An 
-space on a measure
space 
,
for a real number 
with 
,
consists of equivalence classes of measurable
functions 
,
where 
or 
, whose 
th
powers of absolute values
are integrable. Two functions
are identified when they agree almost everywhere
with respect to 
(Axler 2026, pp. 202-203). This generalizes the L2-space
of square integrable functions.
The 
norm is
 |
The 
-space is a Banach
space (Axler 2026, p. 205). For 
, its norm does not arise from
an inner product when the measure
space contains two disjoint sets of finite positive measure. A one-point measure space whose
point has finite positive measure instead gives a one-dimensional Hilbert
space for every such 
.
The 
-space on 
with Lebesgue measure
is the completion of the continuous
functions with compact support using the 
norm. It
is possible for a sequence of functions 
to converge
in 
but not in 
for some other 
. For example, 
converges
in 
to 
, which is not in 
. Although every 
is in 
, the sequence cannot converge there. If a sequence converges in both 
and 
on the same measure space, its limits
agree almost everywhere (Axler 2026, p. 205).
For 
, the dual
normed space to 
is identified with 
,
where 
. More precisely, every bounded linear functional is uniquely of the form
 |
where 
and 
(Axler 2026, p. 275). The integral
exists by Hölder's inequality. In particular,
is self-dual.
For complex functions, the inner
product form 
of this identification is antilinear (Axler 2026,
pp. 233-234). On degenerate measure spaces,
other 
-spaces can also be self-dual,
as in the one-dimensional example above.
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 
.
