Bipolar theorem
In mathematics, the bipolar theorem is a theorem in functional analysis that characterizes the bipolar (that is, the polar of the polar) of a set. 
In convex analysis, the bipolar theorem refers to a necessary and sufficient conditions for a cone to be equal to its bipolar.  The bipolar theorem can be seen as a special case of the Fenchel–Moreau theorem.

Preliminaries

Suppose that  is a topological vector space (TVS) with a continuous dual space  and let  for all  and  
The convex hull of a set  denoted by  is the smallest convex set containing  
The convex balanced hull of a set  is the smallest convex balanced set containing 

The polar of a subset  is defined to be: 
 
while the prepolar of a subset  is:

The bipolar of a subset  often denoted by  is the set

Statement in functional analysis

Let  denote the weak topology on  (that is, the weakest TVS topology on  making all linear functionals in  continuous).

The bipolar theorem: The bipolar of a subset  is equal to the -closure of the convex balanced hull of

Statement in convex analysis

The bipolar theorem: For any nonempty cone  in some linear space  the bipolar set  is given by:

Special case

A subset  is a nonempty closed convex cone if and only if  when  where  denotes the positive dual cone of a set 
Or more generally, if  is a nonempty convex cone then the bipolar cone is given by

Relation to the Fenchel–Moreau theorem

Let 
 
be the indicator function for a cone 
Then the convex conjugate, 
 
is the support function for  and  
Therefore,  if and only if

See also

 
  − A generalization of the bipolar theorem.

References

Bibliography

  
  
  

Convex analysis
Functional analysis
Theorems in analysis
Linear functionals