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[edit]
Assume that is a subset of a vector space The algebraic interior (or radial kernel) of with respect to is the set of all points at which is a radial set. A point is called an internal point of [1][2] and is said to be radial at if for every there exists a real number such that for every This last condition can also be written as where the set
If is a linear subspace of and then this definition can be generalized to the algebraic interior of with respect to is:[4]
Algebraic closure
A point is said to be linearly accessible from a subset if there exists some such that the line segment is contained in [5] The algebraic closure of with respect to , denoted by consists of and all points in that are linearly accessible from [5]
Algebraic Interior (Core)[edit]
In the special case where the set is called the algebraic interior or core of and it is denoted by or Formally, if is a vector space then the algebraic interior of is[6]
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):
If is a Fréchet space, is convex, and is closed in then but in general it is possible to have while is not empty.
Examples[edit]
If then but and
Properties of core[edit]
Suppose
- In general, But if is a convex set then:
- and
- for all then
- is an absorbing subset of a real vector space if and only if [3]
- [7]
- if [7]
Both the core and the algebraic closure of a convex set are again convex.[5] If is convex, and then the line segment is contained in [5]
Relation to topological interior[edit]
Let be a topological vector space, denote the interior operator, and then:
- If is nonempty convex and is finite-dimensional, then [1]
- If is convex with non-empty interior, then [8]
- If is a closed convex set and is a complete metric space, then [9]
Relative algebraic interior[edit]
If then the set is denoted by and it is called the relative algebraic interior of [7] This name stems from the fact that if and only if and (where if and only if ).
Relative interior[edit]
If is a subset of a topological vector space then the relative interior of is the set
Quasi relative interior[edit]
If is a subset of a topological vector space then the quasi relative interior of is the set
In a Hausdorff finite dimensional topological vector space,
See also[edit]
- 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[edit]
- ^ a b Aliprantis & Border 2006, pp. 199–200.
- ^ John Cook (May 21, 1988). "Separation of Convex Sets in Linear Topological Spaces" (PDF). Retrieved November 14, 2012.
- ^ a b Jaschke, Stefan; Kuchler, Uwe (2000). "Coherent Risk Measures, Valuation Bounds, and ()-Portfolio Optimization" (PDF).
- ^ Zălinescu 2002, p. 2.
- ^ a b c d Narici & Beckenstein 2011, p. 109.
- ^ Nikolaĭ Kapitonovich Nikolʹskiĭ (1992). Functional analysis I: linear functional analysis. Springer. ISBN 978-3-540-50584-6.
- ^ a b c Zălinescu 2002, pp. 2–3.
- ^ Kantorovitz, Shmuel (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.
Bibliography[edit]
- Aliprantis, Charalambos D.; Border, Kim C. (2006). Infinite Dimensional Analysis: A Hitchhiker's Guide (Third ed.). Berlin: Springer Science & Business Media. ISBN 978-3-540-29587-7. OCLC 262692874.
- 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. Vol. 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.
- Zălinescu, Constantin (30 July 2002). Convex Analysis in General Vector Spaces. River Edge, N.J. London: World Scientific Publishing. ISBN 978-981-4488-15-0. MR 1921556. OCLC 285163112 – via Internet Archive.