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 $X$ is a linear space then the algebraic interior of $A\subseteq X$ is

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

^[4]

Note that in general $\operatorname {core}(A)\neq \operatorname {core}(\operatorname {core}(A))$ , but if $A$ is a convex set then $\operatorname {core}(A)=\operatorname {core}(\operatorname {core}(A))$ . If $A$ is a convex set then if $x_{0}\in \operatorname {core}(A),y\in A,0<\lambda \leq 1$ then $\lambda x_{0}+(1-\lambda )y\in \operatorname {core}(A)$ .

Example

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

Properties

Let $A,B\subset X$ then:

$A$ is absorbing if and only if $0\in \operatorname {core}(A)$ .^[1]
$A+\operatorname {core}B\subset \operatorname {core}(A+B)$ ^[5]
$A+\operatorname {core}B=\operatorname {core}(A+B)$ if $B=\operatorname {core}B$ ^[5]

Relation to interior

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

$\operatorname {int}A\subseteq \operatorname {core}A$
If $A$ is nonempty convex and $X$ is finite-dimensional, then $\operatorname {int}A=\operatorname {core}A$ ^[2]
If $A$ is convex with non-empty interior, then $\operatorname {int}A=\operatorname {core}A$ ^[6]
If $A$ is a closed convex set and $X$ is a complete metric space, then $\operatorname {int}A=\operatorname {core}A$ ^[7]

References

1 2 Jaschke, Stefan; Kuchler, Uwe (2000). "Coherent Risk Measures, Valuation Bounds, and ( $\mu ,\rho$ )-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 .

Functional analysis

Set / subset types	Absolutely convex Absorbing Balanced Bounded Convex Radial Star-shaped Symmetric Linear cone (subset) Convex cone (subset)

TVS types	Banach Barrelled Bornological Brauner F-space Finite-dimensional Fréchet (tame) Hilbert LF-space Locally convex Mackey Montel Nuclear Normed (norm) Quasinormed Reflexive Riesz Smith Stereotype Webbed Topological tensor product (of Hilbert spaces)

Mapping topologies	Dual Dual space Operator Ultraweak Weak (operator) Mackey Strong (polar operator) Ultrastrong Uniform convergence

Linear operators	Adjoint Bilinear (form) Bounded / Unbounded Closed Continuous / Discontinuous Compact Fredholm Hilbert–Schmidt Functionals (positive) Normal Nuclear Self-adjoint Strictly singular Trace class Transpose Unitary

Set operations	Algebraic interior (core) Interior Minkowski addition Polar

Operator theory	Banach algebras C-algebras Spectrum (C-algebra radius) Spectral theory (of ODEs Spectral theorem) Polar decomposition Singular value decomposition

Theorems	Banach–Alaoglu Banach–Saks Banach–Mazur Bessel's inequality Cauchy–Schwarz inequality Closed range Closed graph Eberlein–Šmulian Freudenthal spectral Gelfand–Mazur Goldstine Hahn–Banach (hyperplane separation) Kakutani fixed-point Lomonosov's invariant subspace Mackey–Arens Mazur's lemma M. Riesz extension Open mapping Parseval's identity Schauder fixed-point

Analysis	Bochner space Differentiation in Fréchet spaces Derivatives (Fréchet Gâteaux functional holomorphic) Integrals (Bochner Dunford Gelfand–Pettis regulated weak) Functional calculus (Borel continuous holomorphic) Inverse function theorem (Nash–Moser theorem) Vector measure Weakly measurable function

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Algebraic interior

Example

Properties

Relation to interior

See also

References