In
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
, a branch of mathematics, the algebraic interior or radial kernel of a subset of a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
is a refinement of the concept of the
interior
Interior may refer to:
Arts and media
* ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas
* ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck
* ''The Interior'' (novel), by Lisa See
* Interior de ...
.
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 In mathematics, a subset A \subseteq X of a linear space X is radial at a given point a_0 \in A if for every x \in X there exists a real t_x > 0 such that for every t \in , t_x a_0 + t x \in A.
Geometrically, this means A is radial at a_0 if for e ...
.
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