Theorems · Theorem · commutative algebra

spectralNorm_eq_iSup_of_finiteDimensional_normal

∀ (K : Type u_2) [inst : NormedField K] (L : Type u_3) [inst_1 : Field L] [inst_2 : Algebra K L]
  [h_fin : FiniteDimensional K L] [hn : Normal K L] {f : AlgebraNorm K L},
  IsPowMul ⇑f →
    IsNonarchimedean ⇑f → (∀ (x : K), f ((algebraMap K L) x) = ‖x‖) → ∀ (x : L), spectralNorm K L x = ⨆ σ, f (σ x)

If L/K is finite and normal, then spectralNorm K L x = supr (λ (σ : Gal(L/K)), f (σ x)).

Defined in

Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Cited by

2 results in Mathlib

Foundations

Depth 201 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedFieldFieldAlgebraFiniteDimensionalNormal