In mathematics, particularly linear algebra, an orthonormal basis for an inner product space with finite dimension is a basis whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, the standard basis for a Euclidean space is an orthonormal basis, where the relevant inner product is the dot product of vectors. The image of the standard basis under a rotation or reflection (or any orthogonal transformation arises in this fashion. An orthonormal basis can be derived from an orthogonal basis via normalization. The choice of an origin and an orthonormal basis forms a coordinate frame known as an orthonormal frame.

an orthonormal basis can be used to define normalized orthogonal coordinates 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 under the 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. an orthonormal basis 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 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 although not necessarily the entire space.

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 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 negative ones.

Examples

• is called the standard basis as the Cartesian product

  1. Proof:.

  , the standard basis and inner product are similarly defined. Any other orthonormal basis is related to the standard basis by an orthogonal transformation in the group O(n).

• For pseudo-Euclidean space, these are Lorentz transformations.

• denotes the exponential function with respect to the 2-norm. This is fundamental to the study of Fourier series.

• Eigenfunctions of a Sturm–Liouville eigenproblem.

• The column vectors of an orthogonal matrix form an orthonormal set.

Basic formula

and the square of the norm

is uncountable, only countably many terms in this sum will be non-zero, and the expression is therefore well-defined. This sum is also called the Fourier expansion and the formula is usually known as Parseval's identity.

is isomorphic in the following sense: there exists a bijective linear

Orthonormal system

. can be regarded as either complete or incomplete itself, being an incomplete when it is a complete orthonormal set.

Existence

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. However, one would have to use the axiom of countable choice.)

Choice of basis as a choice of isomorphism

.

is the Kronecker delta).

which is an isomorphism of inner product spaces: to make this more explicit we can write

.

The inverse is a component map

These definitions make it manifest that there is a bijection

.

.

As a principal homogeneous space

with the standard inner product is a principal homogeneous space or G-torsor for the orthogonal group and is called the Stiefel manifold of orthonormal n-frames.

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. 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.

of incomplete-frames) are still homogeneous spaces for the orthogonal group, but not principal-frame by an orthogonal map, but this map is not uniquely determined.

• .

• .

• .

See also

Notes

References

• (page 218, ch.9)

External links

• This Stack Exchange Post discusses why the set of Dirac Delta functions is not a basis of L2([0,1]).