Theorems · Definition · functional analysis

LinearIsometryEquiv.ofBounds

{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] →
                            (e : E ≃ₛₗ[σ₁₂] E₂) →
                              (∀ (x : E), ‖e x‖ ≤ ‖x‖) → (∀ (y : E₂), ‖e.symm y‖ ≤ ‖y‖) → E ≃ₛₗᵢ[σ₁₂] E₂

Construct a LinearIsometryEquiv from a LinearEquiv and two inequalities: ∀ x, ‖e x‖ ≤ ‖x‖ and ∀ y, ‖e.symm y‖ ≤ ‖y‖.

Defined in

Mathlib.Analysis.Normed.Operator.LinearIsometry

Cited by

0 results in Mathlib

Foundations

Depth 103 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringSemiringRingHomInvPairRingHomInvPairSeminormedAddCommGroupSeminormedAddCommGroupModuleModule