Algebraic interior

Short description: Generalization of topological 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 [math]\displaystyle{ A }[/math] is a subset of a vector space [math]\displaystyle{ X. }[/math] The algebraic interior (or radial kernel) of [math]\displaystyle{ A }[/math] with respect to [math]\displaystyle{ X }[/math] is the set of all points at which [math]\displaystyle{ A }[/math] is a radial set. A point [math]\displaystyle{ a_0 \in A }[/math] is called an internal point of [math]\displaystyle{ A }[/math] and [math]\displaystyle{ A }[/math] is said to be radial at [math]\displaystyle{ a_0 }[/math] if for every [math]\displaystyle{ x \in X }[/math] there exists a real number [math]\displaystyle{ t_x \gt 0 }[/math] such that for every [math]\displaystyle{ t \in [0, t_x], }[/math] [math]\displaystyle{ a_0 + t x \in A. }[/math] This last condition can also be written as [math]\displaystyle{ a_0 + [0, t_x] x \subseteq A }[/math] where the set [math]\displaystyle{ a_0 + [0, t_x] x ~:=~ \left\{a_0 + t x : t \in [0, t_x]\right\} }[/math] is the line segment (or closed interval) starting at [math]\displaystyle{ a_0 }[/math] and ending at [math]\displaystyle{ a_0 + t_x x; }[/math] this line segment is a subset of [math]\displaystyle{ a_0 + [0, \infty) x, }[/math] which is the ray emanating from [math]\displaystyle{ a_0 }[/math] in the direction of [math]\displaystyle{ x }[/math] (that is, parallel to/a translation of [math]\displaystyle{ [0, \infty) x }[/math]). Thus geometrically, an interior point of a subset [math]\displaystyle{ A }[/math] is a point [math]\displaystyle{ a_0 \in A }[/math] with the property that in every possible direction (vector) [math]\displaystyle{ x \neq 0, }[/math] [math]\displaystyle{ A }[/math] contains some (non-degenerate) line segment starting at [math]\displaystyle{ a_0 }[/math] and heading in that direction (i.e. a subset of the ray [math]\displaystyle{ a_0 + [0, \infty) x }[/math]). The algebraic interior of [math]\displaystyle{ A }[/math] (with respect to [math]\displaystyle{ X }[/math]) 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 [math]\displaystyle{ M }[/math] is a linear subspace of [math]\displaystyle{ X }[/math] and [math]\displaystyle{ A \subseteq X }[/math] then this definition can be generalized to the algebraic interior of [math]\displaystyle{ A }[/math] with respect to [math]\displaystyle{ M }[/math] is: [math]\displaystyle{ \operatorname{aint}_M A := \left\{ a \in X : \text{ for all } m \in M, \text{ there exists some } t_m \gt 0 \text{ such that } a + \left[0, t_m\right] \cdot m \subseteq A \right\}. }[/math] where [math]\displaystyle{ \operatorname{aint}_M A \subseteq A }[/math] always holds and if [math]\displaystyle{ \operatorname{aint}_M A \neq \varnothing }[/math] then [math]\displaystyle{ M \subseteq \operatorname{aff} (A - A), }[/math] where [math]\displaystyle{ \operatorname{aff} (A - A) }[/math] is the affine hull of [math]\displaystyle{ A - A }[/math] (which is equal to [math]\displaystyle{ \operatorname{span}(A - A) }[/math]).

Algebraic closure

A point [math]\displaystyle{ x \in X }[/math] is said to be linearly accessible from a subset [math]\displaystyle{ A \subseteq X }[/math] if there exists some [math]\displaystyle{ a \in A }[/math] such that the line segment [math]\displaystyle{ [a, x) := a + [0, 1) x }[/math] is contained in [math]\displaystyle{ A. }[/math] The algebraic closure of [math]\displaystyle{ A }[/math] with respect to [math]\displaystyle{ X }[/math], denoted by [math]\displaystyle{ \operatorname{acl}_X A, }[/math] consists of [math]\displaystyle{ A }[/math] and all points in [math]\displaystyle{ X }[/math] that are linearly accessible from [math]\displaystyle{ A. }[/math]

Algebraic Interior (Core)

In the special case where [math]\displaystyle{ M := X, }[/math] the set [math]\displaystyle{ \operatorname{aint}_X A }[/math] is called the algebraic interior or core of [math]\displaystyle{ A }[/math] and it is denoted by [math]\displaystyle{ A^i }[/math] or [math]\displaystyle{ \operatorname{core} A. }[/math] Formally, if [math]\displaystyle{ X }[/math] is a vector space then the algebraic interior of [math]\displaystyle{ A \subseteq X }[/math] is [math]\displaystyle{ \operatorname{aint}_X A := \operatorname{core}(A) := \left\{ a \in A : \text{ for all } x \in X, \text{ there exists some } t_x \gt 0, \text{ such that for all } t \in \left[0, t_x\right], a + tx \in A \right\}. }[/math]

If [math]\displaystyle{ A }[/math] 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):

[math]\displaystyle{ {}^{ic} A := \begin{cases} {}^i A & \text{ if } \operatorname{aff} A \text{ is a closed set,} \\ \varnothing & \text{ otherwise} \end{cases} }[/math]

[math]\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} }[/math]

If [math]\displaystyle{ X }[/math] is a Fréchet space, [math]\displaystyle{ A }[/math] is convex, and [math]\displaystyle{ \operatorname{aff} A }[/math] is closed in [math]\displaystyle{ X }[/math] then [math]\displaystyle{ {}^{ic} A = {}^{ib} A }[/math] but in general it is possible to have [math]\displaystyle{ {}^{ic} A = \varnothing }[/math] while [math]\displaystyle{ {}^{ib} A }[/math] is not empty.

Examples

If [math]\displaystyle{ A = \{x \in \R^2: x_2 \geq x_1^2 \text{ or } x_2 \leq 0\} \subseteq \R^2 }[/math] then [math]\displaystyle{ 0 \in \operatorname{core}(A), }[/math] but [math]\displaystyle{ 0 \not\in \operatorname{int}(A) }[/math] and [math]\displaystyle{ 0 \not\in \operatorname{core}(\operatorname{core}(A)). }[/math]

Properties of core

Suppose [math]\displaystyle{ A, B \subseteq X. }[/math]

• In general, [math]\displaystyle{ \operatorname{core} A \neq \operatorname{core}(\operatorname{core} A). }[/math] But if [math]\displaystyle{ A }[/math] is a convex set then:
  • [math]\displaystyle{ \operatorname{core} A = \operatorname{core}(\operatorname{core} A), }[/math] and
  • for all [math]\displaystyle{ x_0 \in \operatorname{core} A, y \in A, 0 \lt \lambda \leq 1 }[/math] then [math]\displaystyle{ \lambda x_0 + (1 - \lambda)y \in \operatorname{core} A. }[/math]
• [math]\displaystyle{ A }[/math] is an absorbing subset of a real vector space if and only if [math]\displaystyle{ 0 \in \operatorname{core}(A). }[/math]
• [math]\displaystyle{ A + \operatorname{core} B \subseteq \operatorname{core}(A + B) }[/math]
• [math]\displaystyle{ A + \operatorname{core} B = \operatorname{core}(A + B) }[/math] if [math]\displaystyle{ B = \operatorname{core}B. }[/math]

Both the core and the algebraic closure of a convex set are again convex. If [math]\displaystyle{ C }[/math] is convex, [math]\displaystyle{ c \in \operatorname{core} C, }[/math] and [math]\displaystyle{ b \in \operatorname{acl}_X C }[/math] then the line segment [math]\displaystyle{ [c, b) := c + [0, 1) b }[/math] is contained in [math]\displaystyle{ \operatorname{core} C. }[/math]

Relation to topological interior

Let [math]\displaystyle{ X }[/math] be a topological vector space, [math]\displaystyle{ \operatorname{int} }[/math] denote the interior operator, and [math]\displaystyle{ A \subseteq X }[/math] then:

• [math]\displaystyle{ \operatorname{int}A \subseteq \operatorname{core}A }[/math]
• If [math]\displaystyle{ A }[/math] is nonempty convex and [math]\displaystyle{ X }[/math] is finite-dimensional, then [math]\displaystyle{ \operatorname{int} A = \operatorname{core} A. }[/math]
• If [math]\displaystyle{ A }[/math] is convex with non-empty interior, then [math]\displaystyle{ \operatorname{int}A = \operatorname{core} A. }[/math]
• If [math]\displaystyle{ A }[/math] is a closed convex set and [math]\displaystyle{ X }[/math] is a complete metric space, then [math]\displaystyle{ \operatorname{int} A = \operatorname{core} A. }[/math]

Relative algebraic interior

If [math]\displaystyle{ M = \operatorname{aff} (A - A) }[/math] then the set [math]\displaystyle{ \operatorname{aint}_M A }[/math] is denoted by [math]\displaystyle{ {}^iA := \operatorname{aint}_{\operatorname{aff} (A - A)} A }[/math] and it is called the relative algebraic interior of [math]\displaystyle{ A. }[/math] This name stems from the fact that [math]\displaystyle{ a \in A^i }[/math] if and only if [math]\displaystyle{ \operatorname{aff} A = X }[/math] and [math]\displaystyle{ a \in {}^iA }[/math] (where [math]\displaystyle{ \operatorname{aff} A = X }[/math] if and only if [math]\displaystyle{ \operatorname{aff} (A - A) = X }[/math]).

Relative interior

If [math]\displaystyle{ A }[/math] is a subset of a topological vector space [math]\displaystyle{ X }[/math] then the relative interior of [math]\displaystyle{ A }[/math] is the set [math]\displaystyle{ \operatorname{rint} A := \operatorname{int}_{\operatorname{aff} A} A. }[/math] That is, it is the topological interior of A in [math]\displaystyle{ \operatorname{aff} A, }[/math] which is the smallest affine linear subspace of [math]\displaystyle{ X }[/math] containing [math]\displaystyle{ A. }[/math] The following set is also useful: [math]\displaystyle{ \operatorname{ri} A := \begin{cases} \operatorname{rint} A & \text{ if } \operatorname{aff} A \text{ is a closed subspace of } X \text{,} \\ \varnothing & \text{ otherwise} \end{cases} }[/math]

Quasi relative interior

If [math]\displaystyle{ A }[/math] is a subset of a topological vector space [math]\displaystyle{ X }[/math] then the quasi relative interior of [math]\displaystyle{ A }[/math] is the set [math]\displaystyle{ \operatorname{qri} A := \left\{ a \in A : \overline{\operatorname{cone}} (A - a) \text{ is a linear subspace of } X \right\}. }[/math]

In a Hausdorff finite dimensional topological vector space, [math]\displaystyle{ \operatorname{qri} A = {}^i A = {}^{ic} A = {}^{ib} A. }[/math]

See also

• Bounding point – Mathematical concept related to subsets of vector spaces
• Interior (topology) – Largest open subset of some given set
• Order unit – Element of an ordered vector space
• Quasi-relative interior – Generalization of algebraic interior
• Radial set
• Relative interior – Generalization of topological interior
• Ursescu theorem – Generalization of closed graph, open mapping, and uniform boundedness theorem

References

Bibliography

• Template:Aliprantis Border Infinite Dimensional Analysis A Hitchhiker's Guide Third Edition
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. 
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. 
• Schechter, Eric (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN 978-0-12-622760-4. OCLC 175294365. 
• Template:Zălinescu Convex Analysis in General Vector Spaces 2002