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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> whenever <math>f \neq g</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. Conceptually, the above integral is the equivalent of a vector dot-product; two vectors are mutually independent (orthogonal) if their dot-product is zero.
Suppose <math> \{ f_0, f_1, \ldots\}</math> is a sequence of orthogonal functions of nonzero [[L2-norm|''L''<sup>2</sup>-norm]]s <math> \Vert f_n \Vert _2 = \sqrt{\langle f_n, f_n \rangle} = \left(\int f_n ^2 \ dx \right) ^\frac{1}{2} </math>. It follows that the sequence <math>\left\{ f_n / \Vert f_n \Vert _2 \right\}</math> is of functions of ''L''<sup>2</sup>-norm one, forming an [[orthonormal sequence]]. To have a defined ''L''<sup>2</sup>-norm, the integral must be bounded, which restricts the functions to being [[square-integrable function|square-integrable]].
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