Orthonormality
That is, Many important results in linear algebra deal with collections of two or more orthogonal vectors.The notion of restricting orthogonal pairs of vectors to only those of unit length is important enough to be given a special name.Consider the restrictions on x1, x2, y1, y2 required to make u and v form an orthonormal pair.Using a trigonometric identity to convert the cotangent term gives It is clear that in the plane, orthonormal vectors are simply radii of the unit circle whose difference in angles equals 90°.However, they display certain features that make them fundamental in exploring the notion of diagonalizability of certain operators on vector spaces.Orthonormal sets have certain very appealing properties, which make them particularly easy to work with.Proof of the Gram-Schmidt theorem is constructive, and discussed at length elsewhere.The Gram-Schmidt theorem, together with the axiom of choice, guarantees that every vector space admits an orthonormal basis.This is possibly the most significant use of orthonormality, as this fact permits operators on inner-product spaces to be discussed in terms of their action on the space's orthonormal basis vectors.What results is a deep relationship between the diagonalizability of an operator and how it acts on the orthonormal basis vectors.When referring to real-valued functions, usually the L² inner product is assumed unless otherwise stated.Taking C[−π,π] to be the space of all real-valued functions continuous on the interval [−π,π] and taking the inner product to be it can be shown that forms an orthonormal set.However, this is of little consequence, because C[−π,π] is infinite-dimensional, and a finite set of vectors cannot span it.