Theorems · Theorem · operator theory

ContinuousLinearMap.IsFredholm.closedComplemented_range

∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [inst : NontriviallyNormedField 𝕜] [inst_1 : AddCommGroup E]
  [inst_2 : AddCommGroup F] [inst_3 : Module 𝕜 E] [inst_4 : Module 𝕜 F] [inst_5 : TopologicalSpace E]
  [inst_6 : TopologicalSpace F] [CompleteSpace 𝕜] [IsTopologicalAddGroup F] [ContinuousSMul 𝕜 F] {u : E →L[𝕜] F},
  u.IsFredholm → (↑u).range.ClosedComplemented

A Fredholm operator has (topologically) complemented range.

Defined in

Mathlib.Analysis.Normed.Operator.Fredholm.Basic

Cited by

1 results in Mathlib

Foundations

Depth 173 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldAddCommGroupAddCommGroupModuleModuleTopologicalSpaceTopologicalSpaceCompleteSpaceIsTopologicalAddGroupContinuousSMul