Theorems · Theorem · functional analysis

LinearMap.isClosedEmbedding_of_injective

∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [inst : NontriviallyNormedField 𝕜] [CompleteSpace 𝕜]
  [inst_2 : AddCommGroup E] [inst_3 : TopologicalSpace E] [IsTopologicalAddGroup E] [inst_5 : Module 𝕜 E]
  [ContinuousSMul 𝕜 E] [inst_7 : AddCommGroup F] [inst_8 : TopologicalSpace F] [IsTopologicalAddGroup F]
  [inst_10 : Module 𝕜 F] [ContinuousSMul 𝕜 F] [T2Space E] [FiniteDimensional 𝕜 E] [T2Space F] {f : E →ₗ[𝕜] F},
  f.ker = ⊥ → Topology.IsClosedEmbedding ⇑f

An injective linear map with finite-dimensional domain is a closed embedding.

Defined in

Mathlib.Topology.Algebra.Module.FiniteDimension

Cited by

3 results in Mathlib

Foundations

Depth 174 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldCompleteSpaceAddCommGroupTopologicalSpaceIsTopologicalAddGroupModuleContinuousSMulAddCommGroupTopologicalSpaceIsTopologicalAddGroupModuleContinuousSMulT2SpaceFiniteDimensionalT2Space