eLpNormEssSup_const_smul — Mathlib

English

Scalar multiplication commutes with the essential supremum norm: eLpNormEssSup(c · f) = |c|_e · eLpNormEssSup(f).

Русский

Скалярное умножение commuting с нормой ess sup: eLpNormEssSup(c · f) = |c|ₑ · eLpNormEssSup(f).

LaTeX

$$$$eLpNormEssSup(c \\cdot f) = \\\\|c\\\\|_e \\, eLpNormEssSup(f).$$$$

Lean4

theorem eLpNormEssSup_const_smul (c : 𝕜) (f : α → F) : eLpNormEssSup (c • f) μ = ‖c‖ₑ * eLpNormEssSup f μ := by
  simp_rw [eLpNormEssSup_eq_essSup_enorm, Pi.smul_apply, enorm_smul, ENNReal.essSup_const_mul]

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