Algebraic interior

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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 is a linear subspace of and then the algebraic interior of with respect to is:[4]

where it is clear that and if then , where is the affine hull of (which is equal to ).

Algebraic Interior (Core)[]

The set is called the algebraic interior of or the core of and it is denoted by or . Formally, if is a vector space then the algebraic interior of is

[5]

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's possible to have while is not empty.

Example[]

If then , but and

Properties of core[]

If then:

  • In general,
  • If is a convex set then:
    • and
    • for all then
  • is absorbing if and only if [1]
  • [6]
  • if [6]

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 [7]
  • If is a closed convex set and is a complete metric space, then [8]

Relative algebraic interior[]

If then the set is denoted by and it is called the relative algebraic interior of [6] This name stems from the fact that if and only if and (where if and only if ).

Relative interior[]

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

That is, it is the topological interior of A in , which is the smallest affine linear subspace of containing The following set is also useful:

Quasi relative interior[]

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[]

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

References[]

  1. ^ a b Jaschke, Stefan; Kuchler, Uwe (2000). "Coherent Risk Measures, Valuation Bounds, and ()-Portfolio Optimization". Cite journal requires |journal= (help)
  2. ^ a b 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.
  3. ^ John Cook (May 21, 1988). "Separation of Convex Sets in Linear Topological Spaces" (pdf). Retrieved November 14, 2012.
  4. ^ Zălinescu 2002, p. 2.
  5. ^ Nikolaĭ Kapitonovich Nikolʹskiĭ (1992). Functional analysis I: linear functional analysis. Springer. ISBN 978-3-540-50584-6.
  6. ^ a b c 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.
  7. ^ Shmuel Kantorovitz (2003). Introduction to Modern Analysis. Oxford University Press. p. 134. ISBN 9780198526568.
  8. ^ 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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