An 
-function is a function 
that is square
integrable with respect to a measure 
, meaning that
 |
The quantity 
is its L2-norm. Here 
is a measure space and the
integral is the Lebesgue
integral. The collection of 
-functions on 
, identified when they agree almost
everywhere, is called 
or L2-space, which is a Hilbert
space.
On the unit interval 
, the functions 
are in 
for 
. However, the function 
is not in 
since
 |
diverges.
More generally, there are 
-complex functions obtained by replacing the absolute
value of a real number in the definition with
the norm of the complex number.
In fact, this generalizes to functions from a measure
space 
to any normed space.
-functions play an important role in
many areas of analysis. They also arise in physics,
and especially quantum mechanics, where probabilities
are given as the integral of the squared absolute
value of a wavefunction 
. In this and in the context of energy density, 
-functions arise due to the requirement that these quantities
remain finite.
