| import Mathlib |
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| namespace OAI |
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| noncomputable section |
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| namespace KKConcreteCorner |
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| open scoped InnerProductSpace |
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| noncomputable section |
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| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] |
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| variable (S : Submodule ℂ H) [CompleteSpace S] |
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| def compress (x : H →L[ℂ] H) : S →L[ℂ] S := |
| S.orthogonalProjectionOnto ∘L x ∘L S.subtypeL |
| |
| def extend (x : S →L[ℂ] S) : H →L[ℂ] H := |
| S.subtypeL ∘L x ∘L S.orthogonalProjectionOnto |
| |
| @[simp] theorem compress_extend (x : S →L[ℂ] S) : compress S (extend S x)=x := by |
| ext v |
| simp [compress,extend] |
| |
| @[simp] theorem extend_compress (x : H →L[ℂ] H) : |
| extend S (compress S x)=S.starProjection*x*S.starProjection := by |
| rfl |
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| @[simp] theorem extend_zero : extend S 0=0 := by simp [extend] |
| |
| @[simp] theorem extend_one : extend S 1=S.starProjection := by ext v |
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| @[simp] theorem extend_add (x y : S →L[ℂ] S) : extend S (x+y)=extend S x+extend S y := by |
| simp [extend,ContinuousLinearMap.add_comp,ContinuousLinearMap.comp_add] |
| |
| @[simp] theorem extend_smul (c : ℂ) (x : S →L[ℂ] S) : extend S (c • x)=c • extend S x := by |
| simp [extend] |
| |
| @[simp] theorem extend_mul (x y : S →L[ℂ] S) : extend S (x*y)=extend S x*extend S y := by |
| ext v |
| simp [extend] |
| |
| @[simp] theorem extend_star [CompleteSpace H] (x : S →L[ℂ] S) : extend S (star x)=star (extend S x) := by |
| simp only [extend,ContinuousLinearMap.star_eq_adjoint,ContinuousLinearMap.adjoint_comp, |
| Submodule.adjoint_subtypeL,Submodule.adjoint_orthogonalProjectionOnto,ContinuousLinearMap.comp_assoc] |
| |
| theorem extend_injective : Function.Injective (extend S) := by |
| intro x y h |
| simpa only [compress_extend] using congrArg (compress S) h |
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| theorem extend_supported (x : S →L[ℂ] S) : |
| S.starProjection*extend S x=extend S x ∧ extend S x*S.starProjection=extend S x := by |
| constructor |
| · rw [← extend_one,← extend_mul,one_mul] |
| · rw [← extend_one,← extend_mul,mul_one] |
| |
| @[simp] theorem compress_star [CompleteSpace H] (x : H →L[ℂ] H) : compress S (star x)=star (compress S x) := by |
| simp only [compress,ContinuousLinearMap.star_eq_adjoint,ContinuousLinearMap.adjoint_comp, |
| Submodule.adjoint_subtypeL,Submodule.adjoint_orthogonalProjectionOnto,ContinuousLinearMap.comp_assoc] |
| |
| theorem projection_isStar [CompleteSpace H] : IsStarProjection S.starProjection := |
| isStarProjection_starProjection |
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| theorem extend_norm_le (x : S →L[ℂ] S) : ‖extend S x‖ ≤ ‖x‖ := by |
| apply (extend S x).opNorm_le_bound (norm_nonneg x) |
| intro v |
| change ‖(x (S.orthogonalProjectionOnto v) : H)‖ ≤ ‖x‖*‖v‖ |
| change ‖x (S.orthogonalProjectionOnto v)‖ ≤ _ |
| exact (x.le_opNorm _).trans (mul_le_mul_of_nonneg_left (S.norm_orthogonalProjectionOnto_apply_le v) (norm_nonneg _)) |
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| end |
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| end KKConcreteCorner |
| end |
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| end OAI |
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