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
Assume that
is a subset of a vector space
The algebraic interior (or radial kernel) of
with respect to
is the set of all points at which
is a radial set. A point
is called an internal point of
[2] and
is said to be radial at
if for every
there exists a real number
such that for every
This last condition can also be written as
where the set
is the line segment (or closed interval) starting at
and ending at
this line segment is a subset of
which is the ray emanating from
in the direction of
(that is, parallel to/a translation of
). Thus geometrically, an interior point of a subset
is a point
with the property that in every possible direction (vector)
contains some (non-degenerate) line segment starting at
and heading in that direction (i.e. a subset of the ray
). The algebraic interior of
(with respect to
) 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
is a linear subspace of
and
then this definition can be generalized to the algebraic interior of
with respect to
is:
where
always holds and if
then
where
is the affine hull of
(which is equal to
).
Algebraic closure
A point
is said to be linearly accessible from a subset
if there exists some
such that the line segment
is contained in
The algebraic closure of
with respect to
, denoted by
consists of
and all points in
that are linearly accessible from
Algebraic Interior (Core)
In the special case where
the set
is called the algebraic interior or core of
and it is denoted by
or
Formally, if
is a vector space then the algebraic interior of
is[6]
If
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):
If
is a Fréchet space,
is convex, and
is closed in
then
but in general it is possible to have
while
is not empty.
Examples
If
then
but
and
Properties of core
Suppose
- In general,
But if
is a convex set then:
and - for all
then ![{\displaystyle \lambda x_{0}+(1-\lambda )y\in \operatorname {core} A.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/37175774a96e8378a6f223ec58398937e38dac88)
is an absorbing subset of a real vector space if and only if
[3] ![{\displaystyle A+\operatorname {core} B\subseteq \operatorname {core} (A+B)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fffe18b6ce60c392d688af48389495af4fb3533f)
if ![{\displaystyle B=\operatorname {core} B.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a5a589be26af6930a064ca54dc228d1b51658c9)
Both the core and the algebraic closure of a convex set are again convex. If
is convex,
and
then the line segment
is contained in
Relation to topological interior
Let
be a topological vector space,
denote the interior operator, and
then:
![{\displaystyle \operatorname {int} A\subseteq \operatorname {core} A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c083084f950efa6cd93417da20cc263ade66a823)
- If
is nonempty convex and
is finite-dimensional, then ![{\displaystyle \operatorname {int} A=\operatorname {core} A.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0efc54812b6c87db5527a90cc38ff2840d85e5c)
- If
is convex with non-empty interior, then
[8] - If
is a closed convex set and
is a complete metric space, then
[9]
Relative algebraic interior
If
then the set
is denoted by
and it is called the relative algebraic interior of
This name stems from the fact that
if and only if
and
(where
if and only if
).
Relative interior
If
is a subset of a topological vector space
then the relative interior of
is the set
That is, it is the topological interior of A in
which is the smallest affine linear subspace of
containing
The following set is also useful:
Quasi relative interior
If
is a subset of a topological vector space
then the quasi relative interior of
is the set
In a Hausdorff finite dimensional topological vector space,
See also
- Bounding point – Mathematical concept related to subsets of vector spaces
- Interior (topology) – Largest open subset of some given set
- Order unit – Element of an ordered vector space
- Quasi-relative interior – Generalization of algebraic interior
- Radial set
- Relative interior – Generalization of topological interior
- Ursescu theorem – Generalization of closed graph, open mapping, and uniform boundedness theorem
References
- ^ 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 (
)-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
- 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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