Theorems · Theorem · functional analysis

Submodule.isClosed_sup_finiteDimensional

∀ {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), IsClosed ↑s → ∀ [ht : FiniteDimensional 𝕜 ↥t], IsClosed ↑(s ⊔ t)

The supremum of a closed subspace and a finite dimensional subspace 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