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.
The algebraic interior (or radial kernel) of
is called an internal point of
there exists a real number
is the line segment (or closed interval) starting at
this line segment is a subset of
Thus geometrically, an interior point of a subset
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
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.
then this definition can be generalized to the algebraic interior of
is said to be linearly accessible from a subset
is called the algebraic interior or core of
is a vector space then the algebraic interior of
{\displaystyle \operatorname {aint} _{X}A:=\operatorname {core} (A):=\left\{a\in A:{\text{ for all }}x\in X,{\text{ there exists some }}t_{x}>0,{\text{ such that for all }}t\in \left[0,t_{x}\right],a+tx\in A\right\}.}
We call A algebraically open in X 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):
is a barrelled linear subspace of
{\displaystyle {}^{ib}A:={\begin{cases}{}^{i}A&{\text{ if }}\operatorname {span} (A-a){\text{ is a barrelled linear subspace of }}X{\text{ for any/all }}a\in A{\text{,}}\\\varnothing &{\text{ otherwise}}\end{cases}}}
denote the interior operator, and
{\displaystyle {}^{i}A:=\operatorname {aint} _{\operatorname {aff} (A-A)}A}
and it is called the relative algebraic interior of
is a subset of a topological vector space
{\displaystyle \operatorname {rint} A:=\operatorname {int} _{\operatorname {aff} A}A.}
which is the smallest affine linear subspace of
is a subset of a topological vector space
{\displaystyle \operatorname {qri} A:=\left\{a\in A:{\overline {\operatorname {cone} }}(A-a){\text{ is a linear subspace of }}X\right\}.}
In a Hausdorff finite dimensional topological vector space,
{\displaystyle \operatorname {qri} A={}^{i}A={}^{ic}A={}^{ib}A.}