作為數學 的一個分支，在泛函分析 中，向量空間 子集的代數內部 (英語：Algebraic interior )或徑向核 (英語：Radial kernel )是對內部 概念的細化。 它是給定集合相對於該點是吸收的 的點構成的子集，即集合的徑向 點構成的集合。[ 1] 內點 (英語：Internal point )。 [ 2] [ 3]
正式地，如果
X
{\displaystyle X}
線性空間 ，則
A
⊆
X
{\displaystyle A\subseteq X}
代數內部 是
core
(
A
)
:=
{
x
0
∈
A
:
∀
x
∈
X
,
∃
t
x
>
0
,
∀
t
∈
[
0
,
t
x
]
,
x
0
+
t
x
∈
A
}
{\displaystyle \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] 一般來說，
core
(
A
)
≠
core
(
core
(
A
)
)
{\displaystyle \operatorname {core} (A)\neq \operatorname {core} (\operatorname {core} (A))}
A
{\displaystyle A}
core
(
A
)
=
core
(
core
(
A
)
)
{\displaystyle \operatorname {core} (A)=\operatorname {core} (\operatorname {core} (A))}
A
{\displaystyle A}
x
0
∈
core
(
A
)
,
y
∈
A
,
0
<
λ
≤
1
{\displaystyle x_{0}\in \operatorname {core} (A),y\in A,0<\lambda \leq 1}
λ
x
0
+
(
1
−
λ
)
y
∈
core
(
A
)
{\displaystyle \lambda x_{0}+(1-\lambda )y\in \operatorname {core} (A)}
如果
A
=
{
x
∈
R
2
:
x
2
≥
x
1
2
or
x
2
≤
0
}
⊆
R
2
{\displaystyle A=\{x\in \mathbb {R} ^{2}:x_{2}\geq x_{1}^{2}{\text{ or }}x_{2}\leq 0\}\subseteq \mathbb {R} ^{2}}
0
∈
core
(
A
)
{\displaystyle 0\in \operatorname {core} (A)}
0
∉
int
(
A
)
{\displaystyle 0\not \in \operatorname {int} (A)}
0
∉
core
(
core
(
A
)
)
{\displaystyle 0\not \in \operatorname {core} (\operatorname {core} (A))}
令
A
,
B
⊂
X
{\displaystyle A,B\subset X}
A
{\displaystyle A}
吸收的 若且唯若
0
∈
core
(
A
)
{\displaystyle 0\in \operatorname {core} (A)}
[ 1]
A
+
core
B
⊂
core
(
A
+
B
)
{\displaystyle A+\operatorname {core} B\subset \operatorname {core} (A+B)}
[ 5]
A
+
core
B
=
core
(
A
+
B
)
{\displaystyle A+\operatorname {core} B=\operatorname {core} (A+B)}
B
=
core
B
{\displaystyle B=\operatorname {core} B}
[ 5]
令
X
{\displaystyle X}
拓撲向量空間 ，
int
{\displaystyle \operatorname {int} }
A
⊂
X
{\displaystyle A\subset X}
int
A
⊆
core
A
{\displaystyle \operatorname {int} A\subseteq \operatorname {core} A}
如果
A
{\displaystyle A}
X
{\displaystyle X}
int
A
=
core
A
{\displaystyle \operatorname {int} A=\operatorname {core} A}
[ 2]
如果
A
{\displaystyle A}
int
A
=
core
A
{\displaystyle \operatorname {int} A=\operatorname {core} A}
[ 6]
如果
A
{\displaystyle A}
X
{\displaystyle X}
完備度量空間 ，則有
int
A
=
core
A
{\displaystyle \operatorname {int} A=\operatorname {core} A}
[ 7]
^ 1.0 1.1 Jaschke, Stefan; Kuchler, Uwe. Coherent Risk Measures, Valuation Bounds, and (
μ
,
ρ
{\displaystyle \mu ,\rho }
^ 2.0 2.1 Aliprantis, C.D.; Border, K.C. Infinite Dimensional Analysis: A Hitchhiker's Guide 3rd. Springer. 2007: 199–200. ISBN 978-3-540-32696-0doi:10.1007/3-540-29587-9 ^ John Cook. Separation of Convex Sets in Linear Topological Spaces (pdf) . May 21, 1988 [November 14, 2012] . （原始內容存檔 (PDF) 於2019-02-27）. ^ Nikolaĭ Kapitonovich Nikolʹskiĭ. Functional analysis I: linear functional analysis . Springer. 1992. ISBN 978-3-540-50584-6 ^ 5.0 5.1 Zălinescu, C. Convex analysis in general vector spaces. River Edge, NJ: World Scientific Publishing Co., Inc. 2002: 2–3. ISBN 981-238-067-1MR 1921556 ^ Shmuel Kantorovitz. Introduction to Modern Analysis . Oxford University Press. 2003: 134 . ISBN 9780198526568 ^ Bonnans, J. Frederic; Shapiro, Alexander, Perturbation Analysis of Optimization Problems , Springer series in operations research, Springer, Remark 2.73, p. 56, 2000 [2016-12-19 ] , ISBN 9780387987057存檔 於2019-05-02）
![{\displaystyle \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\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a37add7d1bb74418c604111417fdd15dd281345)
