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 \phi(x) and \psi(x) are orthonormal over the interval a,b if
(1)\quad\langle\phi(x),\psi(x)\rangle = \int_a^b\phi(x)\psi(x)dx = 0,\quad{\rm and}
(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.
An equivalent formulation of the two conditions is done by using the Delta function. A set of vectors (functions, matrices, sequences etc)
\left\{ u_1 , u_2 , ... , u_n , ... \right\}
forms an orthonormal set iff
\forall n,m \ : \quad \left\langle u_n | u_m \right\rangle = \delta_{n,m}
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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