| import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp |
| import Mathlib.Analysis.InnerProductSpace.l2Space |
|
|
| namespace OAI |
|
|
| namespace Invariant233.Fields |
|
|
| open MeasureTheory |
| open scoped ENNReal |
|
|
| variable {X E F : Type*} [MeasurableSpace X] [NormedAddCommGroup E] |
| [NormedSpace ℂ E] [NormedAddCommGroup F] [NormedSpace ℂ F] |
| {μ : Measure X} {p : ℝ≥0∞} [Fact (1 ≤ p)] |
|
|
| theorem lp_operator_ext_indicator (hp : p ≠ ∞) |
| (A B : Lp E p μ →L[ℂ] F) |
| (h : ∀ (v : E) (s : Set X) (hs : MeasurableSet s) (hμs : μ s ≠ ∞), |
| A (indicatorConstLp p hs hμs v) = B (indicatorConstLp p hs hμs v)) : |
| A = B := by |
| ext f |
| apply Lp.induction hp (fun g => A g = B g) _ _ _ f |
| · intro v s hs hμs |
| exact h v s hs hμs.ne |
| · intro g k hg hk hdis hAg hAk |
| simp only [map_add, hAg, hAk] |
| · exact isClosed_eq A.continuous B.continuous |
|
|
| noncomputable def indicatorConstLpCLM (s : Set X) (hs : MeasurableSet s) |
| (hμs : μ s ≠ ∞) : E →L[ℂ] Lp E p μ := |
| LinearMap.mkContinuous |
| { toFun := indicatorConstLp p hs hμs |
| map_add' := fun v w => (indicatorConstLp_add (c := v) (c' := w)).symm |
| map_smul' := by |
| intro c v |
| simp only [indicatorConstLp, ← MemLp.toLp_const_smul] |
| apply MemLp.toLp_congr |
| exact Filter.Eventually.of_forall fun x => by |
| simp only [RingHom.id_apply, Pi.smul_apply, Set.indicator_smul_apply] } |
| (μ.real s ^ (1 / p.toReal)) |
| (fun v => by |
| change ‖indicatorConstLp p hs hμs v‖ ≤ μ.real s ^ (1 / p.toReal) * ‖v‖ |
| simpa only [mul_comm] using |
| (norm_indicatorConstLp_le (p := p) (hs := hs) (hμs := hμs) (c := v))) |
|
|
| theorem lp_operator_ext_basis_indicator {H ι : Type*} |
| [NormedAddCommGroup H] [InnerProductSpace ℂ H] |
| (e : HilbertBasis ι ℂ H) (hp : p ≠ ∞) |
| (A B : Lp H p μ →L[ℂ] F) |
| (h : ∀ (i : ι) (s : Set X) (hs : MeasurableSet s) (hμs : μ s ≠ ∞), |
| A (indicatorConstLp p hs hμs (e i)) = |
| B (indicatorConstLp p hs hμs (e i))) : A = B := by |
| apply lp_operator_ext_indicator hp A B |
| intro v s hs hμs |
| let C : H →L[ℂ] Lp H p μ := indicatorConstLpCLM s hs hμs |
| have hAC : A.comp C = B.comp C := by |
| apply ContinuousLinearMap.ext_on |
| (Submodule.dense_iff_topologicalClosure_eq_top.mpr e.dense_span) |
| rintro _ ⟨i, rfl⟩ |
| exact h i s hs hμs |
| exact DFunLike.congr_fun hAC v |
|
|
| end Invariant233.Fields |
|
|
| end OAI |
|
|
