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 ''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 starting at and ending at
this line segment is a subset of which is the emanating from in the direction of .
Thus geometrically, an interior point of a subset is a point with the property that in every possible direction contains some line segment starting at and heading in that direction.
The algebraic interior of 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 .
Algebraic closure
A point is said to be from a subset if there exists some such that the line segment is contained in
The algebraic closure of with respect to, denoted by consists of 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
We call A ''algebraically open in X'' if
If is non-empty, then these additional subsets are also useful for the statements of many theorems in convex functional analysis :
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
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 .

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,