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. 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 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. If then but and 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.