# 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.[1] The elements of the algebraic interior are often referred to as internal points.[2][3]

Formally, if is a linear space then the algebraic interior of is

[4]

Note that in general , but if is a convex set then . If is a convex set then if then .

## Example

If such that then , but and .

## Properties

Let then:

### Relation to interior

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

• If is nonempty convex and is finite-dimensional, then [2]
• If is convex with non-empty interior, then [6]
• If is a closed convex set and is a complete metric space, then [7]