Orthonormality

Orthonormality is a property of a set of vectors in an inner product space. A set {e1, e2, ..., en} is orthonormal if the inner product ⟨ei, ej⟩ equals 0 for i ≠ j and equals 1 for all i, j with i = j. In real spaces the inner product is typically the dot product; in complex spaces it is a Hermitian inner product, which is conjugate symmetric and positive definite.

An orthonormal set that also spans the space is called an orthonormal basis. In that case every vector v can be expressed as v = sum_i ⟨v, ei⟩ ei, and the coefficient ⟨v, ei⟩ is the i-th coordinate of v in this basis. The norm of v satisfies ||v||^2 = sum_i |⟨v, ei⟩|^2, reflecting the Pythagorean structure of the decomposition.

Key properties include: orthonormality implies simple projection formulas, since the projection of v onto the span of an orthonormal set is sum_i ⟨v, ei⟩ ei; and for any vectors v, w, ⟨v, w⟩ = sum_i ⟨v, ei⟩ ⟨ei, w⟩ when {ei} is an orthonormal basis. In complete spaces (Hilbert spaces), this yields Parseval’s identity and enables stable series expansions.

Gram-Schmidt is a standard method for turning any linearly independent set into an orthonormal set. In finite-dimensional spaces, an orthonormal basis provides convenient, numerically stable representations; in complex spaces such bases relate to unitary (or orthogonal in the real case) matrices.

Applications span quantum mechanics, signal processing (Fourier and wavelet bases), numerical linear algebra, and computer graphics, where orthonormal bases simplify computations and analyses. The concept depends on the chosen inner product; different inner products yield different orthonormal sets.