ortonormert

Ortonormert, in English orthonormal, refers to a set of vectors in an inner product space that are mutually orthogonal and each of unit length. Concretely, a set {e1, e2, …, en} satisfies ⟨ei, ej⟩ = δij, where δij is 1 if i = j and 0 otherwise. The term also extends to infinite sets in Hilbert spaces, where the same inner-product condition holds for all pairs.

An orthonormal set has several important consequences. The norm of any vector v is ||v|| = sqrt(⟨v, v⟩). If {e1, ..., en} is an orthonormal basis for a finite-dimensional space, any vector v can be written uniquely as a linear combination v = ∑i ⟨v, ei⟩ ei, with coordinates given by the inner products ⟨v, ei⟩. In real spaces the matrix whose columns are the basis vectors is orthogonal (M^T M = I); in complex spaces the matrix is unitary (M^* M = I).

Gram–Schmidt is a standard procedure to convert any linearly independent set into an orthonormal set. Orthogonality and normalization simplify many computations, including projections, decompositions, and expansions. In finite dimensions, an orthonormal basis exists for every vector space of dimension n; in infinite dimensions, an orthonormal basis is a complete orthonormal set spanning a Hilbert space.

Related concepts include Parseval’s identity, which equates the norm to the sum of squared inner products with an orthonormal basis, and Bessel’s inequality for incomplete sets. Orthonormal systems are fundamental in linear algebra, signal processing, quantum mechanics, and numerical methods.