Generalization of topological 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 .
Assume that
A
{\displaystyle A}
X
.
{\displaystyle X.}
algebraic interior (or radial kernel ) of
A
{\displaystyle A}
X
{\displaystyle X}
is the set of all points at which
A
{\displaystyle A}
radial set .
A point
a
0
∈
A
{\displaystyle a_{0}\in A}
internal point of
A
{\displaystyle A}
[ 2]
A
{\displaystyle A}
radial at
a
0
{\displaystyle a_{0}}
x
∈
X
{\displaystyle x\in X}
t
x
>
0
{\displaystyle t_{x}>0}
t
∈
[
0
,
t
x
]
,
{\displaystyle t\in [0,t_{x}],}
a
0
+
t
x
∈
A
.
{\displaystyle a_{0}+tx\in A.}
a
0
+
[
0
,
t
x
]
x
⊆
A
{\displaystyle a_{0}+[0,t_{x}]x\subseteq A}
a
0
+
[
0
,
t
x
]
x
:=
{
a
0
+
t
x
:
t
∈
[
0
,
t
x
]
}
{\displaystyle a_{0}+[0,t_{x}]x~:=~\left\{a_{0}+tx:t\in [0,t_{x}]\right\}}
a
0
{\displaystyle a_{0}}
a
0
+
t
x
x
;
{\displaystyle a_{0}+t_{x}x;}
a
0
+
[
0
,
∞
)
x
,
{\displaystyle a_{0}+[0,\infty )x,}
ray emanating from
a
0
{\displaystyle a_{0}}
x
{\displaystyle x}
[
0
,
∞
)
x
{\displaystyle [0,\infty )x}
A
{\displaystyle A}
a
0
∈
A
{\displaystyle a_{0}\in A}
x
≠
0
,
{\displaystyle x\neq 0,}
A
{\displaystyle A}
a
0
{\displaystyle a_{0}}
a
0
+
[
0
,
∞
)
x
{\displaystyle a_{0}+[0,\infty )x}
A
{\displaystyle A}
X
{\displaystyle X}
radial points of the set.[ 3]
If
M
{\displaystyle M}
X
{\displaystyle X}
A
⊆
X
{\displaystyle A\subseteq X}
algebraic interior of
A
{\displaystyle A}
M
{\displaystyle M}
is:
aint
M
A
:=
{
a
∈
X
:
for all
m
∈
M
,
there exists some
t
m
>
0
such that
a
+
[
0
,
t
m
]
⋅
m
⊆
A
}
.
{\displaystyle \operatorname {aint} _{M}A:=\left\{a\in X:{\text{ for all }}m\in M,{\text{ there exists some }}t_{m}>0{\text{ such that }}a+\left[0,t_{m}\right]\cdot m\subseteq A\right\}.}
aint
M
A
⊆
A
{\displaystyle \operatorname {aint} _{M}A\subseteq A}
aint
M
A
≠
∅
{\displaystyle \operatorname {aint} _{M}A\neq \varnothing }
M
⊆
aff
(
A
−
A
)
,
{\displaystyle M\subseteq \operatorname {aff} (A-A),}
aff
(
A
−
A
)
{\displaystyle \operatorname {aff} (A-A)}
affine hull of
A
−
A
{\displaystyle A-A}
span
(
A
−
A
)
{\displaystyle \operatorname {span} (A-A)}
Algebraic closure
A point
x
∈
X
{\displaystyle x\in X}
linearly accessible
A
⊆
X
{\displaystyle A\subseteq X}
a
∈
A
{\displaystyle a\in A}
[
a
,
x
)
:=
a
+
[
0
,
1
)
(
x
−
a
)
{\displaystyle [a,x):=a+[0,1)(x-a)}
A
.
{\displaystyle A.}
algebraic closure of
A
{\displaystyle A}
X
{\displaystyle X}
acl
X
A
,
{\displaystyle \operatorname {acl} _{X}A,}
A
{\displaystyle A}
X
{\displaystyle X}
A
.
{\displaystyle A.}
[ edit ] In the special case where
M
:=
X
,
{\displaystyle M:=X,}
aint
X
A
{\displaystyle \operatorname {aint} _{X}A}
algebraic interior core
A
{\displaystyle A}
A
i
{\displaystyle A^{i}}
core
A
.
{\displaystyle \operatorname {core} A.}
X
{\displaystyle X}
A
⊆
X
{\displaystyle A\subseteq X}
[ 6]
aint
X
A
:=
core
(
A
)
:=
{
a
∈
A
:
for all
x
∈
X
,
there exists some
t
x
>
0
,
such that for all
t
∈
[
0
,
t
x
]
,
a
+
t
x
∈
A
}
.
{\displaystyle \operatorname {aint} _{X}A:=\operatorname {core} (A):=\left\{a\in A:{\text{ for all }}x\in X,{\text{ there exists some }}t_{x}>0,{\text{ such that for all }}t\in \left[0,t_{x}\right],a+tx\in A\right\}.}
We call A algebraically open in X if
A
=
aint
X
A
{\displaystyle A=\operatorname {aint} _{X}A}
If
A
{\displaystyle A}
Ursescu theorem ):
i
c
A
:=
{
i
A
if
aff
A
is a closed set,
∅
otherwise
{\displaystyle {}^{ic}A:={\begin{cases}{}^{i}A&{\text{ if }}\operatorname {aff} A{\text{ is a closed set,}}\\\varnothing &{\text{ otherwise}}\end{cases}}}
i
b
A
:=
{
i
A
if
span
(
A
−
a
)
is a barrelled linear subspace of
X
for any/all
a
∈
A
,
∅
otherwise
{\displaystyle {}^{ib}A:={\begin{cases}{}^{i}A&{\text{ if }}\operatorname {span} (A-a){\text{ is a barrelled linear subspace of }}X{\text{ for any/all }}a\in A{\text{,}}\\\varnothing &{\text{ otherwise}}\end{cases}}}
If
X
{\displaystyle X}
Fréchet space ,
A
{\displaystyle A}
aff
A
{\displaystyle \operatorname {aff} A}
X
{\displaystyle X}
i
c
A
=
i
b
A
{\displaystyle {}^{ic}A={}^{ib}A}
i
c
A
=
∅
{\displaystyle {}^{ic}A=\varnothing }
i
b
A
{\displaystyle {}^{ib}A}
not empty.
If
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)).}
Suppose
A
,
B
⊆
X
.
{\displaystyle A,B\subseteq X.}
In general,
core
A
≠
core
(
core
A
)
.
{\displaystyle \operatorname {core} A\neq \operatorname {core} (\operatorname {core} A).}
A
{\displaystyle A}
convex set then:
core
A
=
core
(
core
A
)
,
{\displaystyle \operatorname {core} A=\operatorname {core} (\operatorname {core} A),}
for all
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.}
A
{\displaystyle A}
absorbing subset of a real vector space if and only if
0
∈
core
(
A
)
.
{\displaystyle 0\in \operatorname {core} (A).}
[ 3]
A
+
core
B
⊆
core
(
A
+
B
)
{\displaystyle A+\operatorname {core} B\subseteq \operatorname {core} (A+B)}
A
+
core
B
=
core
(
A
+
B
)
{\displaystyle A+\operatorname {core} B=\operatorname {core} (A+B)}
B
=
core
B
.
{\displaystyle B=\operatorname {core} B.}
Both the core and the algebraic closure of a convex set are again convex.
If
C
{\displaystyle C}
c
∈
core
C
,
{\displaystyle c\in \operatorname {core} C,}
b
∈
acl
X
C
{\displaystyle b\in \operatorname {acl} _{X}C}
[
c
,
b
)
:=
c
+
[
0
,
1
)
b
{\displaystyle [c,b):=c+[0,1)b}
core
C
.
{\displaystyle \operatorname {core} C.}
Relation to topological interior [ edit ] Let
X
{\displaystyle X}
topological vector space ,
int
{\displaystyle \operatorname {int} }
A
⊆
X
{\displaystyle A\subseteq X}
int
A
⊆
core
A
{\displaystyle \operatorname {int} A\subseteq \operatorname {core} A}
If
A
{\displaystyle A}
X
{\displaystyle X}
int
A
=
core
A
.
{\displaystyle \operatorname {int} A=\operatorname {core} A.}
If
A
{\displaystyle A}
int
A
=
core
A
.
{\displaystyle \operatorname {int} A=\operatorname {core} A.}
[ 8]
If
A
{\displaystyle A}
X
{\displaystyle X}
complete metric space , then
int
A
=
core
A
.
{\displaystyle \operatorname {int} A=\operatorname {core} A.}
[ 9] Relative algebraic interior [ edit ] If
M
=
aff
(
A
−
A
)
{\displaystyle M=\operatorname {aff} (A-A)}
aint
M
A
{\displaystyle \operatorname {aint} _{M}A}
i
A
:=
aint
aff
(
A
−
A
)
A
{\displaystyle {}^{i}A:=\operatorname {aint} _{\operatorname {aff} (A-A)}A}
the relative algebraic interior of
A
.
{\displaystyle A.}
This name stems from the fact that
a
∈
A
i
{\displaystyle a\in A^{i}}
aff
A
=
X
{\displaystyle \operatorname {aff} A=X}
a
∈
i
A
{\displaystyle a\in {}^{i}A}
aff
A
=
X
{\displaystyle \operatorname {aff} A=X}
aff
(
A
−
A
)
=
X
{\displaystyle \operatorname {aff} (A-A)=X}
If
A
{\displaystyle A}
X
{\displaystyle X}
relative interior of
A
{\displaystyle A}
rint
A
:=
int
aff
A
A
.
{\displaystyle \operatorname {rint} A:=\operatorname {int} _{\operatorname {aff} A}A.}
aff
A
,
{\displaystyle \operatorname {aff} A,}
X
{\displaystyle X}
A
.
{\displaystyle A.}
ri
A
:=
{
rint
A
if
aff
A
is a closed subspace of
X
,
∅
otherwise
{\displaystyle \operatorname {ri} A:={\begin{cases}\operatorname {rint} A&{\text{ if }}\operatorname {aff} A{\text{ is a closed subspace of }}X{\text{,}}\\\varnothing &{\text{ otherwise}}\end{cases}}}
Quasi relative interior [ edit ] If
A
{\displaystyle A}
X
{\displaystyle X}
quasi relative interior of
A
{\displaystyle A}
qri
A
:=
{
a
∈
A
:
cone
¯
(
A
−
a
)
is a linear subspace of
X
}
.
{\displaystyle \operatorname {qri} A:=\left\{a\in A:{\overline {\operatorname {cone} }}(A-a){\text{ is a linear subspace of }}X\right\}.}
In a Hausdorff finite dimensional topological vector space,
qri
A
=
i
A
=
i
c
A
=
i
b
A
.
{\displaystyle \operatorname {qri} A={}^{i}A={}^{ic}A={}^{ib}A.}
^ John Cook (May 21, 1988). "Separation of Convex Sets in Linear Topological Spaces" (PDF) . Retrieved November 14, 2012 . ^ a b Jaschke, Stefan; Kuchler, Uwe (2000). "Coherent Risk Measures, Valuation Bounds, and (
μ
,
ρ
{\displaystyle \mu ,\rho }
(PDF) . ^ Nikolaĭ Kapitonovich Nikolʹskiĭ (1992). Functional analysis I: linear functional analysis . Springer. ISBN 978-3-540-50584-6 ^ Kantorovitz, Shmuel (2003). Introduction to Modern Analysis . Oxford University Press . p. 134. ISBN 9780198526568 ^ Bonnans, J. Frederic; Shapiro, Alexander (2000), Perturbation Analysis of Optimization Problems ISBN 9780387987057
Aliprantis, Charalambos D. ; Border, Kim C. (2006). Infinite Dimensional Analysis: A Hitchhiker's Guide ISBN 978-3-540-29587-7 OCLC 262692874 .Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces . Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666 OCLC 144216834 . Schaefer, Helmut H. ; Wolff, Manfred P. (1999). Topological Vector Spaces . GTM . Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0 OCLC 840278135 .Schechter, Eric (1996). Handbook of Analysis and Its Foundations . San Diego, CA: Academic Press. ISBN 978-0-12-622760-4 OCLC 175294365 .Zălinescu, Constantin (30 July 2002). Convex Analysis in General Vector Spaces World Scientific Publishing . ISBN 978-981-4488-15-0 MR 1921556 . OCLC 285163112 – via Internet Archive .
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![{\displaystyle a_{0}+[0,t_{x}]x\subseteq A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d323d0ede0d16ed6bc6a2f9ed7d85d98f74ceec)
![{\displaystyle a_{0}+[0,t_{x}]x~:=~\left\{a_{0}+tx:t\in [0,t_{x}]\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/358073de6ecb8ba88644038fc85a7285175a7939)
![{\displaystyle \operatorname {aint} _{M}A:=\left\{a\in X:{\text{ for all }}m\in M,{\text{ there exists some }}t_{m}>0{\text{ such that }}a+\left[0,t_{m}\right]\cdot m\subseteq A\right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be7d7d28ade91c5aa4bd325442fb718c38360cc6)
![{\displaystyle \operatorname {aint} _{X}A:=\operatorname {core} (A):=\left\{a\in A:{\text{ for all }}x\in X,{\text{ there exists some }}t_{x}>0,{\text{ such that for all }}t\in \left[0,t_{x}\right],a+tx\in A\right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f0b62d222e43648babc5681d1080467daae7f1d)
