Theorems · Definition · functional analysis

ContinuousLinearMap.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'] →
                          [RingHomIsometric σ] →
                            [RingHomIsometric σ'] →
                              [CompleteSpace F] →
                                [CompleteSpace E] →
                                  [inst_11 : RingHomInvPair σ' σ] →
                                    {f : E →SL[σ] F} →
                                      Function.Injective ⇑f → IsClosed (Set.range ⇑f) → E ≃SL[σ] ↥(↑f).range

An injective continuous linear map with a closed range defines a continuous linear equivalence between its domain and its range.

Defined in

Mathlib.Analysis.Normed.Operator.Banach

Cited by

6 results in Mathlib

Foundations

Depth 172 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceRingHomInvPairRingHomIsometricRingHomIsometricCompleteSpaceCompleteSpaceRingHomInvPair