Knowlpedia

Orthonormal basis

A basis whose vectors have unit length and are pairwise orthogonal.

An orthonormal basis of a HH is a family (ei)i∈I(e_i)_{i\in I} whose is HH and which satisfies

⟨ei,ej⟩=δij.\langle e_i,e_j\rangle=\delta_{ij}.
Interpretation

Thus each basis vector has norm one, distinct basis vectors are orthogonal, and every vector in HH can be approximated in norm by finite linear combinations of the eie_i. In finite dimensions, “closed linear span” may be replaced by “linear span.”

Expansion

Every x∈Hx\in H has the norm-convergent expansion

x=∑i∈I⟨x,ei⟩ei.x=\sum_{i\in I}\langle x,e_i\rangle e_i.

The coefficient order uses the convention that the inner product is linear in its first argument. For arbitrary index sets, the sum means the norm limit over finite subsets; each vector has at most countably many nonzero coefficients.

In a finite-dimensional complex space, placing the basis vectors as columns gives a unitary matrix.