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. It is the subset of points contained in a given set with respect to which it is absorbing, i.e. the radial points of the set. The elements of the algebraic interior are often referred to as internal points.
If M is a linear subspace of X and then the algebraic interior of with respect to M is:
where it is clear that and if then, where is the affine hull of .

Algebraic Interior (Core)

The set is called the algebraic interior of A or the core of A and it is denoted by or.
Formally, if is a vector space then the algebraic interior of is
If A is non-empty, then these additional subsets are also useful for the statements of many theorems in convex functional analysis :
If X is a Fréchet space, A is convex, and is closed in X then but in general it's possible to have while is not empty.

Example

If then, but and.

Properties of core

If then:
Let be a topological vector space, denote the interior operator, and then:
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 .

Relative interior

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

Quasi relative interior

If A is a subset of a topological vector space X then the quasi relative interior of A is the set
In a Hausdorff finite dimensional topological vector space,.