In linear algebra, two vectors v and w are said to be orthonormal if they are both orthogonal (according to a given inner product) and normalized. A set of vectors which are pairwise orthonormal is called an orthonormal set. A basis which forms an orthonormal set is called an orthonormal basis.
When referring to functions, usually the Lē-norm is assumed unless otherwise stated, so that two functions φ(x) and ψ(x) are orthonormal over the interval [a,b] if
An equivalent formulation of the two conditions is done by using the Delta function. A set of vectors (functions, matrices, sequences etc)
In mathematics, an orthonormal basis of an inner product space V (i.e., a vector space with an inner product), or in particular of a Hilbert space H, is a set of elements whose span is dense in the space, in which the elements are mutually orthogonal and normal, that is, of magnitude 1.
An orthonormal basis is not generally a "basis", i.e., it is not generally possible to write every member of the space as a linear combination of finitely many members of an orthonormal basis.
Note that in the infinite-dimensional case, an orthonormal basis will not be a basis in the sense of linear algebra; to distinguish the two, the latter bases are also called Hamel bases.
![(2)\quad||\phi(x)||_2 = ||\psi(x)||_2 = \left[\int_a^b|\phi(x)|^2dx\right]^\frac{1}{2} = \left[\int_a^b|\psi(x)|^2dx\right]^\frac{1}{2} = 1.](http://en.wikipedia.org/math/c50cba9420cdf3bfc1dcedf76850ea3c.png)
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