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.
Bonnans, J. Frederic; Shapiro, Alexander (2000), Perturbation Analysis of Optimization Problems, Springer series in operations research, Springer, Remark 2.73, p. 56, ISBN 9780387987057.
Zalinescu, C. (2002). Convex Analysis in General Vector Spaces. World Scientific. ISBN 978-981-238-067-8.
vte
Functional analysis (topics – glossary)
Spaces
Hilbert space Banach space Fréchet space topological vector space
Theorems
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
Operators
bounded operator compact operator adjoint operator unitary operator Hilbert–Schmidt operator trace class unbounded operator
Algebras
Banach algebra C*-algebra spectrum of a C*-algebra operator algebra group algebra of a locally compact group von Neumann algebra
Open problems
invariant subspace problem Mahler's conjecture
Applications
Besov space Hardy space spectral theory of ordinary differential equations heat kernel index theorem calculus of variation functional calculus integral operator Jones polynomial topological quantum field theory noncommutative geometry Riemann hypothesis
Advanced topics
locally convex space approximation property balanced set Schwartz space weak topology barrelled space Banach–Mazur distance Tomita–Takesaki theory
vte
Topological vector spaces (TVSs)
Basic concepts
Banach space Continuous linear operator Functionals Hilbert space Linear operators Locally convex space Homomorphism Topological vector space Vector space
Main results
Closed graph theorem F. Riesz's theorem Hahn–Banach (hyperplane separation Vector-valued Hahn–Banach) Open mapping (Banach–Schauder) (Bounded inverse) Uniform boundedness (Banach–Steinhaus)
Maps
Almost open Bilinear (form operator) and Sesquilinear forms Closed Compact operator Continuous and Discontinuous Linear maps Densely defined Homomorphism Functionals Norm Operator Seminorm Sublinear Transpose
Types of sets
Absolutely convex/disk Absorbing/Radial Affine Balanced/Circled Banach disks Bounding points Bounded Complemented subspace Convex Convex cone (subset) Linear cone (subset) Extreme point Pre-compact/Totally bounded Radial Radially convex/Star-shaped Symmetric
Set operations
Affine hull (Relative) Algebraic interior (core) Convex hull Linear span Minkowski addition Polar (Quasi) Relative interior
Types of TVSs
Asplund B-complete/Ptak Banach (Countably) Barrelled (Ultra-) Bornological Brauner Complete (DF)-space Distinguished F-space Fréchet (tame Fréchet) Grothendieck Hilbert Infrabarreled Interpolation space LB-space LF-space Locally convex space Mackey (Pseudo)Metrizable Montel Quasibarrelled Quasi-complete Quasinormed (Polynomially Semi-) Reflexive Riesz Schwartz Semi-complete Smith Stereotype (B Strictly Uniformly convex (Quasi-) Ultrabarrelled Uniformly smooth Webbed With the approximation property
Undergraduate Texts in Mathematics
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Hellenica World - Scientific Library
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