The set of 
-functions
(where 
) generalizes L2-space.
Instead of square integrable, the measurable
function 
must be 
-integrable for 
to be in 
.
On a measure space 
, the 
norm of a function 
is
 |
The 
-functions are the functions for which
this integral converges. For 
,
the space of 
-functions
is a Banach space which is not a Hilbert
space.
The 
-space on 
, and in most other cases, is the completion
of the continuous functions with compact support
using the 
norm. As in the case of an L2-space, an 
-function is really an equivalence class of functions which
agree almost everywhere. It is possible for
a sequence of functions 
to converge in 
but not in 
for some other 
,
e.g., 
converges in 
but not 
. However, if a sequence converges in 
and in 
, then its limit must be the same in both spaces.
For 
, the dual
vector space to 
is given by integrating against functions in 
, where 
. This makes sense because of Hölder's
inequality for integrals. In particular, the only 
-space which is self-dual is
.
While the use of 
functions is not as common as 
, they are very important in analysis
and partial differential equations.
For instance, some operators are only bounded
in 
for some 
.
