.subtypeL.comp (orthogonalProjection U))
-- IsSelfAdjoint.conj_orthogonalProjection
theorem conj_orthogonalProjection {T : E →L[𝕜] E} (hT : IsSelfAdjoint T) (U : Submodule 𝕜 E)
    [CompleteSpace U] :
    IsSelfAdjoint
      (U.subtypeL ∘L orthogonalProjection U ∘L T ∘L U.subtypeL ∘L orthogonalProjection U) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] {T : E →L[𝕜] E} (hT : IsSelfAdjoint T) (U : Submodule 𝕜 E) [CompleteSpace ↥U]
Goal: IsSelfAdjoint (U.subtypeL.comp ((orthogonalProjection U).comp (T.comp (U.subtypeL.comp (orthogonalProjection U)))))
-- LinearMap.IsSymmetric.coe_toSelfAdjoint
theorem IsSymmetric.coe_toSelfAdjoint (hT : IsSymmetric T) : (hT.toSelfAdjoint : E →ₗ[𝕜] E) = T 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] {T : E →ₗ[𝕜] E} (hT : T.IsSymmetric)
Goal: ↑↑hT.toSelfAdjoint = T
-- LinearMap.IsSymmetric.toSelfAdjoint_apply
theorem IsSymmetric.toSelfAdjoint_apply (hT : IsSymmetric T) {x : E} :
    (hT.toSelfAdjoint : E → E) x = T x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] {T : E →ₗ[𝕜] E} (hT : T.IsSymmetric) {x : E}
Goal: ↑hT.toSelfAdjoint x = T x
-- LinearMap.adjoint_toContinuousLinearMap
theorem adjoint_toContinuousLinearMap (A : E →ₗ[𝕜] F) :
    haveI := FiniteDimensional.complete 𝕜 E
    haveI := FiniteDimensional.complete 𝕜 F
    LinearMap.toContinuousLinearMap (LinearMap.adjoint A) =
      ContinuousLinearMap.adjoint (LinearMap.toContinuousLinearMap A) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 E] [InnerProductSpace 𝕜 F] [FiniteDimensional 𝕜 E] [FiniteDimensional 𝕜 F] (A : E →ₗ[𝕜] F)
Goal: LinearMap.toContinuousLinearMap (LinearMap.adjoint A) = ContinuousLinearMap.adjoint (LinearMap.toContinuousLinearMap A)
-- LinearMap.adjoint_eq_toCLM_adjoint
theorem adjoint_eq_toCLM_adjoint (A : E →ₗ[𝕜] F) :
    haveI := FiniteDimensional.complete 𝕜 E
    haveI := FiniteDimensional.complete 𝕜 F
    LinearMap.adjoint A = ContinuousLinearMap.adjoint (LinearMap.toContinuousLinearMap A) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 E] [InnerProductSpace 𝕜 F] [FiniteDimensional 𝕜 E] [FiniteDimensional 𝕜 F] (A : E →ₗ[𝕜] F)
Goal: LinearMap.adjoint A = ↑(ContinuousLinearMap.adjoint (LinearMap.toContinuousLinearMap A))
-- LinearMap.adjoint_inner_left
theorem adjoint_inner_left (A : E →ₗ[𝕜] F) (x : E) (y : F) : ⟪adjoint A y, x⟫ = ⟪y, A x⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 E] [InnerProductSpace 𝕜 F] [FiniteDimensional 𝕜 E] [FiniteDimensional 𝕜 F] (A : E →ₗ[𝕜] F) (x : E) (y : F)
Goal: ⟪(LinearMap.adjoint A) y, x⟫_𝕜 = ⟪y, A x⟫_𝕜
-- LinearMap.adjoint_inner_right
theorem adjoint_inner_right (A : E →ₗ[𝕜] F) (x : E) (y : F) : ⟪x, adjoint A y⟫ = ⟪A x, y⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 E] [InnerProductSpace 𝕜 F] [FiniteDimensional 𝕜 E] [FiniteDimensional 𝕜 F] (A : E →ₗ[𝕜] F) (x : E) (y : F)
Goal: ⟪x, (LinearMap.adjoint A) y⟫_𝕜 = ⟪A x, y⟫_𝕜
-- LinearMap.adjoint_adjoint
@[simp]
theorem adjoint_adjoint (A : E →ₗ[𝕜] F) : LinearMap.adjoint (LinearMap.adjoint A) = A 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 E] [InnerProductSpace 𝕜 F] [FiniteDimensional 𝕜 E] [FiniteDimensional 𝕜 F] (A : E →ₗ[𝕜] F)
Goal: LinearMap.adjoint (LinearMap.adjoint A) = A
-- LinearMap.adjoint_comp
@[simp]
theorem adjoint_comp (A : F →ₗ[𝕜] G) (B : E →ₗ[𝕜] F) :
    LinearMap.adjoint (A ∘ₗ B) = LinearMap.adjoint B ∘ₗ LinearMap.adjoint A 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [RCLike 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedAddCommGroup G] [InnerProductSpace 𝕜 E] [InnerProductSpace 𝕜 F] [InnerProductSpace 𝕜 G] [FiniteDimensional 𝕜 E] [FiniteDimensional 𝕜 F] [FiniteDimensional 𝕜 G] (A : F →ₗ[𝕜] G) (B : E →ₗ[𝕜] F)
Goal: LinearMap.adjoint (A ∘ₗ B) = LinearMap.adjoint B ∘ₗ LinearMap.adjoint A
-- LinearMap.eq_adjoint_iff
theorem eq_adjoint_iff (A : E →ₗ[𝕜] F) (B : F →ₗ[𝕜] E) :
    A = LinearMap.adjoint B ↔ ∀ x y, ⟪A x, y⟫ = ⟪x, B y⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 E] [InnerProductSpace 𝕜 F] [FiniteDimensional 𝕜 E] [FiniteDimensional 𝕜 F] (A : E →ₗ[𝕜] F) (B : F →ₗ[𝕜] E)
Goal: A = LinearMap.adjoint B ↔ ∀ (x : E) (y : F), ⟪A x, y⟫_𝕜 = ⟪x, B y⟫_𝕜
-- LinearMap.eq_adjoint_iff_basis
theorem eq_adjoint_iff_basis {ι₁ : Type*} {ι₂ : Type*} (b₁ : Basis ι₁ 𝕜 E) (b₂ : Basis ι₂ 𝕜 F)
    (A : E →ₗ[𝕜] F) (B : F →ₗ[𝕜] E) :
    A = LinearMap.adjoint B ↔ ∀ (i₁ : ι₁) (i₂ : ι₂), ⟪A (b₁ i₁), b₂ i₂⟫ = ⟪b₁ i₁, B (b₂ i₂)⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 E] [InnerProductSpace 𝕜 F] [FiniteDimensional 𝕜 E] [FiniteDimensional 𝕜 F] {ι₁ : Type u_5} {ι₂ : Type u_6} (b₁ : Basis ι₁ 𝕜 E) (b₂ : Basis ι₂ 𝕜 F) (A : E →ₗ[𝕜] F) (B : F →ₗ[𝕜] E)
Goal: A = LinearMap.adjoint B ↔ ∀ (i₁ : ι₁) (i₂ : ι₂), ⟪A (b₁ i₁), b₂ i₂⟫_𝕜 = ⟪b₁ i₁, B (b₂ i₂)⟫_𝕜
-- LinearMap.eq_adjoint_iff_basis_left
theorem eq_adjoint_iff_basis_left {ι : Type*} (b : Basis ι 𝕜 E) (A : E →ₗ[𝕜] F) (B : F →ₗ[𝕜] E) :
    A = LinearMap.adjoint B ↔ ∀ i y, ⟪A (b i), y⟫ = ⟪b i, B y⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 E] [InnerProductSpace 𝕜 F] [FiniteDimensional 𝕜 E] [FiniteDimensional 𝕜 F] {ι : Type u_5} (b : Basis ι 𝕜 E) (A : E →ₗ[𝕜] F) (B : F →ₗ[𝕜] E)
Goal: A = LinearMap.adjoint B ↔ ∀ (i : ι) (y : F), ⟪A (b i), y⟫_𝕜 = ⟪b i, B y⟫_𝕜
-- LinearMap.eq_adjoint_iff_basis_right
theorem eq_adjoint_iff_basis_right {ι : Type*} (b : Basis ι 𝕜 F) (A : E →ₗ[𝕜] F) (B : F →ₗ[𝕜] E) :
    A = LinearMap.adjoint B ↔ ∀ i x, ⟪A x, b i⟫ = ⟪x, B (b i)⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 E] [InnerProductSpace 𝕜 F] [FiniteDimensional 𝕜 E] [FiniteDimensional 𝕜 F] {ι : Type u_5} (b : Basis ι 𝕜 F) (A : E →ₗ[𝕜] F) (B : F →ₗ[𝕜] E)
Goal: A = LinearMap.adjoint B ↔ ∀ (i : ι) (x : E), ⟪A x, b i⟫_𝕜 = ⟪x, B (b i)⟫_𝕜
-- LinearMap.star_eq_adjoint
theorem star_eq_adjoint (A : E →ₗ[𝕜] E) : star A = LinearMap.adjoint A 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [FiniteDimensional 𝕜 E] (A : E →ₗ[𝕜] E)
Goal: star A = LinearMap.adjoint A
-- LinearMap.isSelfAdjoint_iff'
theorem isSelfAdjoint_iff' {A : E →ₗ[𝕜] E} : IsSelfAdjoint A ↔ LinearMap.adjoint A = A 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [FiniteDimensional 𝕜 E] {A : E →ₗ[𝕜] E}
Goal: IsSelfAdjoint A ↔ LinearMap.adjoint A = A
-- LinearMap.isSymmetric_iff_isSelfAdjoint
theorem isSymmetric_iff_isSelfAdjoint (A : E →ₗ[𝕜] E) : IsSymmetric A ↔ IsSelfAdjoint A 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [FiniteDimensional 𝕜 E] (A : E →ₗ[𝕜] E)
Goal: A.IsSymmetric ↔ IsSelfAdjoint A
-- LinearMap.isAdjointPair_inner
theorem isAdjointPair_inner (A : E →ₗ[𝕜] F) :
    IsAdjointPair (sesqFormOfInner : E →ₗ[𝕜] E →ₗ⋆[𝕜] 𝕜) (sesqFormOfInner : F →ₗ[𝕜] F →ₗ⋆[𝕜] 𝕜) A
      (LinearMap.adjoint A) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 E] [InnerProductSpace 𝕜 F] [FiniteDimensional 𝕜 E] [FiniteDimensional 𝕜 F] (A : E →ₗ[𝕜] F)
Goal: sesqFormOfInner.IsAdjointPair sesqFormOfInner A (LinearMap.adjoint A)
-- LinearMap.isSymmetric_adjoint_mul_self
theorem isSymmetric_adjoint_mul_self (T : E →ₗ[𝕜] E) : IsSymmetric (LinearMap.adjoint T * T) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [FiniteDimensional 𝕜 E] (T : E →ₗ[𝕜] E)
Goal: (LinearMap.adjoint T * T).IsSymmetric
-- LinearMap.re_inner_adjoint_mul_self_nonneg
theorem re_inner_adjoint_mul_self_nonneg (T : E →ₗ[𝕜] E) (x : E) :
    0 ≤ re ⟪x, (LinearMap.adjoint T * T) x⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [FiniteDimensional 𝕜 E] (T : E →ₗ[𝕜] E) (x : E)
Goal: 0 ≤ RCLike.re ⟪x, (LinearMap.adjoint T * T) x⟫_𝕜
-- LinearMap.im_inner_adjoint_mul_self_eq_zero
@[simp]
theorem im_inner_adjoint_mul_self_eq_zero (T : E →ₗ[𝕜] E) (x : E) :
    im ⟪x, LinearMap.adjoint T (T x)⟫ = 0 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [FiniteDimensional 𝕜 E] (T : E →ₗ[𝕜] E) (x : E)
Goal: RCLike.im ⟪x, (LinearMap.adjoint T) (T x)⟫_𝕜 = 0
-- ContinuousLinearMap.inner_map_map_iff_adjoint_comp_self
theorem inner_map_map_iff_adjoint_comp_self (u : H →L[𝕜] K) :
    (∀ x y : H, ⟪u x, u y⟫_𝕜 = ⟪x, y⟫_𝕜) ↔ adjoint u ∘L u = 1 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {H : Type u_5} [NormedAddCommGroup H] [InnerProductSpace 𝕜 H] [CompleteSpace H] {K : Type u_6} [NormedAddCommGroup K] [InnerProductSpace 𝕜 K] [CompleteSpace K] (u : H →L[𝕜] K)
Goal: (∀ (x y : H), ⟪u x, u y⟫_𝕜 = ⟪x, y⟫_𝕜) ↔ (ContinuousLinearMap.adjoint u).comp u = 1
-- ContinuousLinearMap.norm_map_iff_adjoint_comp_self
theorem norm_map_iff_adjoint_comp_self (u : H →L[𝕜] K) :
    (∀ x : H, ‖u x‖ = ‖x‖) ↔ adjoint u ∘L u = 1 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {H : Type u_5} [NormedAddCommGroup H] [InnerProductSpace 𝕜 H] [CompleteSpace H] {K : Type u_6} [NormedAddCommGroup K] [InnerProductSpace 𝕜 K] [CompleteSpace K] (u : H →L[𝕜] K)
Goal: (∀ (x : H), ‖u x‖ = ‖x‖) ↔ (ContinuousLinearMap.adjoint u).comp u = 1
-- LinearIsometryEquiv.adjoint_eq_symm
@[simp]
lemma _root_.LinearIsometryEquiv.adjoint_eq_symm (e : H ≃ₗᵢ[𝕜] K) :
    adjoint (e : H →L[𝕜] K) = e.symm 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {H : Type u_5} [NormedAddCommGroup H] [InnerProductSpace 𝕜 H] [CompleteSpace H] {K : Type u_6} [NormedAddCommGroup K] [InnerProductSpace 𝕜 K] [CompleteSpace K] (e : H ≃ₗᵢ[𝕜] K)
Goal: ContinuousLinearMap.adjoint ↑{ toLinearEquiv := e.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ } =
  ↑{ toLinearEquiv := e.symm.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ }
-- LinearIsometryEquiv.star_eq_symm
@[simp]
lemma _root_.LinearIsometryEquiv.star_eq_symm (e : H ≃ₗᵢ[𝕜] H) :
    star (e : H →L[𝕜] H) = e.symm 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {H : Type u_5} [NormedAddCommGroup H] [InnerProductSpace 𝕜 H] [CompleteSpace H] (e : H ≃ₗᵢ[𝕜] H)
Goal: star ↑{ toLinearEquiv := e.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ } =
  ↑{ toLinearEquiv := e.symm.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ }
-- ContinuousLinearMap.norm_map_of_mem_unitary
theorem norm_map_of_mem_unitary {u : H →L[𝕜] H} (hu : u ∈ unitary (H →L[𝕜] H)) (x : H) :
    ‖u x‖ = ‖x‖ 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {H : Type u_5} [NormedAddCommGroup H] [InnerProductSpace 𝕜 H] [CompleteSpace H] {u : H →L[𝕜] H} (hu : u ∈ unitary (H →L[𝕜] H)) (x : H)
Goal: ‖u x‖ = ‖x‖
-- ContinuousLinearMap.inner_map_map_of_mem_unitary
theorem inner_map_map_of_mem_unitary {u : H →L[𝕜] H} (hu : u ∈ unitary (H →L[𝕜] H)) (x y : H) :
    ⟪u x, u y⟫_𝕜 = ⟪x, y⟫_𝕜 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {H : Type u_5} [NormedAddCommGroup H] [InnerProductSpace 𝕜 H] [CompleteSpace H] {u : H →L[𝕜] H} (hu : u ∈ unitary (H →L[𝕜] H)) (x y : H)
Goal: ⟪u x, u y⟫_𝕜 = ⟪x, y⟫_𝕜
-- unitary.norm_map
theorem norm_map (u : unitary (H →L[𝕜] H)) (x : H) : ‖(u : H →L[𝕜] H) x‖ = ‖x‖ 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {H : Type u_5} [NormedAddCommGroup H] [InnerProductSpace 𝕜 H] [CompleteSpace H] (u : ↥(unitary (H →L[𝕜] H))) (x : H)
Goal: ‖↑u x‖ = ‖x‖
-- unitary.inner_map_map
theorem inner_map_map (u : unitary (H →L[𝕜] H)) (x y : H) :
    ⟪(u : H →L[𝕜] H) x, (u : H →L[𝕜] H) y⟫_𝕜 = ⟪x, y⟫_𝕜 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {H : Type u_5} [NormedAddCommGroup H] [InnerProductSpace 𝕜 H] [CompleteSpace H] (u : ↥(unitary (H →L[𝕜] H))) (x y : H)
Goal: ⟪↑u x, ↑u y⟫_𝕜 = ⟪x, y⟫_𝕜
-- unitary.linearIsometryEquiv_coe_apply
@[simp]
lemma linearIsometryEquiv_coe_apply (u : unitary (H →L[𝕜] H)) :
    linearIsometryEquiv u = (u : H →L[𝕜] H) 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {H : Type u_5} [NormedAddCommGroup H] [InnerProductSpace 𝕜 H] [CompleteSpace H] (u : ↥(unitary (H →L[𝕜] H)))
Goal: ↑{ toLinearEquiv := (unitary.linearIsometryEquiv u).toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ } = ↑u
-- unitary.linearIsometryEquiv_coe_symm_apply
@[simp]
lemma linearIsometryEquiv_coe_symm_apply (e : H ≃ₗᵢ[𝕜] H) :
    linearIsometryEquiv.symm e = (e : H →L[𝕜] H) 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {H : Type u_5} [NormedAddCommGroup H] [InnerProductSpace 𝕜 H] [CompleteSpace H] (e : H ≃ₗᵢ[𝕜] H)
Goal: ↑(unitary.linearIsometryEquiv.symm e) =
  ↑{ toLinearEquiv := e.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ }
-- Matrix.toLin_conjTranspose
lemma Matrix.toLin_conjTranspose (A : Matrix m n 𝕜) :
    toLin v₂.toBasis v₁.toBasis Aᴴ = adjoint (toLin v₁.toBasis v₂.toBasis A) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 E] [InnerProductSpace 𝕜 F] {m : Type u_5} {n : Type u_6} [Fintype m] [DecidableEq m] [Fintype n] [DecidableEq n] [FiniteDimensional 𝕜 E] [FiniteDimensional 𝕜 F] (v₁ : OrthonormalBasis n 𝕜 E) (v₂ : OrthonormalBasis m 𝕜 F) (A : Matrix m n 𝕜)
Goal: (Matrix.toLin v₂.toBasis v₁.toBasis) A.conjTranspose = LinearMap.adjoint ((Matrix.toLin v₁.toBasis v₂.toBasis) A)
-- LinearMap.toMatrix_adjoint
lemma LinearMap.toMatrix_adjoint (f : E →ₗ[𝕜] F) :
    toMatrix v₂.toBasis v₁.toBasis (adjoint f) = (toMatrix v₁.toBasis v₂.toBasis f)ᴴ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 E] [InnerProductSpace 𝕜 F] {m : Type u_5} {n : Type u_6} [Fintype m] [DecidableEq m] [Fintype n] [DecidableEq n] [FiniteDimensional 𝕜 E] [FiniteDimensional 𝕜 F] (v₁ : OrthonormalBasis n 𝕜 E) (v₂ : OrthonormalBasis m 𝕜 F) (f : E →ₗ[𝕜] F)
Goal: (LinearMap.toMatrix v₂.toBasis v₁.toBasis) (LinearMap.adjoint f) =
  ((LinearMap.toMatrix v₁.toBasis v₂.toBasis) f).conjTranspose
-- LinearMap.toMatrixOrthonormal_apply
@[simps]
def LinearMap.toMatrixOrthonormal : (E →ₗ[𝕜] E) ≃⋆ₐ[𝕜] Matrix n n 𝕜 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {n : Type u_6} [Fintype n] [DecidableEq n] [FiniteDimensional 𝕜 E] (v₁ : OrthonormalBasis n 𝕜 E)
Goal: ∀ (a : E →ₗ[𝕜] E), (LinearMap.toMatrixOrthonormal v₁) a = (↑(LinearMap.toMatrix v₁.toBasis v₁.toBasis)).toFun a
-- LinearMap.toMatrixOrthonormal_symm_apply
@[simps]
def LinearMap.toMatrixOrthonormal : (E →ₗ[𝕜] E) ≃⋆ₐ[𝕜] Matrix n n 𝕜 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {n : Type u_6} [Fintype n] [DecidableEq n] [FiniteDimensional 𝕜 E] (v₁ : OrthonormalBasis n 𝕜 E)
Goal: ∀ (a : Matrix n n 𝕜), (LinearMap.toMatrixOrthonormal v₁).symm a = (LinearMap.toMatrix v₁.toBasis v₁.toBasis).invFun a
-- Matrix.toEuclideanLin_conjTranspose_eq_adjoint
theorem Matrix.toEuclideanLin_conjTranspose_eq_adjoint (A : Matrix m n 𝕜) :
    Matrix.toEuclideanLin A.conjTranspose = LinearMap.adjoint (Matrix.toEuclideanLin A) 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {m : Type u_5} {n : Type u_6} [Fintype m] [DecidableEq m] [Fintype n] [DecidableEq n] (A : Matrix m n 𝕜)
Goal: Matrix.toEuclideanLin A.conjTranspose = LinearMap.adjoint (Matrix.toEuclideanLin A)
-- MeasureTheory.Measure.HaveLebesgueDecomposition.lebesgue_decomposition
class HaveLebesgueDecomposition (μ ν : Measure α) : Prop where
  lebesgue_decomposition :
    ∃ p : Measure α × (α → ℝ≥0∞), Measurable p.2 ∧ p.1 ⟂ₘ ν ∧ μ = p.1 + ν.withDensity p.2
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [self : μ.HaveLebesgueDecomposition ν]
Goal: ∃ p, Measurable p.2 ∧ p.1 ⟂ₘ ν ∧ μ = p.1 + ν.withDensity p.2
-- MeasureTheory.Measure.singularPart_def
noncomputable irreducible_def singularPart (μ ν : Measure α) : Measure α 
Arguments: {α : Type u_3} {m : MeasurableSpace α} (μ ν : MeasureTheory.Measure α)
Goal: μ.singularPart ν = if h : μ.HaveLebesgueDecomposition ν then (Classical.choose ⋯).1 else 0
-- MeasureTheory.Measure.rnDeriv_def
noncomputable irreducible_def rnDeriv (μ ν : Measure α) : α → ℝ≥0∞ 
Arguments: {α : Type u_3} {m : MeasurableSpace α} (μ ν : MeasureTheory.Measure α)
Goal: μ.rnDeriv ν = if h : μ.HaveLebesgueDecomposition ν then (Classical.choose ⋯).2 else 0