Theorems · Theorem · functional analysis

Submodule.isClosed_mono_of_finiteDimensional_quotient

∀ {𝕜 : Type u_1} {E : Type u_2} [inst : NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] [inst_2 : AddCommGroup E]
  [inst_3 : TopologicalSpace E] [IsTopologicalAddGroup E] [inst_5 : Module 𝕜 E] [ContinuousSMul 𝕜 E]
  {s t : Submodule 𝕜 E} [FiniteDimensional 𝕜 (E ⧸ s)], IsClosed ↑s → s ≤ t → IsClosed ↑t

If s is a closed subspace with finite codimension, any subspace containing s is closed.

Defined in

Mathlib.Topology.Algebra.Module.FiniteDimension

Cited by

1 results in Mathlib

Foundations

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

Assumes

NontriviallyNormedFieldCompleteSpaceAddCommGroupTopologicalSpaceIsTopologicalAddGroupModuleContinuousSMulFiniteDimensional