English
Smulgation in MeasureTheory.Lp expresses the product of an Lp function with an Lp function as an Lp function.
Русский
Определение умножения функций в Lp выражает произведение двух функций в Lp как новую функцию в Lp.
LaTeX
$$$f ∈ L^p, g ∈ L^q \\Rightarrow f • g ∈ L^r.$$$
Lean4
protected theorem add_smul (f : Lp 𝕜 p μ) (g₁ g₂ : Lp E q μ) : f • (g₁ + g₂) = f • g₁ + f • g₂ :=
by
simp only [smul_def, ← MemLp.toLp_add]
apply MemLp.toLp_congr _ _ ?_
filter_upwards [AEEqFun.coeFn_add g₁.val g₂.val] with x hx
simp [hx, smul_add]