Orthogonal functions 
and 
in the L2-space ![L^2([a,b],w(x)dx)](/images/equations/OrthogonalFunctions/Inline3.svg)
satisfy
 |
(1)
|
Here 
is a nonnegative measurable
function on the closed interval ![[a,b]](/images/equations/OrthogonalFunctions/Inline5.svg)
, 
, and the bar denotes the complex
conjugate. The inner product is linear in its
first argument (Axler 2026, pp. 213 and 216). For real
functions, this reduces to 
.
Elements of this L2-space are equivalence classes of square integrable functions,
identifying those equal almost everywhere with
respect to 
(Axler 2026, p. 202). Orthogonality allows either function
to represent the zero element. To normalize an orthogonal pair, both functions
must have nonzero norm.
If, in addition,
 |  |  |
(2)
|
 |  |  |
(3)
|
the functions are orthonormal. For real functions, the squared complex moduli reduce to ordinary squares.
