Theorems · Theorem · functional analysis

LinearEquiv.continuous_symm

∀ {𝕜 : Type u_1} {𝕜' : Type u_2} [inst : NontriviallyNormedField 𝕜] [inst_1 : NontriviallyNormedField 𝕜'] {σ : 𝕜 →+* 𝕜'}
  {E : Type u_3} [inst_2 : NormedAddCommGroup E] [inst_3 : NormedSpace 𝕜 E] {F : Type u_4}
  [inst_4 : NormedAddCommGroup F] [inst_5 : NormedSpace 𝕜' F] {σ' : 𝕜' →+* 𝕜} [inst_6 : RingHomInvPair σ σ']
  [RingHomIsometric σ] [RingHomIsometric σ'] [CompleteSpace F] [CompleteSpace E] [inst_11 : RingHomInvPair σ' σ]
  (e : E ≃ₛₗ[σ] F), Continuous ⇑e → Continuous ⇑e.symm

If a bounded linear map is a bijection, then its inverse is also a bounded linear map.

Defined in

Mathlib.Analysis.Normed.Operator.Banach

Cited by

2 results in Mathlib

Foundations

Depth 170 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceRingHomInvPairRingHomIsometricRingHomIsometricCompleteSpaceCompleteSpaceRingHomInvPair