laplacian_eq_iteratedFDeriv_stdOrthonormalBasis

English

Given a finite family of symmetric operators, there exists a basis consisting of joint eigenvectors.

Русский

Существет базис из общих собственных векторов для всейfamily симметричных операторов.

LaTeX

$$DirectSum.IsInternal (fun i => Eig(T_i, i))$$

Lean4

/-- Standard formula, computing the Laplacian from the standard orthonormal basis of a real inner
product space.
-/
theorem laplacian_eq_iteratedFDeriv_stdOrthonormalBasis :
    Δ f = fun x ↦ ∑ i, iteratedFDeriv ℝ 2 f x ![(stdOrthonormalBasis ℝ E) i, (stdOrthonormalBasis ℝ E) i] :=
  laplacian_eq_iteratedFDeriv_orthonormalBasis f (stdOrthonormalBasis ℝ E)

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