查看“︁代数内部”︁的源代码

作为[[数学]]的一个分支，在[[泛函分析]]中，[[向量空间]]子集的'''代数内部'''({{lang-en|Algebraic interior}})或'''径向核'''({{lang-en|Radial kernel}})是对[[内部]]概念的细化。 它是给定集合相对于该点是[[吸收集|吸收的]]的点构成的子集，即集合的[[径向集|径向]]点构成的集合。<ref name="coherent">{{cite journal|title=Coherent Risk Measures, Valuation Bounds, and (<math>\mu,\rho</math>)-Portfolio Optimization|last2=Kuchler|first2=Uwe|date=2000|first1=Stefan|last1=Jaschke}}</ref>代数内部的元素通常被称为'''内点'''({{lang-en|Internal point}})。 <ref name="aliprantis+border">{{cite book|title=Infinite Dimensional Analysis: A Hitchhiker's Guide|last=Aliprantis|first=C.D.|last2=Border|first2=K.C.|date=2007|publisher=Springer|isbn=978-3-540-32696-0|edition=3rd|pages=199–200|doi=10.1007/3-540-29587-9}}</ref><ref name="cook">{{cite web|url=http://www.johndcook.com/SeparationOfConvexSets.pdf|title=Separation of Convex Sets in Linear Topological Spaces|accessdate=November 14, 2012|author=John Cook|date=May 21, 1988|format=pdf|archive-date=2019-02-27|archive-url=https://web.archive.org/web/20190227220547/https://www.johndcook.com/SeparationOfConvexSets.pdf|dead-url=no}}</ref>

正式地，如果<math>X</math>是[[向量空间|线性空间]]，则<math>A \subseteq X</math>的'''代数内部'''是
: <math>\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\}</math>。<ref>{{cite book|title=Functional analysis I: linear functional analysis|url=https://archive.org/details/functionalanalys0000yuil|date=1992|publisher=Springer|isbn=978-3-540-50584-6|author=Nikolaĭ Kapitonovich Nikolʹskiĭ}}</ref>
一般来说，<math>\operatorname{core}(A) \neq \operatorname{core}(\operatorname{core}(A))</math>，但如果<math>A</math>是一个凸集，则有<math>\operatorname{core}(A) = \operatorname{core}(\operatorname{core}(A))</math>。假设<math>A</math>是凸集，则如果<math>x_0 \in \operatorname{core}(A), y \in A, 0 < \lambda \leq 1</math>，就有<math>\lambda x_0 + (1 - \lambda)y \in \operatorname{core}(A)</math>。

== 例子 ==
如果<math>A = \{x \in \mathbb{R}^2: x_2 \geq x_1^2 \text{ or } x_2 \leq 0\} \subseteq \mathbb{R}^2</math>，则有<math>0 \in \operatorname{core}(A)</math>，但<math>0 \not\in \operatorname{int}(A)</math>且<math>0 \not\in \operatorname{core}(\operatorname{core}(A))</math>。

== 性质 ==
令<math>A,B \subset X</math>则：
* <math>A</math>是[[吸收集|吸收的]]当且仅当<math>0 \in \operatorname{core}(A)</math> <ref name="coherent" />
* <math>A + \operatorname{core}B \subset \operatorname{core}(A + B)</math><ref name="Zalinescu">{{cite book|title=Convex analysis in general vector spaces|last=Zălinescu|first=C.|date=2002|publisher=World Scientific Publishing  Co., Inc|isbn=981-238-067-1|location=River Edge, NJ|pages=2–3|mr=1921556}}</ref>
* <math>A + \operatorname{core}B = \operatorname{core}(A + B)</math>如果<math>B = \operatorname{core}B</math><ref name="Zalinescu" />

=== 和内部的关系 ===
令<math>X</math>是[[拓扑向量空间]]，<math>\operatorname{int}</math>表示内部算子，且<math>A \subset X</math>，则有：
* <math>\operatorname{int}A \subseteq \operatorname{core}A</math>
* 如果<math>A</math>是非空凸集且<math>X</math> 有限维的，则有<math>\operatorname{int}A = \operatorname{core}A</math><ref name="aliprantis+border" />
* 如果<math>A</math>是有非空内部的凸集，则有<math>\operatorname{int}A = \operatorname{core}A</math><ref name="kantorovitz">{{cite book|title=Introduction to Modern Analysis|url=https://archive.org/details/introductiontomo00kant_131|date=2003|publisher=Oxford University Press|isbn=9780198526568|page=[https://archive.org/details/introductiontomo00kant_131/page/134 134]|author=Shmuel Kantorovitz}}</ref>
* 如果<math>A</math>是闭凸集且<math>X</math>是[[完备空间|完备度量空间]]，则有<math>\operatorname{int}A = \operatorname{core}A</math><ref>{{citation|title=Perturbation Analysis of Optimization Problems|date=2000|url=https://books.google.com/books?id=ET70F9HgIpIC&pg=PA56|last1=Bonnans|last2=Shapiro|first1=J. Frederic|first2=Alexander|series=Springer series in operations research|at=Remark 2.73, p.&nbsp;56|publisher=Springer|isbn=9780387987057|accessdate=2016-12-19|archive-date=2019-05-02|archive-url=https://web.archive.org/web/20190502114329/https://books.google.com/books?id=ET70F9HgIpIC&pg=PA56|dead-url=no}}.</ref>

== 另请参阅 ==
* [[内部]]
* {{link-en|相对内部|Relative interior}}
* {{link-en|拟相对内部|Quasi-relative interior}}
* {{link-en|有序单位|Order unit}}
* [[边界点]]

== 参考文献 ==
{{Reflist}}
[[Category:泛函分析]]
[[Category:数学分析]]
[[Category:拓扑学]]
{{泛函分析}}