Orthonormalbasen

Orthonormalbasen refers to a set of vectors in an inner product space that is simultaneously a basis and orthonormal. Formally, a finite set {u1, …, un} in a real or complex inner product space V is an orthonormal basis if each vector has unit length and distinct vectors are mutually orthogonal: <ui, uj> = 0 for i ≠ j and <ui, ui> = 1. In real spaces, this uses the standard inner product; in complex spaces it uses a Hermitian inner product, so <ui, uj> = δij with conjugation taken into account.

Properties and consequences. If {u1, …, un} is an orthonormal basis, any v ∈ V can be written uniquely as v = ∑ i=1..n <v, ui> ui. The coefficients are the inner products with the basis vectors, and the norm satisfies ||v||^2 = ∑ i |<v, ui>|^2 (Parseval’s identity in appropriate contexts). The Gram matrix of the basis is the identity, reflecting orthonormality.

Construction and examples. From any basis, the Gram-Schmidt process produces an orthonormal basis for the same span. In R^n with the standard inner product, the standard basis e1, …, en is already orthonormal. In complex spaces, an orthonormal basis may be obtained similarly, using the Hermitian inner product. Unitary transformations preserve orthonormal bases: the image of an orthonormal basis under a unitary map is again orthonormal.

Extensions and usage. In infinite-dimensional Hilbert spaces, a complete orthonormal set forms an orthonormal basis, enabling vector expansions analogous to Fourier series and underpinning projections and least-squares methods. Orthonormal bases simplify computations and analysis by turning inner products and expansions into simple coordinate-like operations.