59 lines
1.7 KiB
Lean4
59 lines
1.7 KiB
Lean4
import Mathlib.Analysis.InnerProductSpace.PiL2
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open TensorProduct
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lemma OrthonormalBasis.sum_repr'
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{𝕜 : Type*} [RCLike 𝕜]
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{E : Type*} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
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[Fintype ι]
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(b : OrthonormalBasis ι 𝕜 E)
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(v : E) :
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v = ∑ i, ⟪b i, v⟫_𝕜 • (b i) := by
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nth_rw 1 [← (b.sum_repr v)]
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simp_rw [b.repr_apply_apply v]
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noncomputable def InnerProductSpace.canonicalTensor
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(E : Type u_2) [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
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: E ⊗[ℝ] E := by
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let v := stdOrthonormalBasis ℝ E
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exact ∑ i, (v i) ⊗ₜ[ℝ] (v i)
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theorem InnerProductSpace.InvariantTensor
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(E : Type*) [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
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(v : OrthonormalBasis (Fin (FiniteDimensional.finrank ℝ E)) ℝ E)
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: InnerProductSpace.canonicalTensor E = ∑ i, (v i) ⊗ₜ[ℝ] (v i) := by
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unfold InnerProductSpace.canonicalTensor
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let v₁ := stdOrthonormalBasis ℝ E
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conv =>
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right
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arg 2
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intro i
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rw [v₁.sum_repr' (v i)]
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simp_rw [TensorProduct.sum_tmul, TensorProduct.tmul_sum, TensorProduct.smul_tmul_smul]
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conv =>
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right
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rw [Finset.sum_comm]
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arg 2
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intro y
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rw [Finset.sum_comm]
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arg 2
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intro x
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rw [← Finset.sum_smul]
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arg 1
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arg 2
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intro i
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rw [← real_inner_comm (v₁ x)]
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simp_rw [OrthonormalBasis.sum_inner_mul_inner v]
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have {x : Fin (FiniteDimensional.finrank ℝ E)} : ∑ x_1 : Fin (FiniteDimensional.finrank ℝ E), ⟪v₁ x, v₁ x_1⟫_ℝ • v₁ x ⊗ₜ[ℝ] v₁ x_1 = v₁ x ⊗ₜ[ℝ] v₁ x := by
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rw [Fintype.sum_eq_single x, orthonormal_iff_ite.1 v₁.orthonormal]; simp
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intro r₁ hr₁
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rw [orthonormal_iff_ite.1 v₁.orthonormal]; simp; tauto
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simp_rw [this]
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