Theorems · Definition · functional analysis

LinearIsometryEquiv.ofLinearIsometry

{R : Type u_1} →
  {R₂ : Type u_2} →
    {E : Type u_5} →
      {E₂ : Type u_6} →
        [inst : Semiring R] →
          [inst_1 : Semiring R₂] →
            {σ₁₂ : R →+* R₂} →
              {σ₂₁ : R₂ →+* R} →
                [inst_2 : RingHomInvPair σ₁₂ σ₂₁] →
                  [inst_3 : RingHomInvPair σ₂₁ σ₁₂] →
                    [inst_4 : SeminormedAddCommGroup E] →
                      [inst_5 : SeminormedAddCommGroup E₂] →
                        [inst_6 : Module R E] →
                          [inst_7 : Module R₂ E₂] →
                            (f : E →ₛₗᵢ[σ₁₂] E₂) →
                              (g : E₂ →ₛₗ[σ₂₁] E) →
                                f.toLinearMap ∘ₛₗ g = LinearMap.id → g ∘ₛₗ f.toLinearMap = LinearMap.id → E ≃ₛₗᵢ[σ₁₂] E₂

If a linear isometry has an inverse, it is a linear isometric equivalence.

Defined in

Mathlib.Analysis.Normed.Operator.LinearIsometry

Cited by

5 results in Mathlib

Foundations

Depth 156 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringSemiringRingHomInvPairRingHomInvPairSeminormedAddCommGroupSeminormedAddCommGroupModuleModule