For a general inner product space an orthonormal basis can be used to define normalized orthogonal coordinates on Under these coordinates, the inner product becomes a dot product of vectors. Thus the presence of an orthonormal basis reduces the study of a finite-dimensional inner product space to the study of under dot product. Every finite-dimensional inner product space has an orthonormal basis, which may be obtained from an arbitrary basis using the Gram–Schmidt process.
In functional analysis, the concept of an orthonormal basis can be generalized to arbitrary (infinite-dimensional) inner product spaces. Given a pre-Hilbert space an orthonormal basis for is an orthonormal set of vectors with the property that every vector in can be written as an infinite linear combination of the vectors in the basis. In this case, the orthonormal basis is sometimes called a Hilbert basis for Note that an orthonormal basis in this sense is not generally a Hamel basis, since infinite linear combinations are required. Specifically, the linear span of the basis must be dense in but it may not be the entire space.
The set of vectors which is called the standard basis, forms an orthonormal basis of .
Proof: A straightforward computation shows that the inner products of these vectors equals zero, and that each of their magnitudes equals one, This means that is an orthonormal set. All vectors can be expressed as a sum of the basis vectors scaled
so spans and hence must be a basis. It may also be shown that the standard basis rotated about an axis through the origin or reflected in a plane through the origin forms an orthonormal basis of .
Notice that an orthogonal transformation of the standard inner-product space can be used to construct other orthogonal bases of .
The set with where denotes the exponential function, forms an orthonormal basis of the space of functions with finite Lebesgue integrals, with respect to the 2-norm. This is fundamental to the study of Fourier series.
The set with if and otherwise forms an orthonormal basis of
If is an orthonormal basis of then is isomorphic to in the following sense: there exists a bijectivelinear map such that
Incomplete orthogonal sets
Given a Hilbert space and a set of mutually orthogonal vectors in we can take the smallest closed linear subspace of containing Then will be an orthogonal basis of which may of course be smaller than itself, being an incomplete orthogonal set, or be when it is a complete orthogonal set.
Using Zorn's lemma and the Gram–Schmidt process (or more simply well-ordering and transfinite recursion), one can show that every Hilbert space admits an orthonormal basis; furthermore, any two orthonormal bases of the same space have the same cardinality (this can be proven in a manner akin to that of the proof of the usual dimension theorem for vector spaces, with separate cases depending on whether the larger basis candidate is countable or not). A Hilbert space is separable if and only if it admits a countable orthonormal basis. (One can prove this last statement without using the axiom of choice.)
In other words, the space of orthonormal bases is like the orthogonal group, but without a choice of base point: given an orthogonal space, there is no natural choice of orthonormal basis, but once one is given one, there is a one-to-one correspondence between bases and the orthogonal group.
Concretely, a linear map is determined by where it sends a given basis: just as an invertible map can take any basis to any other basis, an orthogonal map can take any orthogonal basis to any other orthogonal basis.
The other Stiefel manifolds for of incomplete orthonormal bases (orthonormal -frames) are still homogeneous spaces for the orthogonal group, but not principal homogeneous spaces: any -frame can be taken to any other -frame by an orthogonal map, but this map is not uniquely determined.