Theorems · Theorem · operator theory

ContinuousLinearMap.IsFredholm.of_isInvertible_restrict

∀ {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 E] [ContinuousSMul 𝕜 E]
  [IsTopologicalAddGroup F] [ContinuousSMul 𝕜 F] {u : E →L[𝕜] F} {E₁ : Submodule 𝕜 E},
  IsClosed ↑E₁ →
    ∀ [E₁_coFG : E₁.CoFG] {F₁ : Submodule 𝕜 F},
      IsClosed ↑F₁ →
        ∀ [F₁_coFG : F₁.CoFG] (h_mapsto : Set.MapsTo ⇑u ↑E₁ ↑F₁), (u.restrict h_mapsto).IsInvertible → u.IsFredholm

Assume that u : E →L[𝕜] F restricts to an isomorphism between closed finite codimension subspaces E₁ and F₁. Then u is Fredholm. In fact it is enough to assume that the restriction E₁ →L[𝕜] F₁ is Fredholm, see IsFredholm.of_restrict (not in Mathlib yet).

Defined in

Mathlib.Analysis.Normed.Operator.Fredholm.Basic

Cited by

1 results in Mathlib

Foundations

Depth 181 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldAddCommGroupAddCommGroupModuleModuleTopologicalSpaceTopologicalSpaceCompleteSpaceIsTopologicalAddGroupContinuousSMulIsTopologicalAddGroupContinuousSMulSubmodule.CoFGSubmodule.CoFG