An 
-space
(ell-infinity) is the Banach space of equivalence
classes of essentially bounded measurable
functions, identified when equal almost everywhere.
It generalizes the Lp-spaces
to 
.
No integration is used to define them, and instead, the norm
on 
is given by the essential supremum.
More precisely,
 |
is the norm which makes 
a Banach space. It is
the space of all essentially bounded functions.
For Lebesgue measure on 
, the space of bounded continuous functions is not dense
in 
.
On a discrete space with counting
measure, every bounded function is continuous.
