## Encyclopedia > Orthonormal

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# Orthonormal

Two vectors $\mathbf{v}$ and $\mathbf{w}$ are said to be orthonormal if they are both orthogonal and normalized[?]. A set of vectors which are pairwise orthonormal is called an orthonormal set.

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

$(1)\quad<\phi(x),\psi(x)>\ = \int_a^b\phi(x)\psi(x)dx = 0,\quad{\rm and}$
$(2)\quad||\phi(x)||_2 = ||\psi(x)||_2 = \left[\int_a^b|\phi(x)|^2dx\right]^\frac{1}{2} = \left[\int_a^b|\psi(x)|^2dx\right]^\frac{1}{2} = 1.$

Unfortunately, the word normal is sometimes used synonymously with orthogonal.

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