In
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
and related areas of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
an absorbing set in a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
is a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
which can be "inflated" or "scaled up" to eventually always include any given point of the vector space.
Alternative terms are
radial
Radial is a geometric term of location which may refer to:
Mathematics and Direction
* Vector (geometric)
In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) ...
or absorbent set.
Every
neighborhood of the origin in every
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
is an absorbing subset.
Definition
Suppose that
is a vector space over the
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
of
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s
or
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s
Notation
Products of scalars and vectors
For any
vector
and subset
let
denote the ''open ball'' (respectively, the ''closed ball'') of radius
in
centered at
and let
Similarly, if
and
is a scalar then let
and
Balanced core and balanced hull
A subset
of
is said to be ''
'' if
for all
and all scalars
satisfying
this condition may be written more succinctly as
and it holds if and only if
The (respectively, the ) of a set
denoted by
(respectively, by
), is defined to be the smallest
balanced set
In linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space (over a field \mathbb with an absolute value function , \cdot , ) is a set S such that a S \subseteq S for all scalars a satisfying , a, \le ...
containing
(respectively, the largest
balanced set
In linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space (over a field \mathbb with an absolute value function , \cdot , ) is a set S such that a S \subseteq S for all scalars a satisfying , a, \le ...
contained in
). The balanced hull and core always exist and are unique.
They are given by the formulas
A set
is balanced if and only if it is equal to its balanced hull (
) or to its balanced core (
), in which case all three of these sets are equal:
If
is any scalar then
and if
is non-zero or if
then also
One set absorbing another
If
and
are subsets of
then
is said to absorb
if it satisfies any of the following equivalent conditions:
# ''Definition'': There exists a real
such that
for every scalar
satisfying
Or stated more succinctly,
for some
#* If the scalar field is
then intuitively, "
absorbs
" means that if
is perpetually "scaled up" or "inflated" (referring to
as
) then (for all positive
sufficiently large), all
will contain
and similarly,
must also eventually contain
for all negative
sufficiently large in magnitude.
#* This definition depends on the underlying scalar field's canonical norm (that is, on the
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
), which thus ties this definition to the usual
Euclidean topology
In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean distance, Euclidean metric.
Definition
The Euclidean norm on \R^n is the non-negative f ...
on the scalar field. Consequently, the definition of an ''absorbing set'' (given below) is also tied to this topology.
# There exists a real
such that
for every non-zero
[The requirement that be scalar be non-zero cannot be dropped from this characterization.] scalar
satisfying
Or stated more succinctly,
for some
#* Because this union is equal to
where
is the closed ball with the origin removed, this condition may be restated as:
for some
#* The non-strict inequality
can be replaced with the strict inequality
which is the next characterization.
# There exists a real
such that
for every non-zero
[ scalar satisfying Or stated more succinctly, for some
#* Here is the open ball with the origin removed and
If is a ]balanced set
In linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space (over a field \mathbb with an absolute value function , \cdot , ) is a set S such that a S \subseteq S for all scalars a satisfying , a, \le ...
then this list can be extended to include:
#There exists a non-zero scalar such that
#* If then the requirement may be dropped.
# There exists a non-zero[ scalar such that
If (such as when is a neighborhood of the origin in some topology, as in the definition of "]bounded set
:''"Bounded" and "boundary" are distinct concepts; for the latter see boundary (topology). A circle in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary.
In mathematical analysis and related areas of mat ...
") then this list can be extended to include:
- There exists such that for every scalar satisfying Or stated more succinctly,
- There exists such that for every scalar satisfying Or stated more succinctly,
* This set inclusion is equivalent to (since ). Because this may be rewritten which gives the next statement.
- There exists such that
- There exists such that
- There exists such that
* The next characterizations follow from those above and the fact that for every scalar the
balanced hull
In linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space (over a field \mathbb with an absolute value function , \cdot , ) is a set S such that a S \subseteq S for all scalars a satisfying , a, ...
of satisfies and (since ) its balanced core satisfies
- There exists such that In words, a set is absorbed by if it is contained in some positive scalar multiple of the balanced core of
- There exists such that
- There exists a scalar such that In words, can be scaled to contain the
balanced hull
In linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space (over a field \mathbb with an absolute value function , \cdot , ) is a set S such that a S \subseteq S for all scalars a satisfying , a, ...
of
- There exists a scalar such that
- There exists a scalar such that In words, can be scaled so that its balanced core contains
- There exists a scalar such that
- There exists a non-zero
[ scalar such that In words, the balanced core of contains some non-zero scalar multiple of ]
If or this list can be extended to include:
# absorbs (according to any defining condition of "absorbs" other than this one).
#* In other words, may be replaced by in the characterizations above if (or trivially, if ).
A set absorbing a point
A set is said to absorb a point if it absorbs the singleton set
In mathematics, a singleton, also known as a unit set or one-point set, is a set with exactly one element. For example, the set \ is a singleton whose single element is 0.
Properties
Within the framework of Zermelo–Fraenkel set theory, the ...
A set absorbs the origin if and only if it contains the origin; that is, if and only if
As detailed below, a set is said to be if it absorbs every point of
This notion of one set absorbing another is also used in other definitions:
A subset of a topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
is called if it is absorbed by every neighborhood of the origin.
A set is called if it absorbs every bounded subset.
''Examples''
Every set absorbs the empty set but the empty set does not absorbs any non-empty set. The singleton set containing the origin is the one and only singleton subset that absorbs itself.
Suppose that is equal to either or If is the unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucl ...
(centered at the origin ) together with the origin, then is the one and only non-empty set that absorbs. Moreover, there does exist non-empty subset of that is absorbed by the unit circle In contrast, every neighborhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
of the origin absorbs every bounded subset
:''"Bounded" and "boundary" are distinct concepts; for the latter see boundary (topology). A circle in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary.
In mathematical analysis and related areas of mat ...
of (and so in particular, absorbs every singleton subset/point).
Absorbing set
A subset of a vector space over a field is called an absorbing (or absorbent) subset of and is said to be absorbing in if it satisfies any of the following equivalent conditions (here ordered so that each condition is an easy consequence of the previous one, starting with the definition):
#''Definition'': absorbs every point of that is, for every absorbs
#*So in particular, can not be absorbing if Every absorbing set must contain the origin.
# absorbs every finite subset of
#For every there exists a real such that for any scalar satisfying
#For every there exists a real such that for any scalar satisfying
#For every there exists a real such that
#* Here is the open ball of radius in the scalar field centered at the origin and
#* The closed ball can be used in place of the open ball.
#For every there exists a real such that where
#* Proof: This follows from the previous condition since so that if and only if
#* Connection to topology: If is given its usual Hausdorff Euclidean topology
In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean distance, Euclidean metric.
Definition
The Euclidean norm on \R^n is the non-negative f ...
then the set is a neighborhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
of the origin in thus, there exists a real such that if and only if is a neighborhood of the origin in
#* Every 1-dimensional vector subspace of is of the form for some and if this 1-dimensional space is endowed with the unique Hausdorff vector topology, then the map defined by is necessarily a TVS-isomorphism
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is al ...
(where as usual, has the normed Euclidean topology).
# contains the origin and for every 1-dimensional vector subspace of is a neighborhood of the origin in when is given its unique Hausdorff vector topology
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is al ...
.
#* The Hausdorff vector topology on a 1-dimensional vector space is necessarily TVS-isomorphic to with its usual normed Euclidean topology
In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean distance, Euclidean metric.
Definition
The Euclidean norm on \R^n is the non-negative f ...
.
#* Connection to vector/TVS topologies: This condition shows that it is only natural that any neighborhood of in any topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
(TVS) be absorbing: if is a neighborhood of the origin in then it would be pathological if there existed any 1-dimensional vector subspace in which was not a neighborhood of the origin in at least TVS topology on The only TVS topologies on are the Hausdorff Euclidean topology and the trivial topology, which is a subset of the Euclidean topology. Consequently, this pathology does not occur if and only if to be a neighborhood of in the Euclidean topology for 1-dimensional vector subspaces which is exactly the condition that be absorbing in The fact that all neighborhoods of the origin in all TVSs are necessarily absorbing means that this pathological behavior does not occur. The reason why the Euclidean topology is distinguished ultimately stems from the defining requirement on TVS topologies that scalar multiplication be continuous when the scalar field is given this (Euclidean) topology.
#* This condition is equivalent to: For every is a neighborhood of in when is given its unique Hausdorff TVS topology.
# contains the origin and for every 1-dimensional vector subspace of is absorbing in the
#* Here "absorbing" means absorbing according to any defining condition other than this one.
#* This shows that the property of being absorbing in depends on how behaves with respect to 1 (or 0) dimensional vector subspaces of In contrast, if a finite-dimensional vector subspace of has dimension and is endowed with its unique Hausdorff TVS topology, then being absorbing in is no longer sufficient to guarantee that is a neighborhood of the origin in (although it will still be a necessary condition). For this to happen, it suffices for to be an absorbing set that is also convex balanced and closed in (such a set is called a and it will be a neighborhood of the origin in because like every finite-dimensional Euclidean space, is a barrelled space
In functional analysis and related areas of mathematics, a barrelled space (also written barreled space) is a topological vector space (TVS) for which every barrelled set in the space is a neighbourhood for the zero vector.
A barrelled set or a b ...
).
If then to this list can be appended:
#The algebraic 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 A is a subset of a vector space X.
The ''algebraic i ...
of contains the origin (that is, ).
If is balanced
In telecommunications and professional audio, a balanced line or balanced signal pair is a circuit consisting of two conductors of the same type, both of which have equal impedances along their lengths and equal impedances to ground and to other ...
then to this list can be appended:
# For every there exists a scalar such that (or equivalently, such that ).
# For every there exists a scalar such that
If is convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytope ...
''or'' balanced
In telecommunications and professional audio, a balanced line or balanced signal pair is a circuit consisting of two conductors of the same type, both of which have equal impedances along their lengths and equal impedances to ground and to other ...
then to this list can be appended:
# For every there exists a positive real such that
#* The proof that a ''balanced'' set satisfying this condition is necessarily absorbing in is almost immediate from the definition of a "balanced set".
#* The proof that a ''convex'' set satisfying this condition is necessarily absorbing in is less trivial (but not difficult). A detailed proof is given in this footnote[ and a summary is given below.
#** Summary of proof: By assumption, for non-zero it is possible to pick positive real and such that and so that the convex set contains the open sub-interval which contains the origin ( is called an interval since we identify with and every non-empty convex subset of is an interval). Give its unique Hausdorff vector topology so it remains to show that is a neighborhood of the origin in If then we are done, so assume that The set is a union of two intervals, each of which contains an open sub-interval that contains the origin; moreover, the intersection of these two intervals is precisely the origin. So the convex hull of which is contained in the convex set clearly contains an open ball around the origin. ]
#For every there exists a positive real such that
#* This condition is equivalent to: every belongs to the set This happens if and only if which gives the next characterization.
#
#*It can be shown that for any subset of if and only if for every where
# For every
If (which is necessary for to be absorbing) then it suffices to check any of the above conditions for all non-zero rather than all
Examples and sufficient conditions
For one set to absorb another
Let be a linear map between vector spaces and let and be balanced sets. Then absorbs if and only if absorbs
If a set absorbs another set then any superset of also absorbs
A set absorbs the origin if and only if the origin is an element of
A set absorbs a finite union if and only it absorbs each set individuality (that is, if and only if absorbs for every ). In particular, a set is an absorbing subset of if and only if it absorbs every finite subset of
For a set to be absorbing
In a semi normed vector space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" ...
the unit ball
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* ''Unit'' (alb ...
is absorbing.
More generally, if is a topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
(TVS) then any neighborhood of the origin in is absorbing in This fact is one of the primary motivations for even defining the property "absorbing in "
If is a disk
Disk or disc may refer to:
* Disk (mathematics), a geometric shape
* Disk storage
Music
* Disc (band), an American experimental music band
* ''Disk'' (album), a 1995 EP by Moby
Other uses
* Disk (functional analysis), a subset of a vector sp ...
(convex and balanced) then so that in particular, is an absorbing subset of
Thus if is a disk in then is absorbing in if and only if
This conclusion is not guaranteed if the set is balanced but not convex; for example, the union of the and axes in is a non-convex balanced set that is not absorbing in
Any superset of an absorbing set is absorbing. Thus the union of any family of (one or more) absorbing sets is absorbing.
The image of an absorbing set under a surjective linear operator is again absorbing. The inverse image of an absorbing subset (of the codomain) under a linear operator is again absorbing (in the domain).
The intersection of a finite family of (one or more) absorbing sets is absorbing. If absorbing then the same is true of the symmetric set
In mathematics, a nonempty subset of a group is said to be symmetric if it contains the inverses of all of its elements.
Definition
In set notation a subset S of a group G is called if whenever s \in S then the inverse of s also belongs to ...
Auxiliary normed spaces
If is convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytope ...
and absorbing in then the symmetric set
In mathematics, a nonempty subset of a group is said to be symmetric if it contains the inverses of all of its elements.
Definition
In set notation a subset S of a group G is called if whenever s \in S then the inverse of s also belongs to ...
will be convex and balanced
In telecommunications and professional audio, a balanced line or balanced signal pair is a circuit consisting of two conductors of the same type, both of which have equal impedances along their lengths and equal impedances to ground and to other ...
(also known as an or a ) in addition to being absorbing in
This guarantees that the Minkowski functional
In mathematics, in the field of functional analysis, a Minkowski functional (after Hermann Minkowski) or gauge function is a function that recovers a notion of distance on a linear space.
If K is a subset of a real or complex vector space X, then ...
of will be a seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and ...
on thereby making into a seminormed space In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and ...
that carries its canonical pseduometrizable topology. The set of scalar multiples as ranges over (or over any other set of non-zero scalars having as a limit point) forms a neighborhood basis of absorbing disks at the origin for this locally convex
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ve ...
topology. If is a topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
and if this convex absorbing subset is also a bounded subset
:''"Bounded" and "boundary" are distinct concepts; for the latter see boundary (topology). A circle in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary.
In mathematical analysis and related areas of mat ...
of then the same will be true of the absorbing disk in which case will be a norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
and will form what is known as an auxiliary normed space
In functional analysis, two methods of constructing normed spaces from disks were systematically employed by Alexander Grothendieck to define nuclear operators and nuclear spaces.
One method is used if the disk D is bounded: in this case, the aux ...
. If this normed space is a Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
then is called a .
Properties
Every absorbing set contains the origin.
If is an absorbing disk
Disk or disc may refer to:
* Disk (mathematics), a geometric shape
* Disk storage
Music
* Disc (band), an American experimental music band
* ''Disk'' (album), a 1995 EP by Moby
Other uses
* Disk (functional analysis), a subset of a vector sp ...
in a vector space then there exists an absorbing disk in such that
If is an absorbing subset of then and more generally, for any sequence of scalars such that Consequently, if a topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
is a non-meager subset of itself (or equivalently for TVSs, if it is a Baire space
In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior.
According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are ...
) and if is a closed absorbing subset of then necessarily contains a non-empty open subset of (in other words, 's topological interior
In mathematics, specifically in topology,
the interior of a subset of a topological space is the union of all subsets of that are open in .
A point that is in the interior of is an interior point of .
The interior of is the complement of the ...
will not be empty), which guarantees that is a neighborhood of the origin in
See also
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Notes
Proofs
}
Citations
References
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{{TopologicalVectorSpaces
Functional analysis