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Orthonormality
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 L²-norm is assumed unless otherwise stated, so that two functions φ(x) and ψ(x) are orthonormal over the interval [a,b] if
![(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.](a/../upload/math/c50cba9420cdf3bfc1dcedf76850ea3c.png)
An equivalent formulation of the two conditions is done by using the Delta function. A set of vectors (functions, matrices, sequences etc)
forms an orthonormal set iff
where < | > is the proper inner product defined over the vector space.
Unfortunately, the word normal is sometimes used synonymously with orthogonal.
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