toContinuousLinearMap_norm — Mathlib

English

The coercion of the equivalence between characters and algebra homomorphisms, when applied to a character, yields exactly that character as a function.

Русский

Применение к characer к эквивалентности между характерами и алгебраическими гомоморфизмами возвращает тот же функционал.

LaTeX

$$$\\text{toContinuousLinearMap}$ and the equivalence commute with evaluation:$$

Lean4

@[simp]
theorem toContinuousLinearMap_norm [NormOneClass A] (φ : A →ₐ[𝕜] 𝕜) : ‖φ.toContinuousLinearMap‖ = 1 :=
  ContinuousLinearMap.opNorm_eq_of_bounds zero_le_one
    (fun a => (one_mul ‖a‖).symm ▸ spectrum.norm_le_norm_of_mem (apply_mem_spectrum φ _)) fun _ _ h => by
    simpa only [coe_toContinuousLinearMap, map_one, norm_one, mul_one] using h 1

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