uncenteredCovarianceBilinDual — Mathlib

English

Properties of the uncentered covariance bilinear dual functional.

Русский

Свойства функционала нецентрированной ковариационной билинейной двойственности.

LaTeX

$$$\\text{uncenteredCovarianceBilinDual}(μ) = \\text{bilinear}(L_1, L_2)$$$

Lean4

/-- Continuous bilinear form with value `∫ x, L₁ x * L₂ x ∂μ` on `(L₁, L₂)`.
This is equal to the covariance only if `μ` is centered. -/
noncomputable def uncenteredCovarianceBilinDual (μ : Measure E) : StrongDual ℝ E →L[ℝ] StrongDual ℝ E →L[ℝ] ℝ :=
  ContinuousLinearMap.bilinearComp (isBoundedBilinearMap_inner (𝕜 := ℝ)).toContinuousLinearMap (StrongDual.toLp μ 2)
    (StrongDual.toLp μ 2)

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