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feat(Topology/Algebra/Module/Spaces/CLM): linear map version of smulRightL #43113

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feat(Topology/Algebra/Module/Spaces/CLM): linear map version of smulRightL #43113

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fa3a46a
smulRightL as a linear map
themathqueen Aug 25, 2026
c6f0c9d
fix
themathqueen Aug 25, 2026
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2 changes: 2 additions & 0 deletions Mathlib/Analysis/Normed/Operator/Bilinear.lean
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Expand Up @@ -437,6 +437,8 @@ def smulRightL : StrongDual 𝕜 E →L[𝕜] Fₗ →L[𝕜] E →L[𝕜] Fₗ
simp only [coe_smulRightₗ, one_mul, norm_smulRight_apply, LinearMap.coe_mk, AddHom.coe_mk,
le_refl]

@[simp] lemma toLinearMap_smulRightL : (smulRightL 𝕜 E Fₗ).toLinearMap = smulRightₗ' 𝕜 E Fₗ := rfl

end ContinuousLinearMap

end SemiNormed
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19 changes: 18 additions & 1 deletion Mathlib/Topology/Algebra/Module/Spaces/ContinuousLinearMap.lean
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Expand Up @@ -470,8 +470,9 @@ def prodL : ((E →L[𝕜] F) × (E →L[𝕜] G)) ≃L[S] (E →L[𝕜] F × G)
end Prod

variable {𝕜 E : Type*} [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E]
[TopologicalSpace E] [IsTopologicalAddGroup E] [ContinuousSMul 𝕜 E]
[TopologicalSpace E]

variable [IsTopologicalAddGroup E] [ContinuousSMul 𝕜 E] in
/-- `ContinuousLinearMap.toSpanSingleton` as a continuous linear equivalence. -/
@[simps!]
def toSpanSingletonCLE : E ≃L[𝕜] (𝕜 →L[𝕜] E) where
Expand All @@ -480,6 +481,22 @@ def toSpanSingletonCLE : E ≃L[𝕜] (𝕜 →L[𝕜] E) where
continuous_snd.smul continuous_fst
continuous_invFun := continuous_eval_const 1

variable {F : Type*} [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F]
[IsTopologicalAddGroup F] [ContinuousSMul 𝕜 F]

variable (𝕜 E F) in
/-- `smulRight` as a partly continuous trilinear map.
This is the bundled continuous version of `smulRightₗ`. -/
@[simps! apply_apply_apply] def smulRightₗ' : StrongDual 𝕜 E →ₗ[𝕜] (F →L[𝕜] E →L[𝕜] F) where
toFun c := (c.precomp F) ∘SL toSpanSingletonCLE.toContinuousLinearMap
map_add' _ _ := by ext; simp
map_smul' _ _ := by ext; simp

@[simp] lemma smulRightₗ'_apply_apply (c : StrongDual 𝕜 E) (x) :
c.smulRightₗ' 𝕜 E F x = c.smulRight x := rfl
@[simp] lemma toLinearMap_smulRightₗ'_apply (c : StrongDual 𝕜 E) :
(c.smulRightₗ' 𝕜 E F).toLinearMap = c.smulRightₗ := rfl

end ContinuousLinearMap

open ContinuousLinearMap
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