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feat(MeasureTheory): coercion of a finite sum of scalar multiples in Lp #40827

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feat(MeasureTheory): coercion of a finite sum of scalar multiples in Lp #40827

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feat(MeasureTheory): coercion of a finite sum of scalar multiples in Lp
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12 changes: 12 additions & 0 deletions Mathlib/MeasureTheory/Function/LpSpace/Basic.lean
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Expand Up @@ -431,6 +431,18 @@ instance instModule : Module 𝕜 (Lp E p μ) :=
theorem coeFn_smul (c : 𝕜) (f : Lp E p μ) : ⇑(c • f) =ᵐ[μ] c • ⇑f :=
AEEqFun.coeFn_smul _ _

/-- The coercion of a finite sum of scalar multiples of `MemLp.toLp` functions is almost
everywhere equal to the corresponding sum of those functions. -/
theorem coeFn_sum_smul {ι : Type*} (s : Finset ι) (c : ι → 𝕜) {h : ι → α → E}
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(hmem : ∀ i, MemLp (h i) p μ) :
⇑(∑ i ∈ s, c i • (hmem i).toLp (h i)) =ᵐ[μ] ∑ i ∈ s, c i • h i := by
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induction s using Finset.cons_induction with
| empty => simpa using coeFn_zero E p μ
| cons j t hjt ih =>
rw [Finset.sum_cons, Finset.sum_cons]
exact (coeFn_add _ _).trans
(((coeFn_smul _ _).trans ((hmem j).coeFn_toLp.const_smul (c j))).add ih)

instance instIsCentralScalar [Module 𝕜ᵐᵒᵖ E] [IsBoundedSMul 𝕜ᵐᵒᵖ E] [IsCentralScalar 𝕜 E] :
IsCentralScalar 𝕜 (Lp E p μ) where
op_smul_eq_smul k f := Subtype.ext <| op_smul_eq_smul k (f : α →ₘ[μ] E)
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