ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- lp.summable_inner
theorem summable_inner (f g : lp G 2) : Summable fun i => ⟪f i, g i⟫ 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] {G : ι → Type u_4} [(i : ι) → NormedAddCommGroup (G i)] [(i : ι) → InnerProductSpace 𝕜 (G i)] (f g : ↥(lp G 2))
Goal: Summable fun i => ⟪↑f i, ↑g i⟫_𝕜
-- lp.inner_eq_tsum
theorem inner_eq_tsum (f g : lp G 2) : ⟪f, g⟫ = ∑' i, ⟪f i, g i⟫ 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] {G : ι → Type u_4} [(i : ι) → NormedAddCommGroup (G i)] [(i : ι) → InnerProductSpace 𝕜 (G i)] (f g : ↥(lp G 2))
Goal: ⟪f, g⟫_𝕜 = ∑' (i : ι), ⟪↑f i, ↑g i⟫_𝕜
-- lp.hasSum_inner
theorem hasSum_inner (f g : lp G 2) : HasSum (fun i => ⟪f i, g i⟫) ⟪f, g⟫ 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] {G : ι → Type u_4} [(i : ι) → NormedAddCommGroup (G i)] [(i : ι) → InnerProductSpace 𝕜 (G i)] (f g : ↥(lp G 2))
Goal: HasSum (fun i => ⟪↑f i, ↑g i⟫_𝕜) ⟪f, g⟫_𝕜
-- lp.inner_single_left
theorem inner_single_left (i : ι) (a : G i) (f : lp G 2) : ⟪lp.single 2 i a, f⟫ = ⟪a, f i⟫ 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] {G : ι → Type u_4} [(i : ι) → NormedAddCommGroup (G i)] [(i : ι) → InnerProductSpace 𝕜 (G i)] (i : ι) (a : G i) (f : ↥(lp G 2))
Goal: ⟪lp.single 2 i a, f⟫_𝕜 = ⟪a, ↑f i⟫_𝕜
-- lp.inner_single_right
theorem inner_single_right (i : ι) (a : G i) (f : lp G 2) : ⟪f, lp.single 2 i a⟫ = ⟪f i, a⟫ 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] {G : ι → Type u_4} [(i : ι) → NormedAddCommGroup (G i)] [(i : ι) → InnerProductSpace 𝕜 (G i)] (i : ι) (a : G i) (f : ↥(lp G 2))
Goal: ⟪f, lp.single 2 i a⟫_𝕜 = ⟪↑f i, a⟫_𝕜
-- OrthogonalFamily.summable_of_lp
protected theorem summable_of_lp (f : lp G 2) : Summable fun i => V i (f i) 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] {E : Type u_3} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [cplt : CompleteSpace E] {G : ι → Type u_4} [(i : ι) → NormedAddCommGroup (G i)] [(i : ι) → InnerProductSpace 𝕜 (G i)] {V : (i : ι) → G i →ₗᵢ[𝕜] E} (hV : OrthogonalFamily 𝕜 G V) (f : ↥(lp G 2))
Goal: Summable fun i => (V i) (↑f i)
-- OrthogonalFamily.linearIsometry_apply
protected theorem linearIsometry_apply (f : lp G 2) : hV.linearIsometry f = ∑' i, V i (f i) 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] {E : Type u_3} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [cplt : CompleteSpace E] {G : ι → Type u_4} [(i : ι) → NormedAddCommGroup (G i)] [(i : ι) → InnerProductSpace 𝕜 (G i)] {V : (i : ι) → G i →ₗᵢ[𝕜] E} (hV : OrthogonalFamily 𝕜 G V) (f : ↥(lp G 2))
Goal: hV.linearIsometry f = ∑' (i : ι), (V i) (↑f i)
-- OrthogonalFamily.hasSum_linearIsometry
protected theorem hasSum_linearIsometry (f : lp G 2) :
    HasSum (fun i => V i (f i)) (hV.linearIsometry f) 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] {E : Type u_3} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [cplt : CompleteSpace E] {G : ι → Type u_4} [(i : ι) → NormedAddCommGroup (G i)] [(i : ι) → InnerProductSpace 𝕜 (G i)] {V : (i : ι) → G i →ₗᵢ[𝕜] E} (hV : OrthogonalFamily 𝕜 G V) (f : ↥(lp G 2))
Goal: HasSum (fun i => (V i) (↑f i)) (hV.linearIsometry f)
-- OrthogonalFamily.linearIsometry_apply_single
@[simp]
protected theorem linearIsometry_apply_single {i : ι} (x : G i) :
    hV.linearIsometry (lp.single 2 i x) = V i x 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] {E : Type u_3} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [cplt : CompleteSpace E] {G : ι → Type u_4} [(i : ι) → NormedAddCommGroup (G i)] [(i : ι) → InnerProductSpace 𝕜 (G i)] {V : (i : ι) → G i →ₗᵢ[𝕜] E} (hV : OrthogonalFamily 𝕜 G V) {i : ι} (x : G i)
Goal: hV.linearIsometry (lp.single 2 i x) = (V i) x
-- OrthogonalFamily.linearIsometry_apply_dfinsupp_sum_single
protected theorem linearIsometry_apply_dfinsupp_sum_single (W₀ : Π₀ i : ι, G i) :
    hV.linearIsometry (W₀.sum (lp.single 2)) = W₀.sum fun i => V i 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] {E : Type u_3} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [cplt : CompleteSpace E] {G : ι → Type u_4} [(i : ι) → NormedAddCommGroup (G i)] [(i : ι) → InnerProductSpace 𝕜 (G i)] {V : (i : ι) → G i →ₗᵢ[𝕜] E} (hV : OrthogonalFamily 𝕜 G V) (W₀ : Π₀ (i : ι), G i)
Goal: hV.linearIsometry (W₀.sum (lp.single 2)) = W₀.sum fun i => ⇑(V i)
-- OrthogonalFamily.range_linearIsometry
protected theorem range_linearIsometry [∀ i, CompleteSpace (G i)] :
    LinearMap.range hV.linearIsometry.toLinearMap =
      (⨆ i, LinearMap.range (V i).toLinearMap).topologicalClosure 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] {E : Type u_3} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [cplt : CompleteSpace E] {G : ι → Type u_4} [(i : ι) → NormedAddCommGroup (G i)] [(i : ι) → InnerProductSpace 𝕜 (G i)] {V : (i : ι) → G i →ₗᵢ[𝕜] E} (hV : OrthogonalFamily 𝕜 G V) [∀ (i : ι), CompleteSpace (G i)]
Goal: LinearMap.range hV.linearIsometry.toLinearMap = (⨆ i, LinearMap.range (V i).toLinearMap).topologicalClosure
-- IsHilbertSum.OrthogonalFamily
structure IsHilbertSum : Prop where
  ofSurjective ::
    protected OrthogonalFamily : OrthogonalFamily 𝕜 G V
    protected surjective_isometry : Function.Surjective OrthogonalFamily.linearIsometry
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] {E : Type u_3} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [cplt : CompleteSpace E] {G : ι → Type u_4} [(i : ι) → NormedAddCommGroup (G i)] [(i : ι) → InnerProductSpace 𝕜 (G i)] {V : (i : ι) → G i →ₗᵢ[𝕜] E} (self : IsHilbertSum 𝕜 G V)
Goal: OrthogonalFamily 𝕜 G V
-- IsHilbertSum.surjective_isometry
structure IsHilbertSum : Prop where
  ofSurjective ::
    protected OrthogonalFamily : OrthogonalFamily 𝕜 G V
    protected surjective_isometry : Function.Surjective OrthogonalFamily.linearIsometry
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] {E : Type u_3} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [cplt : CompleteSpace E] {G : ι → Type u_4} [(i : ι) → NormedAddCommGroup (G i)] [(i : ι) → InnerProductSpace 𝕜 (G i)] {V : (i : ι) → G i →ₗᵢ[𝕜] E} (self : IsHilbertSum 𝕜 G V)
Goal: Function.Surjective ⇑⋯.linearIsometry
-- IsHilbertSum.mk
theorem IsHilbertSum.mk [∀ i, CompleteSpace <| G i] (hVortho : OrthogonalFamily 𝕜 G V)
    (hVtotal : ⊤ ≤ (⨆ i, LinearMap.range (V i).toLinearMap).topologicalClosure) :
    IsHilbertSum 𝕜 G V 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] {E : Type u_3} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [cplt : CompleteSpace E] {G : ι → Type u_4} [(i : ι) → NormedAddCommGroup (G i)] [(i : ι) → InnerProductSpace 𝕜 (G i)] {V : (i : ι) → G i →ₗᵢ[𝕜] E} [∀ (i : ι), CompleteSpace (G i)] (hVortho : OrthogonalFamily 𝕜 G V) (hVtotal : ⊤ ≤ (⨆ i, LinearMap.range (V i).toLinearMap).topologicalClosure)
Goal: IsHilbertSum 𝕜 G V
-- IsHilbertSum.mkInternal
theorem IsHilbertSum.mkInternal [∀ i, CompleteSpace <| F i]
    (hFortho : OrthogonalFamily 𝕜 (fun i => F i) fun i => (F i).subtypeₗᵢ)
    (hFtotal : ⊤ ≤ (⨆ i, F i).topologicalClosure) :
    IsHilbertSum 𝕜 (fun i => F i) fun i => (F i).subtypeₗᵢ 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] {E : Type u_3} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [cplt : CompleteSpace E] (F : ι → Submodule 𝕜 E) [∀ (i : ι), CompleteSpace ↥(F i)] (hFortho : OrthogonalFamily 𝕜 (fun i => ↥(F i)) fun i => (F i).subtypeₗᵢ) (hFtotal : ⊤ ≤ (⨆ i, F i).topologicalClosure)
Goal: IsHilbertSum 𝕜 (fun i => ↥(F i)) fun i => (F i).subtypeₗᵢ
-- IsHilbertSum.linearIsometryEquiv_symm_apply
protected theorem IsHilbertSum.linearIsometryEquiv_symm_apply (hV : IsHilbertSum 𝕜 G V)
    (w : lp G 2) : hV.linearIsometryEquiv.symm w = ∑' i, V i (w i) 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] {E : Type u_3} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [cplt : CompleteSpace E] {G : ι → Type u_4} [(i : ι) → NormedAddCommGroup (G i)] [(i : ι) → InnerProductSpace 𝕜 (G i)] {V : (i : ι) → G i →ₗᵢ[𝕜] E} (hV : IsHilbertSum 𝕜 G V) (w : ↥(lp G 2))
Goal: hV.linearIsometryEquiv.symm w = ∑' (i : ι), (V i) (↑w i)
-- IsHilbertSum.hasSum_linearIsometryEquiv_symm
protected theorem IsHilbertSum.hasSum_linearIsometryEquiv_symm (hV : IsHilbertSum 𝕜 G V)
    (w : lp G 2) : HasSum (fun i => V i (w i)) (hV.linearIsometryEquiv.symm w) 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] {E : Type u_3} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [cplt : CompleteSpace E] {G : ι → Type u_4} [(i : ι) → NormedAddCommGroup (G i)] [(i : ι) → InnerProductSpace 𝕜 (G i)] {V : (i : ι) → G i →ₗᵢ[𝕜] E} (hV : IsHilbertSum 𝕜 G V) (w : ↥(lp G 2))
Goal: HasSum (fun i => (V i) (↑w i)) (hV.linearIsometryEquiv.symm w)
-- IsHilbertSum.linearIsometryEquiv_symm_apply_single
@[simp]
protected theorem IsHilbertSum.linearIsometryEquiv_symm_apply_single (hV : IsHilbertSum 𝕜 G V)
    {i : ι} (x : G i) : hV.linearIsometryEquiv.symm (lp.single 2 i x) = V i x 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] {E : Type u_3} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [cplt : CompleteSpace E] {G : ι → Type u_4} [(i : ι) → NormedAddCommGroup (G i)] [(i : ι) → InnerProductSpace 𝕜 (G i)] {V : (i : ι) → G i →ₗᵢ[𝕜] E} (hV : IsHilbertSum 𝕜 G V) {i : ι} (x : G i)
Goal: hV.linearIsometryEquiv.symm (lp.single 2 i x) = (V i) x
-- IsHilbertSum.linearIsometryEquiv_symm_apply_dfinsupp_sum_single
protected theorem IsHilbertSum.linearIsometryEquiv_symm_apply_dfinsupp_sum_single
    (hV : IsHilbertSum 𝕜 G V) (W₀ : Π₀ i : ι, G i) :
    hV.linearIsometryEquiv.symm (W₀.sum (lp.single 2)) = W₀.sum fun i => V i 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] {E : Type u_3} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [cplt : CompleteSpace E] {G : ι → Type u_4} [(i : ι) → NormedAddCommGroup (G i)] [(i : ι) → InnerProductSpace 𝕜 (G i)] {V : (i : ι) → G i →ₗᵢ[𝕜] E} (hV : IsHilbertSum 𝕜 G V) (W₀ : Π₀ (i : ι), G i)
Goal: hV.linearIsometryEquiv.symm (W₀.sum (lp.single 2)) = W₀.sum fun i => ⇑(V i)
-- IsHilbertSum.linearIsometryEquiv_apply_dfinsupp_sum_single
@[simp]
protected theorem IsHilbertSum.linearIsometryEquiv_apply_dfinsupp_sum_single
    (hV : IsHilbertSum 𝕜 G V) (W₀ : Π₀ i : ι, G i) :
    ((W₀.sum (γ := lp G 2) fun a b ↦ hV.linearIsometryEquiv (V a b)) : ∀ i, G i) = W₀ 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] {E : Type u_3} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [cplt : CompleteSpace E] {G : ι → Type u_4} [(i : ι) → NormedAddCommGroup (G i)] [(i : ι) → InnerProductSpace 𝕜 (G i)] {V : (i : ι) → G i →ₗᵢ[𝕜] E} (hV : IsHilbertSum 𝕜 G V) (W₀ : Π₀ (i : ι), G i)
Goal: ↑(W₀.sum fun a b => hV.linearIsometryEquiv ((V a) b)) = ⇑W₀
-- Orthonormal.isHilbertSum
theorem Orthonormal.isHilbertSum {v : ι → E} (hv : Orthonormal 𝕜 v)
    (hsp : ⊤ ≤ (span 𝕜 (Set.range v)).topologicalClosure) :
    IsHilbertSum 𝕜 (fun _ : ι => 𝕜) fun i => LinearIsometry.toSpanSingleton 𝕜 E (hv.1 i) 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] {E : Type u_3} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [cplt : CompleteSpace E] {v : ι → E} (hv : Orthonormal 𝕜 v) (hsp : ⊤ ≤ (Submodule.span 𝕜 (Set.range v)).topologicalClosure)
Goal: IsHilbertSum 𝕜 (fun x => 𝕜) fun i => LinearIsometry.toSpanSingleton 𝕜 E ⋯
-- Submodule.isHilbertSumOrthogonal
theorem Submodule.isHilbertSumOrthogonal (K : Submodule 𝕜 E) [hK : CompleteSpace K] :
    IsHilbertSum 𝕜 (fun b => ↥(cond b K Kᗮ)) fun b => (cond b K Kᗮ).subtypeₗᵢ 
Arguments: {𝕜 : Type u_2} [RCLike 𝕜] {E : Type u_3} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [cplt : CompleteSpace E] (K : Submodule 𝕜 E) [hK : CompleteSpace ↥K]
Goal: IsHilbertSum 𝕜 (fun b => ↥(bif b then K else Kᗮ)) fun b => (bif b then K else Kᗮ).subtypeₗᵢ
-- HilbertBasis.repr_symm_single
protected theorem repr_symm_single (b : HilbertBasis ι 𝕜 E) (i : ι) :
    b.repr.symm (lp.single 2 i (1 : 𝕜)) = b i 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] {E : Type u_3} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (b : HilbertBasis ι 𝕜 E) (i : ι)
Goal: b.repr.symm (lp.single 2 i 1) = (fun i => b.repr.symm (lp.single 2 i 1)) i
-- HilbertBasis.repr_self
protected theorem repr_self (b : HilbertBasis ι 𝕜 E) (i : ι) :
    b.repr (b i) = lp.single 2 i (1 : 𝕜) 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] {E : Type u_3} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (b : HilbertBasis ι 𝕜 E) (i : ι)
Goal: b.repr ((fun i => b.repr.symm (lp.single 2 i 1)) i) = lp.single 2 i 1
-- HilbertBasis.repr_apply_apply
protected theorem repr_apply_apply (b : HilbertBasis ι 𝕜 E) (v : E) (i : ι) :
    b.repr v i = ⟪b i, v⟫ 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] {E : Type u_3} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (b : HilbertBasis ι 𝕜 E) (v : E) (i : ι)
Goal: ↑(b.repr v) i = ⟪(fun i => b.repr.symm (lp.single 2 i 1)) i, v⟫_𝕜
-- HilbertBasis.orthonormal
@[simp]
protected theorem orthonormal (b : HilbertBasis ι 𝕜 E) : Orthonormal 𝕜 b 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] {E : Type u_3} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (b : HilbertBasis ι 𝕜 E)
Goal: Orthonormal 𝕜 fun i => b.repr.symm (lp.single 2 i 1)
-- HilbertBasis.hasSum_repr_symm
protected theorem hasSum_repr_symm (b : HilbertBasis ι 𝕜 E) (f : ℓ²(ι, 𝕜)) :
    HasSum (fun i => f i • b i) (b.repr.symm f) 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] {E : Type u_3} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (b : HilbertBasis ι 𝕜 E) (f : ↥(lp (fun i => 𝕜) 2))
Goal: HasSum (fun i => ↑f i • (fun i => b.repr.symm (lp.single 2 i 1)) i) (b.repr.symm f)
-- HilbertBasis.hasSum_repr
protected theorem hasSum_repr (b : HilbertBasis ι 𝕜 E) (x : E) :
    HasSum (fun i => b.repr x i • b i) x 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] {E : Type u_3} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (b : HilbertBasis ι 𝕜 E) (x : E)
Goal: HasSum (fun i => ↑(b.repr x) i • (fun i => b.repr.symm (lp.single 2 i 1)) i) x
-- HilbertBasis.dense_span
@[simp]
protected theorem dense_span (b : HilbertBasis ι 𝕜 E) :
    (span 𝕜 (Set.range b)).topologicalClosure = ⊤ 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] {E : Type u_3} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (b : HilbertBasis ι 𝕜 E)
Goal: (Submodule.span 𝕜 (Set.range fun i => b.repr.symm (lp.single 2 i 1))).topologicalClosure = ⊤
-- HilbertBasis.hasSum_inner_mul_inner
protected theorem hasSum_inner_mul_inner (b : HilbertBasis ι 𝕜 E) (x y : E) :
    HasSum (fun i => ⟪x, b i⟫ * ⟪b i, y⟫) ⟪x, y⟫ 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] {E : Type u_3} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (b : HilbertBasis ι 𝕜 E) (x y : E)
Goal: HasSum (fun i => ⟪x, (fun i => b.repr.symm (lp.single 2 i 1)) i⟫_𝕜 * ⟪(fun i => b.repr.symm (lp.single 2 i 1)) i, y⟫_𝕜)
  ⟪x, y⟫_𝕜
-- HilbertBasis.summable_inner_mul_inner
protected theorem summable_inner_mul_inner (b : HilbertBasis ι 𝕜 E) (x y : E) :
    Summable fun i => ⟪x, b i⟫ * ⟪b i, y⟫ 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] {E : Type u_3} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (b : HilbertBasis ι 𝕜 E) (x y : E)
Goal: Summable fun i => ⟪x, (fun i => b.repr.symm (lp.single 2 i 1)) i⟫_𝕜 * ⟪(fun i => b.repr.symm (lp.single 2 i 1)) i, y⟫_𝕜
-- HilbertBasis.tsum_inner_mul_inner
protected theorem tsum_inner_mul_inner (b : HilbertBasis ι 𝕜 E) (x y : E) :
    ∑' i, ⟪x, b i⟫ * ⟪b i, y⟫ = ⟪x, y⟫ 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] {E : Type u_3} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (b : HilbertBasis ι 𝕜 E) (x y : E)
Goal: ∑' (i : ι), ⟪x, (fun i => b.repr.symm (lp.single 2 i 1)) i⟫_𝕜 * ⟪(fun i => b.repr.symm (lp.single 2 i 1)) i, y⟫_𝕜 =
  ⟪x, y⟫_𝕜
-- HilbertBasis.coe_toOrthonormalBasis
@[simp]
theorem coe_toOrthonormalBasis [Fintype ι] (b : HilbertBasis ι 𝕜 E) :
    (b.toOrthonormalBasis : ι → E) = b 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] {E : Type u_3} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [Fintype ι] (b : HilbertBasis ι 𝕜 E)
Goal: ⇑b.toOrthonormalBasis = fun i => b.repr.symm (lp.single 2 i 1)
-- HilbertBasis.hasSum_orthogonalProjection
protected theorem hasSum_orthogonalProjection {U : Submodule 𝕜 E} [CompleteSpace U]
    (b : HilbertBasis ι 𝕜 U) (x : E) :
    HasSum (fun i => ⟪(b i : E), x⟫ • b i) (orthogonalProjection U x) 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] {E : Type u_3} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {U : Submodule 𝕜 E} [CompleteSpace ↥U] (b : HilbertBasis ι 𝕜 ↥U) (x : E)
Goal: HasSum (fun i => ⟪↑((fun i => b.repr.symm (lp.single 2 i 1)) i), x⟫_𝕜 • (fun i => b.repr.symm (lp.single 2 i 1)) i)
  ((orthogonalProjection U) x)
-- HilbertBasis.finite_spans_dense
theorem finite_spans_dense (b : HilbertBasis ι 𝕜 E) :
    (⨆ J : Finset ι, span 𝕜 (J.image b : Set E)).topologicalClosure = ⊤ 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] {E : Type u_3} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (b : HilbertBasis ι 𝕜 E)
Goal: (⨆ J, Submodule.span 𝕜 ↑(Finset.image (fun i => b.repr.symm (lp.single 2 i 1)) J)).topologicalClosure = ⊤
-- Orthonormal.linearIsometryEquiv_symm_apply_single_one
theorem _root_.Orthonormal.linearIsometryEquiv_symm_apply_single_one (h i) :
    (hv.isHilbertSum h).linearIsometryEquiv.symm (lp.single 2 i 1) = v i 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] {E : Type u_3} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [cplt : CompleteSpace E] {v : ι → E} (hv : Orthonormal 𝕜 v) (h : ⊤ ≤ (Submodule.span 𝕜 (Set.range v)).topologicalClosure) (i : ι)
Goal: ⋯.linearIsometryEquiv.symm (lp.single 2 i 1) = v i
-- HilbertBasis.coe_mk
@[simp]
protected theorem coe_mk (hsp : ⊤ ≤ (span 𝕜 (Set.range v)).topologicalClosure) :
    ⇑(HilbertBasis.mk hv hsp) = v 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] {E : Type u_3} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [cplt : CompleteSpace E] {v : ι → E} (hv : Orthonormal 𝕜 v) (hsp : ⊤ ≤ (Submodule.span 𝕜 (Set.range v)).topologicalClosure)
Goal: (fun i => (HilbertBasis.mk hv hsp).repr.symm (lp.single 2 i 1)) = v
-- HilbertBasis.coe_mkOfOrthogonalEqBot
@[simp]
protected theorem coe_mkOfOrthogonalEqBot (hsp : (span 𝕜 (Set.range v))ᗮ = ⊥) :
    ⇑(HilbertBasis.mkOfOrthogonalEqBot hv hsp) = v 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] {E : Type u_3} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [cplt : CompleteSpace E] {v : ι → E} (hv : Orthonormal 𝕜 v) (hsp : (Submodule.span 𝕜 (Set.range v))ᗮ = ⊥)
Goal: (fun i => (HilbertBasis.mkOfOrthogonalEqBot hv hsp).repr.symm (lp.single 2 i 1)) = v
-- OrthonormalBasis.coe_toHilbertBasis
@[simp]
theorem _root_.OrthonormalBasis.coe_toHilbertBasis [Fintype ι] (b : OrthonormalBasis ι 𝕜 E) :
    (b.toHilbertBasis : ι → E) = b 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] {E : Type u_3} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [cplt : CompleteSpace E] [Fintype ι] (b : OrthonormalBasis ι 𝕜 E)
Goal: (fun i => b.toHilbertBasis.repr.symm (lp.single 2 i 1)) = ⇑b
-- Orthonormal.exists_hilbertBasis_extension
theorem _root_.Orthonormal.exists_hilbertBasis_extension {s : Set E}
    (hs : Orthonormal 𝕜 ((↑) : s → E)) :
    ∃ (w : Set E) (b : HilbertBasis w 𝕜 E), s ⊆ w ∧ ⇑b = ((↑) : w → E) 
Arguments: {𝕜 : Type u_2} [RCLike 𝕜] {E : Type u_3} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [cplt : CompleteSpace E] {s : Set E} (hs : Orthonormal 𝕜 Subtype.val)
Goal: ∃ w b, s ⊆ w ∧ (fun i => b.repr.symm (lp.single 2 i 1)) = Subtype.val
-- exists_hilbertBasis
theorem _root_.exists_hilbertBasis : ∃ (w : Set E) (b : HilbertBasis w 𝕜 E), ⇑b = ((↑) : w → E) 
Arguments: (𝕜 : Type u_2) [RCLike 𝕜] (E : Type u_3) [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [cplt : CompleteSpace E]
Goal: ∃ w b, (fun i => b.repr.symm (lp.single 2 i 1)) = Subtype.val
-- BoxIntegral.Prepartition.pairwiseDisjoint
structure Prepartition (I : Box ι) where
    boxes : Finset (Box ι)
    le_of_mem' : ∀ J ∈ boxes, J ≤ I
    pairwiseDisjoint : Set.Pairwise (↑boxes) (Disjoint on ((↑) : Box ι → Set (ι → ℝ)))
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} (self : BoxIntegral.Prepartition I)
Goal: (↑self.boxes).Pairwise (Disjoint on BoxIntegral.Box.toSet)
-- BoxIntegral.Prepartition.le_of_mem'
structure Prepartition (I : Box ι) where
    boxes : Finset (Box ι)
    le_of_mem' : ∀ J ∈ boxes, J ≤ I
    pairwiseDisjoint : Set.Pairwise (↑boxes) (Disjoint on ((↑) : Box ι → Set (ι → ℝ)))
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} (self : BoxIntegral.Prepartition I) (J : BoxIntegral.Box ι)
Goal: J ∈ self.boxes → J ≤ I
-- BoxIntegral.Prepartition.mem_boxes
@[simp]
theorem mem_boxes : J ∈ π.boxes ↔ J ∈ π 
Arguments: {ι : Type u_1} {I J : BoxIntegral.Box ι} (π : BoxIntegral.Prepartition I)
Goal: J ∈ π.boxes ↔ J ∈ π
-- BoxIntegral.Prepartition.mem_mk
@[simp]
theorem mem_mk {s h₁ h₂} : J ∈ (mk s h₁ h₂ : Prepartition I) ↔ J ∈ s 
Arguments: {ι : Type u_1} {I J : BoxIntegral.Box ι} {s : Finset (BoxIntegral.Box ι)} {h₁ : ∀ J ∈ s, J ≤ I} {h₂ : (↑s).Pairwise (Disjoint on BoxIntegral.Box.toSet)}
Goal: J ∈ { boxes := s, le_of_mem' := h₁, pairwiseDisjoint := h₂ } ↔ J ∈ s
-- BoxIntegral.Prepartition.disjoint_coe_of_mem
theorem disjoint_coe_of_mem (h₁ : J₁ ∈ π) (h₂ : J₂ ∈ π) (h : J₁ ≠ J₂) :
    Disjoint (J₁ : Set (ι → ℝ)) J₂ 
Arguments: {ι : Type u_1} {I J₁ J₂ : BoxIntegral.Box ι} (π : BoxIntegral.Prepartition I) (h₁ : J₁ ∈ π) (h₂ : J₂ ∈ π) (h : J₁ ≠ J₂)
Goal: Disjoint ↑J₁ ↑J₂
-- BoxIntegral.Prepartition.eq_of_mem_of_mem
theorem eq_of_mem_of_mem (h₁ : J₁ ∈ π) (h₂ : J₂ ∈ π) (hx₁ : x ∈ J₁) (hx₂ : x ∈ J₂) : J₁ = J₂ 
Arguments: {ι : Type u_1} {I J₁ J₂ : BoxIntegral.Box ι} (π : BoxIntegral.Prepartition I) {x : ι → ℝ} (h₁ : J₁ ∈ π) (h₂ : J₂ ∈ π) (hx₁ : x ∈ J₁) (hx₂ : x ∈ J₂)
Goal: J₁ = J₂
-- BoxIntegral.Prepartition.eq_of_le_of_le
theorem eq_of_le_of_le (h₁ : J₁ ∈ π) (h₂ : J₂ ∈ π) (hle₁ : J ≤ J₁) (hle₂ : J ≤ J₂) : J₁ = J₂ 
Arguments: {ι : Type u_1} {I J J₁ J₂ : BoxIntegral.Box ι} (π : BoxIntegral.Prepartition I) (h₁ : J₁ ∈ π) (h₂ : J₂ ∈ π) (hle₁ : J ≤ J₁) (hle₂ : J ≤ J₂)
Goal: J₁ = J₂
-- BoxIntegral.Prepartition.eq_of_le
theorem eq_of_le (h₁ : J₁ ∈ π) (h₂ : J₂ ∈ π) (hle : J₁ ≤ J₂) : J₁ = J₂ 
Arguments: {ι : Type u_1} {I J₁ J₂ : BoxIntegral.Box ι} (π : BoxIntegral.Prepartition I) (h₁ : J₁ ∈ π) (h₂ : J₂ ∈ π) (hle : J₁ ≤ J₂)
Goal: J₁ = J₂
-- BoxIntegral.Prepartition.le_of_mem
theorem le_of_mem (hJ : J ∈ π) : J ≤ I 
Arguments: {ι : Type u_1} {I J : BoxIntegral.Box ι} (π : BoxIntegral.Prepartition I) (hJ : J ∈ π)
Goal: J ≤ I
-- BoxIntegral.Prepartition.lower_le_lower
theorem lower_le_lower (hJ : J ∈ π) : I.lower ≤ J.lower 
Arguments: {ι : Type u_1} {I J : BoxIntegral.Box ι} (π : BoxIntegral.Prepartition I) (hJ : J ∈ π)
Goal: I.lower ≤ J.lower
-- BoxIntegral.Prepartition.upper_le_upper
theorem upper_le_upper (hJ : J ∈ π) : J.upper ≤ I.upper 
Arguments: {ι : Type u_1} {I J : BoxIntegral.Box ι} (π : BoxIntegral.Prepartition I) (hJ : J ∈ π)
Goal: J.upper ≤ I.upper
-- BoxIntegral.Prepartition.injective_boxes
theorem injective_boxes : Function.Injective (boxes : Prepartition I → Finset (Box ι)) 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι}
Goal: Function.Injective BoxIntegral.Prepartition.boxes
-- BoxIntegral.Prepartition.ext
@[ext]
theorem ext (h : ∀ J, J ∈ π₁ ↔ J ∈ π₂) : π₁ = π₂ 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} {π₁ π₂ : BoxIntegral.Prepartition I} (h : ∀ (J : BoxIntegral.Box ι), J ∈ π₁ ↔ J ∈ π₂)
Goal: π₁ = π₂
-- BoxIntegral.Prepartition.single_boxes
@[simps]
def single (I J : Box ι) (h : J ≤ I) : Prepartition I 
Arguments: {ι : Type u_1} (I J : BoxIntegral.Box ι) (h : J ≤ I)
Goal: (BoxIntegral.Prepartition.single I J h).boxes = {J}
-- BoxIntegral.Prepartition.mem_single
@[simp]
theorem mem_single {J'} (h : J ≤ I) : J' ∈ single I J h ↔ J' = J 
Arguments: {ι : Type u_1} {I J J' : BoxIntegral.Box ι} (h : J ≤ I)
Goal: J' ∈ BoxIntegral.Prepartition.single I J h ↔ J' = J
-- BoxIntegral.Prepartition.le_def
theorem le_def : π₁ ≤ π₂ ↔ ∀ J ∈ π₁, ∃ J' ∈ π₂, J ≤ J' 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} {π₁ π₂ : BoxIntegral.Prepartition I}
Goal: π₁ ≤ π₂ ↔ ∀ J ∈ π₁, ∃ J' ∈ π₂, J ≤ J'
-- BoxIntegral.Prepartition.mem_top
@[simp]
theorem mem_top : J ∈ (⊤ : Prepartition I) ↔ J = I 
Arguments: {ι : Type u_1} {I J : BoxIntegral.Box ι}
Goal: J ∈ ⊤ ↔ J = I
-- BoxIntegral.Prepartition.top_boxes
@[simp]
theorem top_boxes : (⊤ : Prepartition I).boxes = {I} 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι}
Goal: ⊤.boxes = {I}
-- BoxIntegral.Prepartition.not_mem_bot
@[simp]
theorem not_mem_bot : J ∉ (⊥ : Prepartition I) 
Arguments: {ι : Type u_1} {I J : BoxIntegral.Box ι}
Goal: J ∉ ⊥
-- BoxIntegral.Prepartition.bot_boxes
@[simp]
theorem bot_boxes : (⊥ : Prepartition I).boxes = ∅ 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι}
Goal: ⊥.boxes = ∅