constL — Mathlib

English

The composition rules for const linear maps ensure that (Lp.const p μ) respects linear combinations in its argument E.

Русский

Правила композиции линейных отображений сохраняют линейность аргумента E для Lp.const.

LaTeX

$$Empirical composition rule: (Lp.const p μ) is linear in the input and respects addition and scalar multiplication.$$

Lean4

/-- `MeasureTheory.Lp.const` as a `ContinuousLinearMap`. -/
@[simps! apply]
protected def constL (𝕜 : Type*) [NormedRing 𝕜] [Module 𝕜 E] [IsBoundedSMul 𝕜 E] [Fact (1 ≤ p)] : E →L[𝕜] Lp E p μ :=
  (Lp.constₗ p μ 𝕜).mkContinuous (μ.real Set.univ ^ (1 / p.toReal)) fun _ ↦
    (Lp.norm_const_le _ _ _).trans_eq (mul_comm _ _)

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