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. It is the subset of points contained in a given set with respect to which it is absorbing, i.e. the radial points of the set.[1] The elements of the algebraic interior are often referred to as internal points.[2][3]

If M is a linear subspace of X and \( A\subseteq X \) then the algebraic interior ofΑ with respect to M is:[4]

\( {\displaystyle \operatorname {aint} _{M}A:=\left\{a\in X:\forall m\in M,\exists t_{m}>0{\text{ s.t. }}a+[0,t_{m}]\cdot m\subseteq A\right\}.}

where it is clear that \( {\displaystyle \operatorname {aint} _{M}A\subseteq A} \) and if \( {\displaystyle \operatorname {aint} _{M}A\neq \emptyset } \) then \( {\displaystyle M\subseteq \operatorname {aff} (A-A)} \) , where \( {\displaystyle \operatorname {aff} (A-A)} \) is the affine hull of \( {\displaystyle A-A} \) (which is equal to \( {\displaystyle \operatorname {span} (A-A)}). \)

Algebraic Interior (Core)

The set \( {\displaystyle \operatorname {aint} _{X}A} \) is called the algebraic interior of A or the core of A and it is denoted by \( {\displaystyle A^{i}} \) or \( {\displaystyle \operatorname {core} A} \) . Formally, if X is a vector space then the algebraic interior of \( A\subseteq X \) is

\( {\displaystyle \operatorname {aint} _{X}A:=\operatorname {core} (A):=\left\{a\in A:\forall x\in X,\exists t_{x}>0,\forall t\in [0,t_{x}],a+tx\in A\right\}.} \)[5]

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):

\( {\displaystyle {}^{ic}A:={\begin{cases}{}^{i}A&{\text{ if }}\operatorname {aff} A{\text{ is a closed set,}}\\\emptyset &{\text{ otherwise}}\end{cases}}} \)

\( {\displaystyle {}^{ib}A:={\begin{cases}{}^{i}A&{\text{ if }}\operatorname {span} (A-a){\text{ is a barrelled linear subspace of }}X{\text{ for any/all }}a\in A{\text{,}}\\\emptyset &{\text{ otherwise}}\end{cases}}} \)

If X is a Fréchet space, A is convex, and \( {\displaystyle \operatorname {aff} A} \) is closed in X then \) {\displaystyle {}^{ic}A={}^{ib}A} \) but in general it's possible to have i c A = ∅ {\displaystyle {}^{ic}A=\emptyset } {\displaystyle {}^{ic}A=\emptyset } while \( {\displaystyle {}^{ib}A} \) is not empty.

Example

If \( {\displaystyle A=\{x\in \mathbb {R} ^{2}:x_{2}\geq x_{1}^{2}{\text{ or }}x_{2}\leq 0\}\subseteq \mathbb {R} ^{2}} \) then \( 0\in \operatorname {core}(A) \) , but \( 0\not \in \operatorname {int}(A) \) and \( 0\not \in \operatorname {core}(\operatorname {core}(A)) \).

Properties of core

If \( A,B\subset X \) then:

In general, \( (\operatorname {core} (A))} \operatorname {core}(A)\neq \operatorname {core}(\operatorname {core}(A)). \)

If A is a convex set then:

\( (\operatorname {core} (A))} \operatorname {core}(A)=\operatorname {core}(\operatorname {core}(A)) \), and

for all \( {\displaystyle x_{0}\in \operatorname {core} A,y\in A,0<\lambda \leq 1} \) then \( {\displaystyle \lambda x_{0}+(1-\lambda )y\in \operatorname {core} A} \)

Α is absorbing if and only if \( 0\in \operatorname {core}(A) \).[1]

\( A+\operatorname {core}B\subset \operatorname {core}(A+B) \)[6]

\( A+\operatorname {core}B=\operatorname {core}(A+B)\) if \( B=\operatorname {core}B \)[6]

Relation to interior

Let Χ be a topological vector space, \( \operatorname {int} \) denote the interior operator, and \( A\subset X \) then:

\( \operatorname {int}A\subseteq \operatorname {core}A \)

If Α is nonempty convex and Χ is finite-dimensional, then\( \operatorname {int}A=\operatorname {core}A \)[2]

If Α is convex with non-empty interior, then \( \operatorname {int}A=\operatorname {core}A \) [7]

If Α is a closed convex set and Χ is a complete metric space, then \( \operatorname {int}A=\operatorname {core}A \) [8]

Relative algebraic interior

If \( {\displaystyle M=\operatorname {aff} (A-A)} \) then the set \( {\displaystyle \operatorname {aint} _{M}A} \) is denoted by \( {\displaystyle {}^{i}A:=\operatorname {aint} _{\operatorname {aff} (A-A)}A} \) and it is called the relative algebraic interior ofΑ.[6] This name stems from the fact that \( {\displaystyle a\in A^{i}} \) if and only if aff \( {\displaystyle \operatorname {aff} A=X} \) and \( {\displaystyle a\in {}^{i}A} \) (where aff \( {\displaystyle \operatorname {aff} A=X} \) if and only if \( {\displaystyle \operatorname {aff} \left(A-A\right)=X}). \)

Relative interior

If A is a subset of a topological vector space X then the relative interior of A is the set

\( {\displaystyle \operatorname {rint} A:=\operatorname {int} _{\operatorname {aff} A}A}. \)

That is, it is the topological interior of A in \( {\displaystyle \operatorname {aff} A} \), which is the smallest affine linear subspace of X containing A. The following set is also useful:

\( {\displaystyle \operatorname {ri} A:={\begin{cases}\operatorname {rint} A&{\text{ if }}\operatorname {aff} A{\text{ is a closed subspace of }}X{\text{,}}\\\emptyset &{\text{ otherwise}}\end{cases}}} \)

Quasi relative interior

If A is a subset of a topological vector space X then the quasi relative interior of A is the set

\( {\displaystyle \operatorname {qri} A:=\left\{a\in A:{\overline {\operatorname {cone} }}(A-a){\text{ is a linear subspace of }}X\right\}}. \)

In a Hausdorff finite dimensional topological vector space,\( {\displaystyle \operatorname {qri} A={}^{i}A={}^{ic}A={}^{ib}A}. \)

See also

Bounding point

Interior (topology)

Quasi-relative interior

Relative interior

Order unit

Ursescu theorem

References

Jaschke, Stefan; Kuchler, Uwe (2000). "Coherent Risk Measures, Valuation Bounds, and ( μ , ρ )-Portfolio Optimization".

Aliprantis, C.D.; Border, K.C. (2007). Infinite Dimensional Analysis: A Hitchhiker's Guide (3rd ed.). Springer. pp. 199–200. doi:10.1007/3-540-29587-9. ISBN 978-3-540-32696-0.

John Cook (May 21, 1988). "Separation of Convex Sets in Linear Topological Spaces" (pdf). Retrieved November 14, 2012.

Zalinescu 2002, p. 2.

Nikolaĭ Kapitonovich Nikolʹskiĭ (1992). Functional analysis I: linear functional analysis. Springer. ISBN 978-3-540-50584-6.

Zălinescu, C. (2002). Convex analysis in general vector spaces. River Edge, NJ: World Scientific Publishing Co., Inc. pp. 2–3. ISBN 981-238-067-1. MR 1921556.

Shmuel Kantorovitz (2003). Introduction to Modern Analysis. Oxford University Press. p. 134. ISBN 9780198526568.

vte

Functional analysis (topics – glossary)

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Hahn–Banach theorem closed graph theorem uniform boundedness principle Kakutani fixed-point theorem Krein–Milman theorem min-max theorem Gelfand–Naimark theorem Banach–Alaoglu theorem

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