linguistic:voice_change, quality:correction, information:classification
Pythagorean identity

Two vectors u and v in a Hilbert space H are orthogonal when ⟨ u, v ⟩ = 0. The notation for this is u ⊥ v. More generally, when S is a subset in H, the notation u ⊥ S means that u is orthogonal to every element from S.

When u and v are orthogonal, one has ‖ u + v ‖ 2 = ⟨ u + v, u + v ⟩ = ⟨ u, u ⟩ + 2 Re ⟨ u, v ⟩ + ⟨ v, v ⟩ = ‖ u ‖ 2 + ‖ v ‖ 2. {\displaystyle \|u+v\|^{2}=\langle u+v,u+v\rangle =\langle u,u\rangle +2\,\operatorname {Re} \langle u,v\rangle +\langle v,v\rangle =\|u\|^{2}+\|v\|^{2}\,.}

By induction on n, this is extended to any family u,..., u of n orthogonal vectors, ‖ u 1 + ⋯ + u n ‖ 2 = ‖ u 1 ‖ 2 + ⋯ + ‖ u n ‖ 2. {\displaystyle \left\|u_{1}+\cdots +u_{n}\right\|^{2}=\left\|u_{1}\right\|^{2}+\cdots +\left\|u_{n}\right\|^{2}.}

Whereas the Pythagorean identity as stated is valid in any inner product space, completeness is required for the extension of the Pythagorean identity to series. A series Σ u of orthogonal vectors converges in H if and only if the series of squares of norms converges, and ‖ ∑ k = 0 ∞ u k ‖ 2 = ∑ k = 0 ∞ ‖ u k ‖ 2. {\displaystyle {\Biggl \|}\sum _{k=0}^{\infty }u_{k}{\Biggr \|}^{2}=\sum _{k=0}^{\infty }\left\|u_{k}\right\|^{2}\,.} Furthermore, the sum of a series of orthogonal vectors is independent of the order in which it is taken.