Algebraic interior
In functional analysis, a branch of mathematics, the algebraic interior or radial kernel of a subset of a vector space is a refinement of the concept of the interior. Definition

Assume that  is a subset of a vector space  
The algebraic interior (or radial kernel) of  with respect to  is the set of all points at which  is a radial set. A point  is called an  of  and  is said to be  if for every  there exists a real number  such that for every   
This last condition can also be written as  where the set 
 
is the line segment (or closed interval) starting at  and ending at  
this line segment is a subset of  which is the  emanating from  in the direction of  (that is, parallel to/a translation of ). Thus geometrically, an interior point of a subset  is a point  with the property that in every possible direction (vector)   contains some (non-degenerate) line segment starting at  and heading in that direction (i.e. a subset of the ray ). The algebraic interior of  (with respect to ) is the set of all such points. That is to say, it is the subset of points contained in a given set with respect to which it is radial points of the set. If  is a linear subspace of  and  then this definition can be generalized to the algebraic interior of  with respect to  is:

where  always holds and if  then  where  is the affine hull of  (which is equal to ). Algebraic closure

A point  is said to be  from a subset  if there exists some  such that the line segment  is contained in  
The , denoted by  consists of  and all points in  that are linearly accessible from

Algebraic Interior (Core)

In the special case where  the set  is called the  or  of  and it is denoted by  or  
Formally, if  is a vector space then the algebraic interior of  is 

If  is non-empty, then these additional subsets are also useful for the statements of many theorems in convex functional analysis (such as the Ursescu theorem):

If  is a Fréchet space,  is convex, and  is closed in  then  but in general it is possible to have  while  is  empty. Examples

If  then  but  and

Properties of core

Suppose  
 In general,  But if  is a convex set then:
  and
 for all  then 
  is an absorbing subset of a real vector space if and only if 
 
  if 

Both the core and the algebraic closure of a convex set are again convex. If  is convex,  and  then the line segment  is contained in

Relation to topological interior

Let  be a topological vector space,  denote the interior operator, and  then: 
 
 If  is nonempty convex and  is finite-dimensional, then 
 If  is convex with non-empty interior, then 
 If  is a closed convex set and  is a complete metric space, then

Relative algebraic interior

If  then the set  is denoted by  and it is called the relative algebraic interior of  This name stems from the fact that  if and only if  and  (where  if and only if ). Relative interior

If  is a subset of a topological vector space  then the relative interior of  is the set
 
That is, it is the topological interior of A in  which is the smallest affine linear subspace of  containing  The following set is also useful:

Quasi relative interior

If  is a subset of a topological vector space  then the quasi relative interior of  is the set

In a Hausdorff finite dimensional topological vector space,

See also

References

Bibliography

  
  
  
  
  

Convex analysis
Functional analysis
Mathematical analysis
Topology