Orthonormal functions form a family 
of orthogonal
functions with unit norm. On a closed
interval ![[a,b]](/images/equations/OrthonormalFunctions/Inline2.svg)
, 
, work in the L2-space ![L^2([a,b],w(x)dx)](/images/equations/OrthonormalFunctions/Inline4.svg)
, where 
is a nonnegative measurable
function. For distinct indices 
, the functions
are orthogonal and normalized so that
 |  |  |
(1)
|
 |  |  |
(2)
|
Together with orthogonality, these conditions can be written for every pair of indices 
, including 
, as
 |
(3)
|
where 
is the Kronecker delta (Axler 2026, p. 237).
The functions are identified when equal almost everywhere with respect to 
, and the bar denotes the
complex conjugate. For real
functions, the conjugation can be omitted and the squared complex
moduli in the normalizations are ordinary squares (Axler 2026, pp. 202 and
213).
