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 .
Definition [ edit ] Assume that A {\displaystyle A} is a subset of a vector space X . {\displaystyle X.} The algebraic interior (or radial kernel ) of A {\displaystyle A} with respect to X {\displaystyle X} is the set of all points at which A {\displaystyle A} is a radial set . A point a 0 ∈ A {\displaystyle a_{0}\in A} is called an internal point of A {\displaystyle A} [2] and A {\displaystyle A} is said to be radial at a 0 {\displaystyle a_{0}} if for every x ∈ X {\displaystyle x\in X} there exists a real number t x > 0 {\displaystyle t_{x}>0} such that for every t ∈ [ 0 , t x ] , {\displaystyle t\in [0,t_{x}],} a 0 + t x ∈ A . {\displaystyle a_{0}+tx\in A.} This last condition can also be written as a 0 + [ 0 , t x ] x ⊆ A {\displaystyle a_{0}+[0,t_{x}]x\subseteq A} where the set
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\}} is the line segment (or closed interval) starting at
a 0 {\displaystyle a_{0}} and ending at
a 0 + t x x ; {\displaystyle a_{0}+t_{x}x;} this line segment is a subset of
a 0 + [ 0 , ∞ ) x , {\displaystyle a_{0}+[0,\infty )x,} which is the
ray emanating from
a 0 {\displaystyle a_{0}} in the direction of
x {\displaystyle x} (that is, parallel to/a translation of
[ 0 , ∞ ) x {\displaystyle [0,\infty )x} ). Thus geometrically, an interior point of a subset
A {\displaystyle A} is a point
a 0 ∈ A {\displaystyle a_{0}\in A} with the property that in every possible direction (vector)
x ≠ 0 , {\displaystyle x\neq 0,} A {\displaystyle A} contains some (non-degenerate) line segment starting at
a 0 {\displaystyle a_{0}} and heading in that direction (i.e. a subset of the ray
a 0 + [ 0 , ∞ ) x {\displaystyle a_{0}+[0,\infty )x} ). The algebraic interior of
A {\displaystyle A} (with respect to
X {\displaystyle X} ) is the set of all such points. That is to say, it is the subset of points contained in a given set with respect to which it is
radial points of the set.
[3] If M {\displaystyle M} is a linear subspace of X {\displaystyle X} and A ⊆ X {\displaystyle A\subseteq X} then this definition can be generalized to the algebraic interior of A {\displaystyle A} with respect to 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\}.} where
aint M A ⊆ A {\displaystyle \operatorname {aint} _{M}A\subseteq A} always holds and if
aint M A ≠ ∅ {\displaystyle \operatorname {aint} _{M}A\neq \varnothing } then
M ⊆ aff ( A − A ) , {\displaystyle M\subseteq \operatorname {aff} (A-A),} where
aff ( A − A ) {\displaystyle \operatorname {aff} (A-A)} is the
affine hull of
A − A {\displaystyle A-A} (which is equal to
span ( A − A ) {\displaystyle \operatorname {span} (A-A)} ).
Algebraic closure
A point x ∈ X {\displaystyle x\in X} is said to be linearly accessible from a subset A ⊆ X {\displaystyle A\subseteq X} if there exists some a ∈ A {\displaystyle a\in A} such that the line segment [ a , x ) := a + [ 0 , 1 ) x {\displaystyle [a,x):=a+[0,1)x} is contained in A . {\displaystyle A.} The algebraic closure of A {\displaystyle A} with respect to X {\displaystyle X} , denoted by acl X A , {\displaystyle \operatorname {acl} _{X}A,} consists of A {\displaystyle A} and all points in X {\displaystyle X} that are linearly accessible from A . {\displaystyle A.}
Algebraic Interior (Core) [ edit ] In the special case where M := X , {\displaystyle M:=X,} the set aint X A {\displaystyle \operatorname {aint} _{X}A} is called the algebraic interior or core of A {\displaystyle A} and it is denoted by A i {\displaystyle A^{i}} or core A . {\displaystyle \operatorname {core} A.} Formally, if X {\displaystyle X} is a vector space then the algebraic interior of A ⊆ X {\displaystyle A\subseteq X} is[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\}.} If A {\displaystyle A} 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 ):
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} is a Fréchet space , A {\displaystyle A} is convex, and aff A {\displaystyle \operatorname {aff} A} is closed in X {\displaystyle X} then i c A = i b A {\displaystyle {}^{ic}A={}^{ib}A} but in general it is possible to have i c A = ∅ {\displaystyle {}^{ic}A=\varnothing } while i b A {\displaystyle {}^{ib}A} is not empty.
Examples [ edit ] 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}} then 0 ∈ core ( A ) , {\displaystyle 0\in \operatorname {core} (A),} but 0 ∉ int ( A ) {\displaystyle 0\not \in \operatorname {int} (A)} and 0 ∉ core ( core ( A ) ) . {\displaystyle 0\not \in \operatorname {core} (\operatorname {core} (A)).}
Properties of core [ edit ] 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).} But if A {\displaystyle A} is a convex set then: core A = core ( core A ) , {\displaystyle \operatorname {core} A=\operatorname {core} (\operatorname {core} A),} and 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} then λ x 0 + ( 1 − λ ) y ∈ core A . {\displaystyle \lambda x_{0}+(1-\lambda )y\in \operatorname {core} A.} A {\displaystyle A} is an 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)} if 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} is convex, c ∈ core C , {\displaystyle c\in \operatorname {core} C,} and b ∈ acl X C {\displaystyle b\in \operatorname {acl} _{X}C} then the line segment [ c , b ) := c + [ 0 , 1 ) b {\displaystyle [c,b):=c+[0,1)b} is contained in core C . {\displaystyle \operatorname {core} C.}
Relation to topological interior [ edit ] Let X {\displaystyle X} be a topological vector space , int {\displaystyle \operatorname {int} } denote the interior operator, and A ⊆ X {\displaystyle A\subseteq X} then:
int A ⊆ core A {\displaystyle \operatorname {int} A\subseteq \operatorname {core} A} If A {\displaystyle A} is nonempty convex and X {\displaystyle X} is finite-dimensional, then int A = core A . {\displaystyle \operatorname {int} A=\operatorname {core} A.} If A {\displaystyle A} is convex with non-empty interior, then int A = core A . {\displaystyle \operatorname {int} A=\operatorname {core} A.} [8] If A {\displaystyle A} is a closed convex set and X {\displaystyle X} is a 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)} then the set aint M A {\displaystyle \operatorname {aint} _{M}A} is denoted by i A := aint aff ( A − A ) A {\displaystyle {}^{i}A:=\operatorname {aint} _{\operatorname {aff} (A-A)}A} and it is called the relative algebraic interior of A . {\displaystyle A.} This name stems from the fact that a ∈ A i {\displaystyle a\in A^{i}} if and only if aff A = X {\displaystyle \operatorname {aff} A=X} and a ∈ i A {\displaystyle a\in {}^{i}A} (where aff A = X {\displaystyle \operatorname {aff} A=X} if and only if aff ( A − A ) = X {\displaystyle \operatorname {aff} (A-A)=X} ).
Relative interior [ edit ] If A {\displaystyle A} is a subset of a topological vector space X {\displaystyle X} then the relative interior of A {\displaystyle A} is the set
rint A := int aff A A . {\displaystyle \operatorname {rint} A:=\operatorname {int} _{\operatorname {aff} A}A.} That is, it is the topological interior of A in
aff A , {\displaystyle \operatorname {aff} A,} which is the smallest affine linear subspace of
X {\displaystyle X} containing
A . {\displaystyle A.} The following set is also useful:
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} is a subset of a topological vector space X {\displaystyle X} then the quasi relative interior of A {\displaystyle A} is the set
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.}
See also [ edit ] References [ edit ]
^ 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 } )-Portfolio Optimization" (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 , Springer series in operations research, Springer, Remark 2.73, p. 56, ISBN 9780387987057 . Bibliography [ edit ] Aliprantis, Charalambos D. ; Border, Kim C. (2006). Infinite Dimensional Analysis: A Hitchhiker's Guide (Third ed.). Berlin: Springer Science & Business Media. 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 . River Edge, N.J. London: World Scientific Publishing . ISBN 978-981-4488-15-0 . MR 1921556 . OCLC 285163112 – via Internet Archive .
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