Theorems · Definition · functional analysis

LinearIsometryEquiv.ofSurjective

{R : Type u_1} →
  {R₂ : Type u_2} →
    {E₂ : Type u_6} →
      {F : Type u_9} →
        [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 E₂] →
                      [inst_5 : Module R₂ E₂] →
                        [inst_6 : NormedAddCommGroup F] →
                          [inst_7 : Module R F] → (f : F →ₛₗᵢ[σ₁₂] E₂) → Function.Surjective ⇑f → F ≃ₛₗᵢ[σ₁₂] E₂

Construct a linear isometry equiv from a surjective linear isometry.

Defined in

Mathlib.Analysis.Normed.Operator.LinearIsometry

Cited by

2 results in Mathlib

Foundations

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

Assumes

SemiringSemiringRingHomInvPairRingHomInvPairSeminormedAddCommGroupModuleNormedAddCommGroupModule