English
If (s_i) is a directed family of subsets of E, each producing an orthonormal set, then the union of these subsets (as a subspace of E) is also orthonormal.
Русский
Если (s_i) образует направленную пару подмножеств пространства E, каждое образование ортонормированного набора, то объединение этих подмножеств образует ортонормированный набор.
LaTeX
$$$\operatorname{Orthonormal} 𝕜 (\lambda x. x) = \text{ (directed union) }\Rightarrow \operatorname{Orthonormal} 𝕜 (\cup_i s_i)"$$
Lean4
/-- The double sum of weighted inner products of pairs of vectors from an orthonormal sequence is the
sum of the weights.
-/
theorem inner_left_right_finset {s : Finset ι} {v : ι → E} (hv : Orthonormal 𝕜 v) {a : ι → ι → 𝕜} :
(∑ i ∈ s, ∑ j ∈ s, a i j • ⟪v j, v i⟫) = ∑ k ∈ s, a k k := by classical simp [orthonormal_iff_ite.mp hv]
