File size: 2,426 Bytes
9eeb9c7 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 | 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
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