Theorems · Definition · functional analysis

Unitization.normedRingAux

{𝕜 : Type u_1} →
  {A : Type u_2} →
    [inst : NontriviallyNormedField 𝕜] →
      [inst_1 : NonUnitalNormedRing A] →
        [inst_2 : NormedSpace 𝕜 A] →
          [inst_3 : IsScalarTower 𝕜 A A] →
            [inst_4 : SMulCommClass 𝕜 A A] → [RegularNormedAlgebra 𝕜 A] → NormedRing (Unitization 𝕜 A)

Pull back the normed ring structure from 𝕜 × (A →L[𝕜] A) to Unitization 𝕜 A using the algebra homomorphism Unitization.splitMul 𝕜 A. This does not give us the desired topology, uniformity or bornology on Unitization 𝕜 A (which we want to agree with Prod), so we only use it as a local instance to build the real one.

Defined in

Mathlib.Analysis.Normed.Algebra.Unitization

Cited by

8 results in Mathlib

Foundations

Depth 183 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNonUnitalNormedRingNormedSpaceIsScalarTowerSMulCommClassRegularNormedAlgebra