Theorems · Theorem · functional analysis

Unitization.norm_eq_sup

∀ {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]
  [inst_5 : RegularNormedAlgebra 𝕜 A] (x : Unitization 𝕜 A),
  ‖x‖ = max ‖x.toProd.1‖ ‖(algebraMap 𝕜 (A →L[𝕜] A)) x.toProd.1 + (ContinuousLinearMap.mul 𝕜 A) x.toProd.2‖

This is often the more useful lemma to rewrite the norm as opposed to Unitization.norm_def.

Defined in

Mathlib.Analysis.Normed.Algebra.Unitization

Cited by

4 results in Mathlib

Foundations

Depth 185 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNonUnitalNormedRingNormedSpaceIsScalarTowerSMulCommClassRegularNormedAlgebra