le_radius_compContinuousLinearMap

English

Scaling by norm does not affect radius; i.e., r•p keeps the same radius as p.

Русский

Нормовое масштабирование не изменяет радиус: radius(r•p) = radius(p).

LaTeX

$$$radius(p) = radius(r\\cdot p).$$$

Lean4

theorem le_radius_compContinuousLinearMap (p : FormalMultilinearSeries 𝕜 F G) (u : E →ₗᵢ[𝕜] F) :
    p.radius ≤ (p.compContinuousLinearMap u.toContinuousLinearMap).radius :=
  by
  obtain (h | h) := subsingleton_or_nontrivial E
  · simp [Subsingleton.elim u.toContinuousLinearMap 0]
  · simpa [u.norm_toContinuousLinearMap] using div_le_radius_compContinuousLinearMap p u.toContinuousLinearMap

Похожие утверждения