Theorems · Theorem · functional analysis

ContinuousLinearMap.coe_equivRange

∀ {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']
  [inst_7 : RingHomIsometric σ] [inst_8 : RingHomIsometric σ'] [inst_9 : CompleteSpace F] [inst_10 : CompleteSpace E]
  [inst_11 : RingHomInvPair σ' σ] {f : E →SL[σ] F} (hinj : Function.Injective ⇑f) (hclo : IsClosed (Set.range ⇑f)),
  ⇑(ContinuousLinearMap.equivRange hinj hclo) = ⇑f.rangeRestrict

Defined in

Mathlib.Analysis.Normed.Operator.Banach

Cited by

0 results in Mathlib

Foundations

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

Assumes

NontriviallyNormedFieldNontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceRingHomInvPairRingHomIsometricRingHomIsometricCompleteSpaceCompleteSpaceRingHomInvPair