In mathematics, particularly linear algebra, an orthonormal basis for an inner product space
[1][2][3] For example, the standard basis for a Euclidean space
The image of the standard basis under a rotation or reflection (or any orthogonal transformation) is also orthonormal, and every orthonormal basis for
an orthonormal basis can be used to define normalized orthogonal coordinates on
In functional analysis, the concept of an orthonormal basis can be generalized to arbitrary (infinite-dimensional) inner product spaces.
can be written as an infinite linear combination of the vectors in the basis.
Note that an orthonormal basis in this sense is not generally a Hamel basis, since infinite linear combinations are required.
[5] Specifically, the linear span of the basis must be dense in
If we go on to Hilbert spaces, a non-orthonormal set of vectors having the same linear span as an orthonormal basis may not be a basis at all.
For instance, any square-integrable function on the interval
can be expressed (almost everywhere) as an infinite sum of Legendre polynomials (an orthonormal basis), but not necessarily as an infinite sum of the monomials
equipped with a non-degenerate symmetric bilinear form known as the metric tensor.
In such a basis, the metric takes the form
is uncountable, only countably many terms in this sum will be non-zero, and the expression is therefore well-defined.
and the formula is usually known as Parseval's identity.
in the following sense: there exists a bijective linear map
of mutually orthonormal vectors in a Hilbert space
can be regarded as either complete or incomplete with respect to
That is, we can take the smallest closed linear subspace
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;[7] 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.
For concreteness we discuss orthonormal bases for a real,
with a positive definite symmetric bilinear form
One way to view an orthonormal basis with respect to
The inverse is a component map These definitions make it manifest that there is a bijection The space of isomorphisms admits actions of orthogonal groups at either the
For concreteness we fix the isomorphisms to point in the direction
This space admits a left action by the group of isometries of
This space also admits a right action by the group of isometries of
with the standard inner product is a principal homogeneous space or G-torsor for the orthogonal group
[8] In other words, the space of orthonormal bases is like the orthogonal group, but without a choice of base point: given the space of orthonormal bases, 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.