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Mathlib.Topology.Instances.RealVectorSpace

Continuous additive maps are ℝ-linear #

In this file we prove that a continuous map f : E →+ F between two topological vector spaces over ℝ is ℝ-linear

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theorem map_real_smul {E : Type u_1} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [ContinuousSMul ℝ E] {F : Type u_2} [AddCommGroup F] [Module ℝ F] [TopologicalSpace F] [ContinuousSMul ℝ F] [T2Space F] {G : Type u_3} [FunLike G E F] [AddMonoidHomClass G E F] (f : G) (hf : Continuous ⇑f) (c : ℝ) (x : E) :
f (c • x) = c • f x

A continuous additive map between two vector spaces over ℝ is ℝ-linear.

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def AddMonoidHom.toRealLinearMap {E : Type u_1} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [ContinuousSMul ℝ E] {F : Type u_2} [AddCommGroup F] [Module ℝ F] [TopologicalSpace F] [ContinuousSMul ℝ F] [T2Space F] (f : E →+ F) (hf : Continuous ⇑f) :
E →L[ℝ] F

Reinterpret a continuous additive homomorphism between two real vector spaces as a continuous real-linear map.

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    @[simp]
    theorem AddMonoidHom.coe_toRealLinearMap {E : Type u_1} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [ContinuousSMul ℝ E] {F : Type u_2} [AddCommGroup F] [Module ℝ F] [TopologicalSpace F] [ContinuousSMul ℝ F] [T2Space F] (f : E →+ F) (hf : Continuous ⇑f) :
    ⇑(AddMonoidHom.toRealLinearMap f hf) = ⇑f
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    def AddEquiv.toRealLinearEquiv {E : Type u_1} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [ContinuousSMul ℝ E] {F : Type u_2} [AddCommGroup F] [Module ℝ F] [TopologicalSpace F] [ContinuousSMul ℝ F] [T2Space F] (e : E ≃+ F) (h₁ : Continuous ⇑e) (h₂ : Continuous ⇑(AddEquiv.symm e)) :
    E ≃L[ℝ] F

    Reinterpret a continuous additive equivalence between two real vector spaces as a continuous real-linear map.

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      instance Real.isScalarTower {E : Type u_1} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [ContinuousSMul ℝ E] [T2Space E] {A : Type u_3} [TopologicalSpace A] [Ring A] [Algebra ℝ A] [Module A E] [ContinuousSMul ℝ A] [ContinuousSMul A E] :
      IsScalarTower ℝ A E

      A topological group carries at most one structure of a topological ℝ-module, so for any topological ℝ-algebra A (e.g. A = ℂ) and any topological group that is both a topological ℝ-module and a topological A-module, these structures agree.

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