ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContinuousMultilinearMap.le_of_opNNNorm_le
theorem le_of_opNNNorm_le {C : ℝ≥0} (h : ‖f‖₊ ≤ C) : ‖f m‖₊ ≤ C * ∏ i, ‖m i‖₊ 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) (m : (i : ι) → E i) {C : ℝ≥0} (h : ‖f‖₊ ≤ C)
Goal: ‖f m‖₊ ≤ C * ∏ i : ι, ‖m i‖₊
-- ContinuousMultilinearMap.opNNNorm_le_iff
theorem opNNNorm_le_iff {C : ℝ≥0} : ‖f‖₊ ≤ C ↔ ∀ m, ‖f m‖₊ ≤ C * ∏ i, ‖m i‖₊ 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) {C : ℝ≥0}
Goal: ‖f‖₊ ≤ C ↔ ∀ (m : (i : ι) → E i), ‖f m‖₊ ≤ C * ∏ i : ι, ‖m i‖₊
-- ContinuousMultilinearMap.isLeast_opNNNorm
theorem isLeast_opNNNorm : IsLeast {C : ℝ≥0 | ∀ m, ‖f m‖₊ ≤ C * ∏ i, ‖m i‖₊} ‖f‖₊ 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G)
Goal: IsLeast {C | ∀ (m : (i : ι) → E i), ‖f m‖₊ ≤ C * ∏ i : ι, ‖m i‖₊} ‖f‖₊
-- ContinuousMultilinearMap.opNNNorm_prod
theorem opNNNorm_prod (f : ContinuousMultilinearMap 𝕜 E G) (g : ContinuousMultilinearMap 𝕜 E G') :
    ‖f.prod g‖₊ = max ‖f‖₊ ‖g‖₊ 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} {G' : Type wG'} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [SeminormedAddCommGroup G'] [NormedSpace 𝕜 G'] (f : ContinuousMultilinearMap 𝕜 E G) (g : ContinuousMultilinearMap 𝕜 E G')
Goal: ‖f.prod g‖₊ = max ‖f‖₊ ‖g‖₊
-- ContinuousMultilinearMap.opNorm_prod
theorem opNorm_prod (f : ContinuousMultilinearMap 𝕜 E G) (g : ContinuousMultilinearMap 𝕜 E G') :
    ‖f.prod g‖ = max ‖f‖ ‖g‖ 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} {G' : Type wG'} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [SeminormedAddCommGroup G'] [NormedSpace 𝕜 G'] (f : ContinuousMultilinearMap 𝕜 E G) (g : ContinuousMultilinearMap 𝕜 E G')
Goal: ‖f.prod g‖ = max ‖f‖ ‖g‖
-- ContinuousMultilinearMap.opNNNorm_pi
theorem opNNNorm_pi
    [∀ i', SeminormedAddCommGroup (E' i')] [∀ i', NormedSpace 𝕜 (E' i')]
    (f : ∀ i', ContinuousMultilinearMap 𝕜 E (E' i')) : ‖pi f‖₊ = ‖f‖₊ 
Arguments: {𝕜 : Type u} {ι : Type v} {ι' : Type v'} {E : ι → Type wE} {E' : ι' → Type wE'} [Fintype ι] [Fintype ι'] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [(i' : ι') → SeminormedAddCommGroup (E' i')] [(i' : ι') → NormedSpace 𝕜 (E' i')] (f : (i' : ι') → ContinuousMultilinearMap 𝕜 E (E' i'))
Goal: ‖ContinuousMultilinearMap.pi f‖₊ = ‖f‖₊
-- ContinuousMultilinearMap.opNorm_pi
theorem opNorm_pi {ι' : Type v'} [Fintype ι'] {E' : ι' → Type wE'}
    [∀ i', SeminormedAddCommGroup (E' i')] [∀ i', NormedSpace 𝕜 (E' i')]
    (f : ∀ i', ContinuousMultilinearMap 𝕜 E (E' i')) :
    ‖pi f‖ = ‖f‖ 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] {ι' : Type v'} [Fintype ι'] {E' : ι' → Type wE'} [(i' : ι') → SeminormedAddCommGroup (E' i')] [(i' : ι') → NormedSpace 𝕜 (E' i')] (f : (i' : ι') → ContinuousMultilinearMap 𝕜 E (E' i'))
Goal: ‖ContinuousMultilinearMap.pi f‖ = ‖f‖
-- ContinuousMultilinearMap.norm_ofSubsingleton
@[simp]
theorem norm_ofSubsingleton [Subsingleton ι] (i : ι) (f : G →L[𝕜] G') :
    ‖ofSubsingleton 𝕜 G G' i f‖ = ‖f‖ 
Arguments: {𝕜 : Type u} {ι : Type v} {G : Type wG} {G' : Type wG'} [Fintype ι] [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [SeminormedAddCommGroup G'] [NormedSpace 𝕜 G'] [Subsingleton ι] (i : ι) (f : G →L[𝕜] G')
Goal: ‖(ContinuousMultilinearMap.ofSubsingleton 𝕜 G G' i) f‖ = ‖f‖
-- ContinuousMultilinearMap.nnnorm_ofSubsingleton
@[simp]
theorem nnnorm_ofSubsingleton [Subsingleton ι] (i : ι) (f : G →L[𝕜] G') :
    ‖ofSubsingleton 𝕜 G G' i f‖₊ = ‖f‖₊ 
Arguments: {𝕜 : Type u} {ι : Type v} {G : Type wG} {G' : Type wG'} [Fintype ι] [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [SeminormedAddCommGroup G'] [NormedSpace 𝕜 G'] [Subsingleton ι] (i : ι) (f : G →L[𝕜] G')
Goal: ‖(ContinuousMultilinearMap.ofSubsingleton 𝕜 G G' i) f‖₊ = ‖f‖₊
-- ContinuousMultilinearMap.ofSubsingletonₗᵢ_symm_apply
@[simps apply symm_apply]
def ofSubsingletonₗᵢ [Subsingleton ι] (i : ι) :
    (G →L[𝕜] G') ≃ₗᵢ[𝕜] ContinuousMultilinearMap 𝕜 (fun _ : ι ↦ G) G' 
Arguments: (𝕜 : Type u) {ι : Type v} (G : Type wG) {G' : Type wG'} [Fintype ι] [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [SeminormedAddCommGroup G'] [NormedSpace 𝕜 G'] [Subsingleton ι] (i : ι)
Goal: ∀ (a : ContinuousMultilinearMap 𝕜 (fun x => G) G'),
  (ContinuousMultilinearMap.ofSubsingletonₗᵢ 𝕜 G i).symm a = (ContinuousMultilinearMap.ofSubsingleton 𝕜 G G' i).invFun a
-- ContinuousMultilinearMap.ofSubsingletonₗᵢ_apply
@[simps apply symm_apply]
def ofSubsingletonₗᵢ [Subsingleton ι] (i : ι) :
    (G →L[𝕜] G') ≃ₗᵢ[𝕜] ContinuousMultilinearMap 𝕜 (fun _ : ι ↦ G) G' 
Arguments: (𝕜 : Type u) {ι : Type v} (G : Type wG) {G' : Type wG'} [Fintype ι] [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [SeminormedAddCommGroup G'] [NormedSpace 𝕜 G'] [Subsingleton ι] (i : ι)
Goal: ∀ (a : G →L[𝕜] G'),
  (ContinuousMultilinearMap.ofSubsingletonₗᵢ 𝕜 G i) a = (ContinuousMultilinearMap.ofSubsingleton 𝕜 G G' i).toFun a
-- ContinuousMultilinearMap.norm_ofSubsingleton_id_le
theorem norm_ofSubsingleton_id_le [Subsingleton ι] (i : ι) :
    ‖ofSubsingleton 𝕜 G G i (.id _ _)‖ ≤ 1 
Arguments: (𝕜 : Type u) {ι : Type v} (G : Type wG) [Fintype ι] [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [Subsingleton ι] (i : ι)
Goal: ‖(ContinuousMultilinearMap.ofSubsingleton 𝕜 G G i) (ContinuousLinearMap.id 𝕜 G)‖ ≤ 1
-- ContinuousMultilinearMap.nnnorm_ofSubsingleton_id_le
theorem nnnorm_ofSubsingleton_id_le [Subsingleton ι] (i : ι) :
    ‖ofSubsingleton 𝕜 G G i (.id _ _)‖₊ ≤ 1 
Arguments: (𝕜 : Type u) {ι : Type v} (G : Type wG) [Fintype ι] [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [Subsingleton ι] (i : ι)
Goal: ‖(ContinuousMultilinearMap.ofSubsingleton 𝕜 G G i) (ContinuousLinearMap.id 𝕜 G)‖₊ ≤ 1
-- ContinuousMultilinearMap.norm_constOfIsEmpty
@[simp]
theorem norm_constOfIsEmpty [IsEmpty ι] (x : G) : ‖constOfIsEmpty 𝕜 E x‖ = ‖x‖ 
Arguments: (𝕜 : Type u) {ι : Type v} (E : ι → Type wE) {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [IsEmpty ι] (x : G)
Goal: ‖ContinuousMultilinearMap.constOfIsEmpty 𝕜 E x‖ = ‖x‖
-- ContinuousMultilinearMap.nnnorm_constOfIsEmpty
@[simp]
theorem nnnorm_constOfIsEmpty [IsEmpty ι] (x : G) : ‖constOfIsEmpty 𝕜 E x‖₊ = ‖x‖₊ 
Arguments: (𝕜 : Type u) {ι : Type v} (E : ι → Type wE) {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [IsEmpty ι] (x : G)
Goal: ‖ContinuousMultilinearMap.constOfIsEmpty 𝕜 E x‖₊ = ‖x‖₊
-- ContinuousMultilinearMap.norm_restrictScalars
@[simp]
theorem norm_restrictScalars : ‖f.restrictScalars 𝕜'‖ = ‖f‖ 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜' 𝕜] [NormedSpace 𝕜' G] [IsScalarTower 𝕜' 𝕜 G] [(i : ι) → NormedSpace 𝕜' (E i)] [∀ (i : ι), IsScalarTower 𝕜' 𝕜 (E i)]
Goal: ‖ContinuousMultilinearMap.restrictScalars 𝕜' f‖ = ‖f‖
-- ContinuousMultilinearMap.continuous_restrictScalars
theorem continuous_restrictScalars :
    Continuous
      (restrictScalars 𝕜' : ContinuousMultilinearMap 𝕜 E G → ContinuousMultilinearMap 𝕜' E G) 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜' 𝕜] [NormedSpace 𝕜' G] [IsScalarTower 𝕜' 𝕜 G] [(i : ι) → NormedSpace 𝕜' (E i)] [∀ (i : ι), IsScalarTower 𝕜' 𝕜 (E i)]
Goal: Continuous (ContinuousMultilinearMap.restrictScalars 𝕜')
-- ContinuousMultilinearMap.norm_image_sub_le'
theorem norm_image_sub_le' [DecidableEq ι] (m₁ m₂ : ∀ i, E i) :
    ‖f m₁ - f m₂‖ ≤ ‖f‖ * ∑ i, ∏ j, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) [DecidableEq ι] (m₁ m₂ : (i : ι) → E i)
Goal: ‖f m₁ - f m₂‖ ≤ ‖f‖ * ∑ i : ι, ∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖
-- ContinuousMultilinearMap.norm_image_sub_le
theorem norm_image_sub_le (m₁ m₂ : ∀ i, E i) :
    ‖f m₁ - f m₂‖ ≤ ‖f‖ * Fintype.card ι * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1) * ‖m₁ - m₂‖ 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) (m₁ m₂ : (i : ι) → E i)
Goal: ‖f m₁ - f m₂‖ ≤ ‖f‖ * ↑(Fintype.card ι) * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1) * ‖m₁ - m₂‖
-- MultilinearMap.mkContinuous_norm_le
theorem MultilinearMap.mkContinuous_norm_le (f : MultilinearMap 𝕜 E G) {C : ℝ} (hC : 0 ≤ C)
    (H : ∀ m, ‖f m‖ ≤ C * ∏ i, ‖m i‖) : ‖f.mkContinuous C H‖ ≤ C 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : MultilinearMap 𝕜 E G) {C : ℝ} (hC : 0 ≤ C) (H : ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i : ι, ‖m i‖)
Goal: ‖f.mkContinuous C H‖ ≤ C
-- MultilinearMap.mkContinuous_norm_le'
theorem MultilinearMap.mkContinuous_norm_le' (f : MultilinearMap 𝕜 E G) {C : ℝ}
    (H : ∀ m, ‖f m‖ ≤ C * ∏ i, ‖m i‖) : ‖f.mkContinuous C H‖ ≤ max C 0 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : MultilinearMap 𝕜 E G) {C : ℝ} (H : ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i : ι, ‖m i‖)
Goal: ‖f.mkContinuous C H‖ ≤ max C 0
-- ContinuousMultilinearMap.norm_restr
theorem norm_restr {k n : ℕ} (f : G[×n]→L[𝕜] G') (s : Finset (Fin n)) (hk : s.card = k) (z : G) :
    ‖f.restr s hk z‖ ≤ ‖f‖ * ‖z‖ ^ (n - k) 
Arguments: {𝕜 : Type u} {G : Type wG} {G' : Type wG'} [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [SeminormedAddCommGroup G'] [NormedSpace 𝕜 G'] {k n : ℕ} (f : ContinuousMultilinearMap 𝕜 (fun i => G) G') (s : Finset (Fin n)) (hk : s.card = k) (z : G)
Goal: ‖f.restr s hk z‖ ≤ ‖f‖ * ‖z‖ ^ (n - k)
-- ContinuousMultilinearMap.norm_mkPiAlgebra_le
@[simp]
theorem norm_mkPiAlgebra_le [Nonempty ι] : ‖ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A‖ ≤ 1 
Arguments: {𝕜 : Type u} {ι : Type v} [Fintype ι] [NontriviallyNormedField 𝕜] {A : Type u_1} [NormedCommRing A] [NormedAlgebra 𝕜 A] [Nonempty ι]
Goal: ‖ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A‖ ≤ 1
-- ContinuousMultilinearMap.norm_mkPiAlgebra_of_empty
theorem norm_mkPiAlgebra_of_empty [IsEmpty ι] :
    ‖ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A‖ = ‖(1 : A)‖ 
Arguments: {𝕜 : Type u} {ι : Type v} [Fintype ι] [NontriviallyNormedField 𝕜] {A : Type u_1} [NormedCommRing A] [NormedAlgebra 𝕜 A] [IsEmpty ι]
Goal: ‖ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A‖ = ‖1‖
-- ContinuousMultilinearMap.norm_mkPiAlgebra
@[simp]
theorem norm_mkPiAlgebra [NormOneClass A] : ‖ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A‖ = 1 
Arguments: {𝕜 : Type u} {ι : Type v} [Fintype ι] [NontriviallyNormedField 𝕜] {A : Type u_1} [NormedCommRing A] [NormedAlgebra 𝕜 A] [NormOneClass A]
Goal: ‖ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A‖ = 1
-- ContinuousMultilinearMap.norm_mkPiAlgebraFin_succ_le
theorem norm_mkPiAlgebraFin_succ_le : ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 n.succ A‖ ≤ 1 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {n : ℕ} {A : Type u_1} [NormedRing A] [NormedAlgebra 𝕜 A]
Goal: ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 n.succ A‖ ≤ 1
-- ContinuousMultilinearMap.norm_mkPiAlgebraFin_le_of_pos
theorem norm_mkPiAlgebraFin_le_of_pos (hn : 0 < n) :
    ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 n A‖ ≤ 1 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {n : ℕ} {A : Type u_1} [NormedRing A] [NormedAlgebra 𝕜 A] (hn : 0 < n)
Goal: ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 n A‖ ≤ 1
-- ContinuousMultilinearMap.norm_mkPiAlgebraFin_zero
theorem norm_mkPiAlgebraFin_zero : ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 0 A‖ = ‖(1 : A)‖ 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {A : Type u_1} [NormedRing A] [NormedAlgebra 𝕜 A]
Goal: ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 0 A‖ = ‖1‖
-- ContinuousMultilinearMap.norm_mkPiAlgebraFin
@[simp]
theorem norm_mkPiAlgebraFin [NormOneClass A] :
    ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 n A‖ = 1 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {n : ℕ} {A : Type u_1} [NormedRing A] [NormedAlgebra 𝕜 A] [NormOneClass A]
Goal: ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 n A‖ = 1
-- ContinuousMultilinearMap.nnnorm_smulRight
@[simp]
theorem nnnorm_smulRight (f : ContinuousMultilinearMap 𝕜 E 𝕜) (z : G) :
    ‖f.smulRight z‖₊ = ‖f‖₊ * ‖z‖₊ 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E 𝕜) (z : G)
Goal: ‖f.smulRight z‖₊ = ‖f‖₊ * ‖z‖₊
-- ContinuousMultilinearMap.norm_smulRight
@[simp]
theorem norm_smulRight (f : ContinuousMultilinearMap 𝕜 E 𝕜) (z : G) :
    ‖f.smulRight z‖ = ‖f‖ * ‖z‖ 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E 𝕜) (z : G)
Goal: ‖f.smulRight z‖ = ‖f‖ * ‖z‖
-- ContinuousMultilinearMap.norm_mkPiRing
@[simp]
theorem norm_mkPiRing (z : G) : ‖ContinuousMultilinearMap.mkPiRing 𝕜 ι z‖ = ‖z‖ 
Arguments: {𝕜 : Type u} {ι : Type v} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (z : G)
Goal: ‖ContinuousMultilinearMap.mkPiRing 𝕜 ι z‖ = ‖z‖
-- ContinuousMultilinearMap.smulRightL_apply
@[simp] lemma smulRightL_apply (f : ContinuousMultilinearMap 𝕜 E 𝕜) (z : G) :
  smulRightL 𝕜 E G f z = f.smulRight z 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E 𝕜) (z : G)
Goal: ((ContinuousMultilinearMap.smulRightL 𝕜 E G) f) z = f.smulRight z
-- ContinuousMultilinearMap.norm_smulRightL_le
lemma norm_smulRightL_le : ‖smulRightL 𝕜 E G‖ ≤ 1 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G]
Goal: ‖ContinuousMultilinearMap.smulRightL 𝕜 E G‖ ≤ 1
-- ContinuousLinearMap.norm_compContinuousMultilinearMap_le
theorem norm_compContinuousMultilinearMap_le (g : G →L[𝕜] G') (f : ContinuousMultilinearMap 𝕜 E G) :
    ‖g.compContinuousMultilinearMap f‖ ≤ ‖g‖ * ‖f‖ 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} {G' : Type wG'} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [SeminormedAddCommGroup G'] [NormedSpace 𝕜 G'] (g : G →L[𝕜] G') (f : ContinuousMultilinearMap 𝕜 E G)
Goal: ‖g.compContinuousMultilinearMap f‖ ≤ ‖g‖ * ‖f‖
-- ContinuousLinearEquiv.compContinuousMultilinearMapL_symm
@[simp]
theorem _root_.ContinuousLinearEquiv.compContinuousMultilinearMapL_symm (g : G ≃L[𝕜] G') :
    (g.compContinuousMultilinearMapL E).symm = g.symm.compContinuousMultilinearMapL E 
Arguments: {𝕜 : Type u} {ι : Type v} (E : ι → Type wE) {G : Type wG} {G' : Type wG'} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [SeminormedAddCommGroup G'] [NormedSpace 𝕜 G'] (g : G ≃L[𝕜] G')
Goal: (ContinuousLinearEquiv.compContinuousMultilinearMapL E g).symm =
  ContinuousLinearEquiv.compContinuousMultilinearMapL E g.symm
-- ContinuousLinearEquiv.compContinuousMultilinearMapL_apply
@[simp]
theorem _root_.ContinuousLinearEquiv.compContinuousMultilinearMapL_apply (g : G ≃L[𝕜] G')
    (f : ContinuousMultilinearMap 𝕜 E G) :
    g.compContinuousMultilinearMapL E f = (g : G →L[𝕜] G').compContinuousMultilinearMap f 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} {G' : Type wG'} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [SeminormedAddCommGroup G'] [NormedSpace 𝕜 G'] (g : G ≃L[𝕜] G') (f : ContinuousMultilinearMap 𝕜 E G)
Goal: (ContinuousLinearEquiv.compContinuousMultilinearMapL E g) f = (↑g).compContinuousMultilinearMap f
-- ContinuousLinearMap.flipMultilinear_apply_apply
@[simps! apply_apply]
def flipMultilinear (f : G →L[𝕜] ContinuousMultilinearMap 𝕜 E G') :
    ContinuousMultilinearMap 𝕜 E (G →L[𝕜] G') 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} {G' : Type wG'} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [SeminormedAddCommGroup G'] [NormedSpace 𝕜 G'] (f : G →L[𝕜] ContinuousMultilinearMap 𝕜 E G') (m : (i : ι) → E i) (x : G)
Goal: (f.flipMultilinear m) x = (f x) m
-- LinearIsometry.norm_compContinuousMultilinearMap
theorem LinearIsometry.norm_compContinuousMultilinearMap (g : G →ₗᵢ[𝕜] G')
    (f : ContinuousMultilinearMap 𝕜 E G) :
    ‖g.toContinuousLinearMap.compContinuousMultilinearMap f‖ = ‖f‖ 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} {G' : Type wG'} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [SeminormedAddCommGroup G'] [NormedSpace 𝕜 G'] (g : G →ₗᵢ[𝕜] G') (f : ContinuousMultilinearMap 𝕜 E G)
Goal: ‖g.toContinuousLinearMap.compContinuousMultilinearMap f‖ = ‖f‖
-- MultilinearMap.mkContinuousLinear_norm_le'
theorem mkContinuousLinear_norm_le' (f : G →ₗ[𝕜] MultilinearMap 𝕜 E G') (C : ℝ)
    (H : ∀ x m, ‖f x m‖ ≤ C * ‖x‖ * ∏ i, ‖m i‖) : ‖mkContinuousLinear f C H‖ ≤ max C 0 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} {G' : Type wG'} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [SeminormedAddCommGroup G'] [NormedSpace 𝕜 G'] (f : G →ₗ[𝕜] MultilinearMap 𝕜 E G') (C : ℝ) (H : ∀ (x : G) (m : (i : ι) → E i), ‖(f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖)
Goal: ‖MultilinearMap.mkContinuousLinear f C H‖ ≤ max C 0
-- MultilinearMap.mkContinuousLinear_norm_le
theorem mkContinuousLinear_norm_le (f : G →ₗ[𝕜] MultilinearMap 𝕜 E G') {C : ℝ} (hC : 0 ≤ C)
    (H : ∀ x m, ‖f x m‖ ≤ C * ‖x‖ * ∏ i, ‖m i‖) : ‖mkContinuousLinear f C H‖ ≤ C 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} {G' : Type wG'} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [SeminormedAddCommGroup G'] [NormedSpace 𝕜 G'] (f : G →ₗ[𝕜] MultilinearMap 𝕜 E G') {C : ℝ} (hC : 0 ≤ C) (H : ∀ (x : G) (m : (i : ι) → E i), ‖(f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖)
Goal: ‖MultilinearMap.mkContinuousLinear f C H‖ ≤ C
-- MultilinearMap.mkContinuousMultilinear_apply
@[simp]
theorem mkContinuousMultilinear_apply (f : MultilinearMap 𝕜 E (MultilinearMap 𝕜 E' G)) {C : ℝ}
    (H : ∀ m₁ m₂, ‖f m₁ m₂‖ ≤ (C * ∏ i, ‖m₁ i‖) * ∏ i, ‖m₂ i‖) (m : ∀ i, E i) :
    ⇑(mkContinuousMultilinear f C H m) = f m 
Arguments: {𝕜 : Type u} {ι : Type v} {ι' : Type v'} {E : ι → Type wE} {E' : ι' → Type wE'} {G : Type wG} [Fintype ι] [Fintype ι'] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [(i : ι') → SeminormedAddCommGroup (E' i)] [(i : ι') → NormedSpace 𝕜 (E' i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : MultilinearMap 𝕜 E (MultilinearMap 𝕜 E' G)) {C : ℝ} (H : ∀ (m₁ : (i : ι) → E i) (m₂ : (i : ι') → E' i), ‖(f m₁) m₂‖ ≤ (C * ∏ i : ι, ‖m₁ i‖) * ∏ i : ι', ‖m₂ i‖) (m : (i : ι) → E i)
Goal: ⇑((f.mkContinuousMultilinear C H) m) = ⇑(f m)
-- MultilinearMap.mkContinuousMultilinear_norm_le'
theorem mkContinuousMultilinear_norm_le' (f : MultilinearMap 𝕜 E (MultilinearMap 𝕜 E' G)) (C : ℝ)
    (H : ∀ m₁ m₂, ‖f m₁ m₂‖ ≤ (C * ∏ i, ‖m₁ i‖) * ∏ i, ‖m₂ i‖) :
    ‖mkContinuousMultilinear f C H‖ ≤ max C 0 
Arguments: {𝕜 : Type u} {ι : Type v} {ι' : Type v'} {E : ι → Type wE} {E' : ι' → Type wE'} {G : Type wG} [Fintype ι] [Fintype ι'] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [(i : ι') → SeminormedAddCommGroup (E' i)] [(i : ι') → NormedSpace 𝕜 (E' i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : MultilinearMap 𝕜 E (MultilinearMap 𝕜 E' G)) (C : ℝ) (H : ∀ (m₁ : (i : ι) → E i) (m₂ : (i : ι') → E' i), ‖(f m₁) m₂‖ ≤ (C * ∏ i : ι, ‖m₁ i‖) * ∏ i : ι', ‖m₂ i‖)
Goal: ‖f.mkContinuousMultilinear C H‖ ≤ max C 0
-- MultilinearMap.mkContinuousMultilinear_norm_le
theorem mkContinuousMultilinear_norm_le (f : MultilinearMap 𝕜 E (MultilinearMap 𝕜 E' G)) {C : ℝ}
    (hC : 0 ≤ C) (H : ∀ m₁ m₂, ‖f m₁ m₂‖ ≤ (C * ∏ i, ‖m₁ i‖) * ∏ i, ‖m₂ i‖) :
    ‖mkContinuousMultilinear f C H‖ ≤ C 
Arguments: {𝕜 : Type u} {ι : Type v} {ι' : Type v'} {E : ι → Type wE} {E' : ι' → Type wE'} {G : Type wG} [Fintype ι] [Fintype ι'] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [(i : ι') → SeminormedAddCommGroup (E' i)] [(i : ι') → NormedSpace 𝕜 (E' i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : MultilinearMap 𝕜 E (MultilinearMap 𝕜 E' G)) {C : ℝ} (hC : 0 ≤ C) (H : ∀ (m₁ : (i : ι) → E i) (m₂ : (i : ι') → E' i), ‖(f m₁) m₂‖ ≤ (C * ∏ i : ι, ‖m₁ i‖) * ∏ i : ι', ‖m₂ i‖)
Goal: ‖f.mkContinuousMultilinear C H‖ ≤ C
-- ContinuousMultilinearMap.norm_compContinuousLinearMap_le
theorem norm_compContinuousLinearMap_le (g : ContinuousMultilinearMap 𝕜 E₁ G)
    (f : ∀ i, E i →L[𝕜] E₁ i) : ‖g.compContinuousLinearMap f‖ ≤ ‖g‖ * ∏ i, ‖f i‖ 
Arguments: {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {E₁ : ι → Type wE₁} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [(i : ι) → SeminormedAddCommGroup (E₁ i)] [(i : ι) → NormedSpace 𝕜 (E₁ i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (g : ContinuousMultilinearMap 𝕜 E₁ G) (f : (i : ι) → E i →L[𝕜] E₁ i)
Goal: ‖g.compContinuousLinearMap f‖ ≤ ‖g‖ * ∏ i : ι, ‖f i‖