Theorems · Theorem · functional analysis

map_real_smul

∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] [inst_2 : TopologicalSpace E] [ContinuousSMul ℝ E]
  {F : Type u_2} [inst_4 : AddCommGroup F] [inst_5 : Module ℝ F] [inst_6 : TopologicalSpace F] [ContinuousSMul ℝ F]
  [T2Space F] {G : Type u_3} [inst_9 : FunLike G E F] [AddMonoidHomClass G E F] (f : G),
  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.

Defined in

Mathlib.Topology.Instances.RealVectorSpace

Cited by

2 results in Mathlib

Foundations

Depth 158 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

AddCommGroupModuleTopologicalSpaceContinuousSMulAddCommGroupModuleTopologicalSpaceContinuousSMulT2SpaceFunLikeAddMonoidHomClass