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 
.
The Riesz-Markov-Kakutani form concerns a locally compact T2-space 
.
Every bounded linear functional 
on the space 
of continuous complex-valued functions vanishing at infinity
is represented by a unique regular complex Borel measure 
such that
 |
Moreover, 
,
where 
is the total variation of 
. The functional 
is positive iff 
is a positive measure.
