Theorems · Theorem · functional analysis

Submodule.topologicalClosure_map

∀ {R₁ : Type u_1} {R₂ : Type u_2} [inst : Semiring R₁] [inst_1 : Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4}
  [inst_2 : TopologicalSpace M₁] [inst_3 : AddCommMonoid M₁] {M₂ : Type u_6} [inst_4 : TopologicalSpace M₂]
  [inst_5 : AddCommMonoid M₂] [inst_6 : Module R₁ M₁] [inst_7 : Module R₂ M₂] [inst_8 : RingHomSurjective σ₁₂]
  [inst_9 : TopologicalSpace R₁] [inst_10 : TopologicalSpace R₂] [inst_11 : ContinuousSMul R₁ M₁]
  [inst_12 : ContinuousAdd M₁] [inst_13 : ContinuousSMul R₂ M₂] [inst_14 : ContinuousAdd M₂] (f : M₁ →SL[σ₁₂] M₂)
  (s : Submodule R₁ M₁), Submodule.map (↑f) s.topologicalClosure ≤ (Submodule.map (↑f) s).topologicalClosure

Under a continuous linear map, the image of the TopologicalClosure of a submodule is contained in the TopologicalClosure of its image.

Defined in

Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic

Cited by

2 results in Mathlib

Foundations

Depth 82 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringSemiringTopologicalSpaceAddCommMonoidTopologicalSpaceAddCommMonoidModuleModuleRingHomSurjectiveTopologicalSpaceTopologicalSpaceContinuousSMulContinuousAddContinuousSMulContinuousAdd