An 
-space
(ell-infinity) on a measure space 
is the Banach space of
equivalence classes of essentially bounded measurable functions 
, where 
or 
, identified when equal almost
everywhere with respect to 
. It generalizes the Lp-spaces
to 
.
No integration is used in its definition. Instead,
the norm is
 |
The space consists of those equivalence classes with 
(Axler 2026, pp. 194 and 202-205). When 
, this norm is the essential
supremum of 
.
The restriction to nonnegative bounds also gives
when 
,
in which case every measurable function represents
the zero class. The unrestricted essential supremum
would instead be 
on such a measure space.
For Lebesgue measure on 
, the space of bounded continuous functions is not dense
in 
.
For example, the indicator function of 
has distance 
from this space: every continuous
function has essential error at least 
, and the constant function 
attains this value. On a discrete
space with counting measure, every bounded
function is continuous.
