On a measure space 
, the set of square integrable L2-functions is an 
-space. Taken together with the L2-inner
product with respect to a measure 
,
 |
(1)
|
the 
-space forms a Hilbert
space. The functions in an 
-space satisfy
 |
(2)
|
and
 |
(3)
|
 |
(4)
|
 |
(5)
|
 |
(6)
|
 |
(7)
|
The inequality (7) is called Schwarz's inequality.
The basic example is when 
with Lebesgue measure. Another important example
is when 
is the positive integers, in which case
it is denoted as 
,
or "little ell-two." These are the square summable series.
Strictly speaking, 
-space
really consists of equivalence classes of functions.
Two functions represent the same 
-function if the set where they differ has measure zero.
It is not hard to see that this makes 
an inner product, because 
if and only if 
almost everywhere.
A good way to think of an 
-function
is as a density function, so only its integral on sets with positive measure matter.
In practice, this does not cause much trouble, except that some care has to be taken with boundary conditions in differential equations.
The problem is that for any particular point 
, the value 
isn't well-defined for
an 
-function 
.
If an 
-function in Euclidean
space can be represented by a continuous function 
, then 
is the only continuous representative. In such a case, it is not harmful to consider
the 
-function as the continuous function
. Also, it is often convenient to think
of 
as the completion
of the continuous functions with respect to the L2-norm.
