wInner_cWeight_const_right — Mathlib

English

If weights w, and pointwise nonnegative f and g are considered, then the weighted inner product is nonnegative: 0 ≤ ⟪f, g⟫_{𝕜, w}.

Русский

Если веса и функции неотрицательны по каждой координате, то взвешенное скалярное произведение неотрицательно.

LaTeX

$$0 ≤ ⟪f, g⟫_{𝕜, w} (when w ≥ 0, f ≥ 0, g ≥ 0)$$

Lean4

@[simp]
theorem wInner_cWeight_const_right (f : ι → 𝕜) (a : 𝕜) : ⟪f, const _ a⟫ₙ_[𝕜] = a * (𝔼 i, conj (f i)) := by
  simp [wInner_cWeight_eq_expect, mul_expect]

Похожие утверждения