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

theorem map_real_smul {E : Type u_1} [] [] [] [] {F : Type u_2} [] [] [] [] [] {G : Type u_3} [FunLike G E F] [] (f : G) (hf : ) (c : ) (x : E) :
f (c x) = c f x

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

def AddMonoidHom.toRealLinearMap {E : Type u_1} [] [] [] [] {F : Type u_2} [] [] [] [] [] (f : E →+ F) (hf : ) :

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

Equations
• f.toRealLinearMap hf = { toFun := f, map_add' := , map_smul' := , cont := hf }
Instances For
@[simp]
theorem AddMonoidHom.coe_toRealLinearMap {E : Type u_1} [] [] [] [] {F : Type u_2} [] [] [] [] [] (f : E →+ F) (hf : ) :
(f.toRealLinearMap hf) = f
def AddEquiv.toRealLinearEquiv {E : Type u_1} [] [] [] [] {F : Type u_2} [] [] [] [] [] (e : E ≃+ F) (h₁ : ) (h₂ : Continuous e.symm) :

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

Equations
• e.toRealLinearEquiv h₁ h₂ = { toFun := e.toFun, map_add' := , map_smul' := , invFun := e.invFun, left_inv := , right_inv := , continuous_toFun := , continuous_invFun := }
Instances For
@[instance 900]
instance Real.isScalarTower {E : Type u_1} [] [] [] [] [] {A : Type u_3} [] [Ring A] [] [Module 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.

Equations
• =