norm_const' — Mathlib

English

The Lp-constant map with a fixed E-valued parameter c is a continuous linear map in the scalar c.

Русский

Карта константы Lp по параметру c непрерывно линейна.

LaTeX

$$The map c ↦ indicatorConstLp p hs hμs c is a continuous linear map from E to Lp E p μ.$$

Lean4

theorem norm_const' (hp_zero : p ≠ 0) (hp_top : p ≠ ∞) : ‖Lp.const p μ c‖ = ‖c‖ * μ.real Set.univ ^ (1 / p.toReal) :=
  by
  rw [← MemLp.toLp_const, Lp.norm_toLp, eLpNorm_const'] <;> try assumption
  rw [measureReal_def, ENNReal.toReal_mul, toReal_enorm, ← ENNReal.toReal_rpow]

Похожие утверждения