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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, an orthonormal basis of an inner product space
Inner product space

In mathematics, an inner product space is a vector space with the additional Mathematical structure of inner product. This additional structure associates each pair of vectors in the space with a Scalar quantity known as the inner product of the vectors....
 V (i.e., a vector space
Vector space

File:Vector addition ans scaling.pngA vector space is a mathematical structure formed by a collection of vectors: objects that may be Vector addition together and Scalar multiplication by numbers, called scalar s in this context....
 with an inner product), is a set of mutually orthogonal
Orthogonality

In mathematics, two vectors are orthogonal if they are perpendicular, i.e., they form a right angle. The word comes from the Greek language ' , meaning "straight", and ' , meaning "angle"....
 vectors of magnitude 1 (unit vector
Unit vector

In mathematics, a unit vector in a normed vector space is a Vector space whose Norm is 1 . A unit vector is often denoted by a lowercase letter with a superscribed caret or ?hat?, like this: ....
s) that span the space when infinite linear combination
Linear combination

In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics.Most of this article deals with linear combinations in the context of a vector space over a field , with some generalisations given at the end of the article....
s are allowed. (In some contexts, especially in linear algebra
Linear algebra

Linear algebra is the branch of mathematics concerned with the study of Euclidean vectors, vector spaces , linear maps , and system of linear equations....
, the concept of basis (linear algebra)
Basis (linear algebra)

In linear algebra, a basis is a set of vectors that, in a linear combination, can represent every vector in a given vector space or free module, and such that no element of the set can be represented as a linear combination of the others....
 means a set of vectors that span a space when only finite linear combinations are allowed.) Such an infinite linear combination is an infinite series, and the concept of convergence relied upon is defined in terms of the space's inner product.

Elements in an orthogonal basis do not have to be unit vectors, but must be mutually perpendicular.






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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, an orthonormal basis of an inner product space
Inner product space

In mathematics, an inner product space is a vector space with the additional Mathematical structure of inner product. This additional structure associates each pair of vectors in the space with a Scalar quantity known as the inner product of the vectors....
 V (i.e., a vector space
Vector space

File:Vector addition ans scaling.pngA vector space is a mathematical structure formed by a collection of vectors: objects that may be Vector addition together and Scalar multiplication by numbers, called scalar s in this context....
 with an inner product), is a set of mutually orthogonal
Orthogonality

In mathematics, two vectors are orthogonal if they are perpendicular, i.e., they form a right angle. The word comes from the Greek language ' , meaning "straight", and ' , meaning "angle"....
 vectors of magnitude 1 (unit vector
Unit vector

In mathematics, a unit vector in a normed vector space is a Vector space whose Norm is 1 . A unit vector is often denoted by a lowercase letter with a superscribed caret or ?hat?, like this: ....
s) that span the space when infinite linear combination
Linear combination

In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics.Most of this article deals with linear combinations in the context of a vector space over a field , with some generalisations given at the end of the article....
s are allowed. (In some contexts, especially in linear algebra
Linear algebra

Linear algebra is the branch of mathematics concerned with the study of Euclidean vectors, vector spaces , linear maps , and system of linear equations....
, the concept of basis (linear algebra)
Basis (linear algebra)

In linear algebra, a basis is a set of vectors that, in a linear combination, can represent every vector in a given vector space or free module, and such that no element of the set can be represented as a linear combination of the others....
 means a set of vectors that span a space when only finite linear combinations are allowed.) Such an infinite linear combination is an infinite series, and the concept of convergence relied upon is defined in terms of the space's inner product.

Elements in an orthogonal basis do not have to be unit vectors, but must be mutually perpendicular. It is easy to change the vectors in an orthogonal basis by scalar multiples to get an orthonormal basis, and indeed this is a typical way that an orthonormal basis is constructed.

The standard basis
Standard basis

In mathematics, the standard basis of the -dimension Euclidean space Rn is the basis obtained by taking the basis vectorswhere is the vector with a in the th coordinate and elsewhere....
 of the n-dimensional Euclidean space
Euclidean space

Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
 Rn is an example of orthonormal (and ordered) basis.

For a finite-dimensional space, every orthonormal basis is a Hamel basis (a basis as defined in linear algebra
Linear algebra

Linear algebra is the branch of mathematics concerned with the study of Euclidean vectors, vector spaces , linear maps , and system of linear equations....
, that spans
Linear span

In the mathematics subfield of linear algebra, the linear span, also called the linear hull, of a Set of vector space in a vector space is the intersection of all Linear subspace containing that set....
 the entire space), but most Hamel bases are not orthonormal bases. For an (infinite-dimensional) Hilbert space
Hilbert space

The mathematics concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra from the two-dimensional plane and three-dimensional space to infinite-dimensional spaces....
, an orthonormal basis is not a Hamel basis, i.e., it is not possible to write every member of the space as a linear combination of finitely many members of an orthonormal basis. In the infinite-dimensional case the distinction matters. An orthonormal basis of a Hilbert space H is required to have a dense
Dense set

In topology and related areas of mathematics, a subset A of a topological space X is called dense if, intuitively, any point in X can be "well-approximated" by points in A....
 linear span
Linear span

In the mathematics subfield of linear algebra, the linear span, also called the linear hull, of a Set of vector space in a vector space is the intersection of all Linear subspace containing that set....
 in H, but its linear space is not the entire space.

An orthonormal basis of a vector space V makes no sense unless V is given an inner product. In a Banach space
Banach space

In mathematics, Banach spaces are one of the central objects of study in functional analysis. They are topological vector spaces that have many interesting properties associated with them....
 that is not an inner-product space it makes no sense to speak of whether a set of vectors is orthonormal.

Examples

  • The set (the standard basis) forms an orthonormal basis of R3.
Proof: A straightforward computation shows that <e1, e2> = <e1, e3> = <e2, e3> = 0 and that ||e1|| = ||e2|| = ||e3|| = 1. So is an orthonormal set. For all (x, y, z) in R3 we have

so spans R3 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 R3.
  • The set with fn(x) = exp
    Exponential function

    The exponential function is a function in mathematics. The application of this function to a value x is written as exp. Equivalently, this can be written in the form ex, where e is the mathematical constant that is the base of the natural logarithm and that is also known as Euler's number....
    (2πinx) forms an orthonormal basis of the complex space L2([0,1]). This is fundamental to the study of Fourier series
    Fourier series

    In mathematics, a Fourier series decomposes a periodic function into a sum of simple oscillating functions, namely sine wave . The study of Fourier series is a branch of Fourier analysis....
    .
  • The set with eb(c) = 1 if b = c and 0 otherwise forms an orthonormal basis of l 2(B).
  • Eigenfunctions of a Sturm-Liouville eigenproblem.


Basic formula

If B is an orthogonal basis of H, then every element x of H may be written as

When B is orthonormal, we have instead

and the norm
Norm (mathematics)

In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector....
 of x can be given by

Even if B 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 of x, and the formula is usually known as Parseval's identity
Parseval's identity

In mathematical analysis, Parseval's identity is a fundamental result on the summability of the Fourier series of a function. Geometrically, it is the...
. See also Generalized Fourier series
Generalized Fourier series

In mathematical analysis, many generalizations of Fourier series have proved to be useful.They are all special cases of decompositions over an orthonormal basis of an inner product space....
.

If B is an orthonormal basis of H, then H is isomorphic to l 2(B) in the following sense: there exists a bijective linear map Φ : H -> l 2(B) such that

for all x and y in H.

Incomplete orthogonal sets

Given a Hilbert space H and a set S of mutually orthogonal vectors in H, we can take the smallest closed linear subspace V of H containing S. Then S will be an orthogonal basis of V; which may of course be smaller than H itself, being an incomplete orthogonal set, or be H, when it is a complete orthogonal set.

Existence

Using Zorn's lemma and the Gram-Schmidt process, one can show that every Hilbert space admits a basis and thus an orthonormal basis; furthermore, any two orthonormal bases of the same space have the same cardinality
Cardinal number

In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of Set ....
. A Hilbert space is separable if and only if it admits a countable orthonormal basis.

See also

  • Basis (linear algebra)
    Basis (linear algebra)

    In linear algebra, a basis is a set of vectors that, in a linear combination, can represent every vector in a given vector space or free module, and such that no element of the set can be represented as a linear combination of the others....
  • Schauder basis
    Schauder basis

    In mathematics, a Schauder basis or countable basis is similar to the usual basis . The difference is that for Hamel bases, linear combinations are assumed to be finite sums, while for Schauder bases they may be infinite....
  • Gram-Schmidt