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. 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]}
Formally, if is a linear space then the algebraic interior of is
- ^{[4]}
Note that in general , but if is a convex set then . If is a convex set then if then .
Example
If such that then , but and .
Properties
Let then:
- is absorbing if and only if .^{[1]}
- ^{[5]}
- if ^{[5]}
Relation to interior
Let be a topological vector space, denote the interior operator, and then:
- If is nonempty convex and is finite-dimensional, then ^{[2]}
- If is convex with non-empty interior, then ^{[6]}
- If is a closed convex set and is a complete metric space, then ^{[7]}
See also
References
- 1 2 Jaschke, Stefan; Kuchler, Uwe (2000). "Coherent Risk Measures, Valuation Bounds, and ()-Portfolio Optimization".
- 1 2 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.
- ↑ Nikolaĭ Kapitonovich Nikolʹskiĭ (1992). Functional analysis I: linear functional analysis. Springer. ISBN 978-3-540-50584-6.
- 1 2 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.
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