Theorems · Definition · functional analysis

LinearIsometry.comp

{R : Type u_1} →
  {R₂ : Type u_2} →
    {R₃ : Type u_3} →
      {E : Type u_5} →
        {E₂ : Type u_6} →
          {E₃ : Type u_7} →
            [inst : Semiring R] →
              [inst_1 : Semiring R₂] →
                [inst_2 : Semiring R₃] →
                  {σ₁₂ : R →+* R₂} →
                    {σ₁₃ : R →+* R₃} →
                      {σ₂₃ : R₂ →+* R₃} →
                        [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] →
                          [inst_4 : SeminormedAddCommGroup E] →
                            [inst_5 : SeminormedAddCommGroup E₂] →
                              [inst_6 : SeminormedAddCommGroup E₃] →
                                [inst_7 : Module R E] →
                                  [inst_8 : Module R₂ E₂] →
                                    [inst_9 : Module R₃ E₃] → (E₂ →ₛₗᵢ[σ₂₃] E₃) → (E →ₛₗᵢ[σ₁₂] E₂) → E →ₛₗᵢ[σ₁₃] E₃

Composition of linear isometries.

Defined in

Mathlib.Analysis.Normed.Operator.LinearIsometry

Cited by

9 results in Mathlib

Foundations

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

Assumes

SemiringSemiringSemiringRingHomCompTripleSeminormedAddCommGroupSeminormedAddCommGroupSeminormedAddCommGroupModuleModuleModule