Definition 3

In a prehilbertian space, a family (e_i)_{i∈I} of vectors is said to be orthogonal if e_i and e_k are orthogonals for all i ≠ k, and is said to be orthonormal, if in addition $\|e_i\| = 1$ for all $i \in I$.

A subset S of E such that the family defined by the identity mapping from S onto itself is orthonormal is said to be an orthonormal set. If (e_i)_{i∈I} is an orthonormal family, the mapping $i \mapsto e_i$ is injective; we can then talk indifferently of an orthonormal family or an orthonormal set.

If (e_i)_{i∈I} is an orthonormal family, the complete one dimensional vector subspaces D_i = Ke_i are two by two orthogonal. For every x ∈ E, the orthogonal projection of x on D_i is $\lambda_i e_i$ with $⟨e_i|x - \lambda_i e_i⟩ = 0$, which gives $⟨e_i|x⟩ = \lambda_i ⟨e_i|e_i⟩ = \lambda_i$. The results of No. 2 applied to the subspaces D_i imply the following propositions: