The functions <math>f</math> and <math>g</math> are [[bilinear form#Reflexivity and orthogonality|orthogonal]] when this integral is zero:, i.e. <math>\langle f, \ g \rangle = 0.</math>
As with a [[basis (linear algebra)|basis]] of vectors in a finite-dimensional space, orthogonal functions can form an infinite basis for a function space.
Suppose <math> \{ f_n \}f_0, n = 0, 1, 2f_1, \ldots\}</math> is a sequence of orthogonal functions. If <math>f_n</math> has positive [[support (mathematics)|support]] then <math> m_n = \sqrt{\langle f_n, f_n \rangle} = \left(\int f_n ^2 \ dx =\right) m_n^\frac{1}{2} </math> is the [[L2-norm|''L''<sup>2</sup>-norm]] of <math>f_n</math>, and the sequence <math>\left\{ \frac {f_n} / {m_n} \right\}</math> has functions of ''L''<sup>2</sup>-norm one, forming an [[orthonormal sequence]]. The possibility that an integral is unbounded must be avoided, hence attention is restricted to [[square-integrable function]]s.
