Theorems · Definition · functional analysis

LinearIsometry.equivRange

{E : Type u_5} →
  {F : Type u_9} →
    [inst : SeminormedAddCommGroup E] →
      [inst_1 : NormedAddCommGroup F] →
        {R : Type u_11} →
          {S : Type u_12} →
            [inst_2 : Semiring R] →
              [inst_3 : Ring S] →
                [inst_4 : Module S E] →
                  [inst_5 : Module R F] →
                    {σ₁₂ : R →+* S} →
                      {σ₂₁ : S →+* R} →
                        [inst_6 : RingHomInvPair σ₁₂ σ₂₁] →
                          [inst_7 : RingHomInvPair σ₂₁ σ₁₂] → (f : F →ₛₗᵢ[σ₁₂] E) → F ≃ₛₗᵢ[σ₁₂] ↥f.range

Reinterpret a LinearIsometry as a LinearIsometryEquiv to the range.

Defined in

Mathlib.Analysis.Normed.Operator.LinearIsometry

Cited by

3 results in Mathlib

Foundations

Depth 159 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SeminormedAddCommGroupNormedAddCommGroupSemiringRingModuleModuleRingHomInvPairRingHomInvPair