Informally, an 
-function
is a function 
that is square integrable, i.e.,
 |
with respect to the measure 
, exists (and is finite), in which case 
is its L2-norm. Here 
is a measure space and the
integral is the Lebesgue integral. The collection
of 
functions on 
is called 
(ell-two) of L2-space, which
is a Hilbert space.
On the unit interval 
,
the functions 
are in 
for 
. However, the function 
is not in 
since
 |
does not exist.
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 absolute square of a wavefunction 
. In this and in the context of energy density, 
-functions arise due to the requirement that these quantities
remain finite.
