Theorems · Theorem · functional analysis

ContinuousLinearMap.IsIdempotentElem.range_mem_invtSubmodule_iff

∀ {R : Type u_1} {M : Type u_2} [inst : Ring R] [inst_1 : TopologicalSpace M] [inst_2 : AddCommGroup M]
  [inst_3 : Module R M] {f T : M →L[R] M},
  IsIdempotentElem f → ((↑f).range ∈ Module.End.invtSubmodule ↑T ↔ f ∘SL T ∘SL f = T ∘SL f)

range f is invariant under T if and only if f ∘L T ∘L f = T ∘L f, for idempotent f.

Defined in

Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Idempotent

Cited by

3 results in Mathlib

Foundations

Depth 68 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RingTopologicalSpaceAddCommGroupModule