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 $A$ is a subset of a vector space $X.$
The ''algebraic interior'' (or ''radial kernel'') ''of $A$ with respect to $X$'' is the set of all points at which $A$ is a radial set.
A point $a_0 \in A$ is called an of $A$ and $A$ is said to be if for every $x \in X$ there exists a real number $t_x > 0$ such that for every $t \in$, t_x $a_0 + t x \in A.$
This last condition can also be written as $a_0 +$, t_xx \subseteq A where the set
$a_0 +$, t_xx ~:=~ \left\
is the line segment (or closed interval) starting at $a_0$ and ending at $a_0 + t_x x;$
this line segment is a subset of $a_0 +$radial points of the set.

If $M$ is a linear subspace of $X$ and $A \subseteq X$ then this definition can be generalized to the ''algebraic interior of $A$ with respect to $M$'' is:
$\operatorname_M A := \left\.$
where $\operatorname_M A \subseteq A$ always holds and if $\operatorname_M A \neq \varnothing$ then $M \subseteq \operatorname (A - A),$ where $\operatorname (A - A)$ is the affine hull of $A - A$ (which is equal to $\operatorname(A - A)$).

Algebraic closure

A point $x \in X$ is said to be from a subset $A \subseteq X$ if there exists some $a \in A$ such that the line segment $[a, x) := a + [0, 1) x$ is contained in $A.$
The , denoted by $\operatorname_X A,$ consists of $A$ and all points in $X$ that are linearly accessible from $A.$

Algebraic Interior (Core)

In the special case where $M := X,$ the set $\operatorname_X A$ is called the ' or '' of $A$'' and it is denoted by $A^i$ or $\operatorname A.$
Formally, if $X$ is a vector space then the algebraic interior of $A \subseteq X$ is
$\operatorname_X A := \operatorname(A) := \left\.$

If $A$ 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):

$^ A := \begin ^i A & \text \operatorname A \text \\ \varnothing & \text \end$

$^ A := \begin ^i A & \text \operatorname (A - a) \text X \text a \in A \text \\ \varnothing & \text \end$

If $X$ is a Fréchet space, $A$ is convex, and $\operatorname A$ is closed in $X$ then $^ A = ^ A$ but in general it is possible to have $^ A = \varnothing$ while $^ A$ is empty.

Examples

If $A = \ \subseteq \R^2$ then $0 \in \operatorname(A),$ but $0 \not\in \operatorname(A)$ and $0 \not\in \operatorname(\operatorname(A)).$

Properties of core

Suppose $A, B \subseteq X.$
* In general, $\operatorname A \neq \operatorname(\operatorname A).$ But if $A$ is a convex set then:
** $\operatorname A = \operatorname(\operatorname A),$ and
** for all $x_0 \in \operatorname A, y \in A, 0 < \lambda \leq 1$ then $\lambda x_0 + (1 - \lambda)y \in \operatorname A.$
* $A$ is an absorbing subset of a real vector space if and only if $0 \in \operatorname(A).$
* $A + \operatorname B \subseteq \operatorname(A + B)$
* $A + \operatorname B = \operatorname(A + B)$ if $B = \operatornameB.$

Both the core and the algebraic closure of a convex set are again convex.
If $C$ is convex, $c \in \operatorname C,$ and $b \in \operatorname_X C$ then the line segment , b) := c + [0, 1) b is contained in $\operatorname C.$

Relation to topological interior

Let $X$ be a topological vector space, $\operatorname$ denote the interior operator, and $A \subseteq X$ then:
* $\operatornameA \subseteq \operatornameA$
* If $A$ is nonempty convex and $X$ is finite-dimensional, then $\operatorname A = \operatorname A.$
* If $A$ is convex with non-empty interior, then $\operatornameA = \operatorname A.$
* If $A$ is a closed convex set and $X$ is a complete metric space, then $\operatorname A = \operatorname A.$.

Relative algebraic interior

If $M = \operatorname (A - A)$ then the set $\operatorname_M A$ is denoted by $^iA := \operatorname_ A$ and it is called ''the relative algebraic interior of $A.$'' This name stems from the fact that $a \in A^i$ if and only if $\operatorname A = X$ and $a \in ^iA$ (where $\operatorname A = X$ if and only if $\operatorname (A - A) = X$).

Relative interior

If $A$ is a subset of a topological vector space $X$ then the ''relative interior'' of $A$ is the set
$\operatorname A := \operatorname_ A.$
That is, it is the topological interior of A in $\operatorname A,$ which is the smallest affine linear subspace of $X$ containing $A.$ The following set is also useful:
$\operatorname A := \begin \operatorname A & \text \operatorname A \text X \text \\ \varnothing & \text \end$

Quasi relative interior

If $A$ is a subset of a topological vector space $X$ then the ''quasi relative interior'' of $A$ is the set
$\operatorname A := \left\.$

In a Hausdorff finite dimensional topological vector space, $\operatorname A = ^i A = ^ A = ^ A.$

See also

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Reference

Bibliography

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Convex analysis
Functional analysis
Mathematical analysis
Topology