The Riesz representation theorem has two standard forms. For a Hilbert space 
,
every bounded linear
functional 
on 
has a unique representing vector 
such that 
for every 
,
and 
.
Here the inner product is linear in its first argument.
The Riesz-Markov-Kakutani form concerns a locally compact T2-space 
. Every bounded linear
functional 
on the function space 
of continuous complex-valued
functions vanishing at infinity, equipped
with the supremum norm, is represented by a unique
regular complex measure 
on the Borel sets of 
such that
 |
Moreover, 
,
where 
is the total variation of 
. The linear functional 
is positive iff 
is a positive measure.
Here regularity means that 
is a regular Borel
measure, using both outer approximation by open sets
and inner approximation by compact sets. In particular,
(Rudin 1987, pp. 129-131).
