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In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for ''V'' 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 R''n'' 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) is also orthonormal, and every orthonormal basis for R''n'' arises in this fashion. For a general inner product space ''V'', an orthonormal basis can be used to define normalized orthogonal coordinates on ''V''. Under these coordinates, the inner product becomes 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 R''n'' 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 (or pre-Hilbert spaces). Given a pre-Hilbert space ''H'', an orthonormal basis for ''H'' is an orthonormal set of vectors with the property that every vector in ''H'' 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 ''H''. 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 ''H'', but it may not be 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 on the interval (1 ) can be expressed (almost everywhere) as an infinite sum of Legendre polynomials (an orthonomal basis), but not necessarily as an infinite sum of the monomials ''xn''. ==Examples== * The set of vectors (the standard basis) forms an orthonormal basis of R3. ::Proof: A straightforward computation shows that the inner products of these vectors equals zero, and that each of their magnitudes equals one, ||''e''1|| = ||''e''2|| = ||''e''3|| = 1. This means that is an orthonormal set. All vectors in R3 can be expressed as a sum of the basis vectors scaled ::: ::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 forms an orthonormal basis of the space of functions with finite Lebesgue integrals, L2(()), with respect to the 2-norm. This is fundamental to the study of Fourier series. * The set with if and 0 otherwise forms an orthonormal basis of ℓ2(''B''). * Eigenfunctions of a Sturm–Liouville eigenproblem. * An orthogonal matrix is a matrix whose column vectors form an orthonormal set. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「orthonormal basis」の詳細全文を読む スポンサード リンク
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