In functional analysis and related areas of mathematics an absorbing set in a vector space is a set which can be "inflated" or "scaled up" to eventually always include any given point of the vector space. Alternative terms are radial or absorbent set. Every neighborhood of the origin in every topological vector space is an absorbing subset.
Notation for scalars
Suppose that is a vector space over the field of real numbers or complex numbers and for any let
denote the open ball (respectively, the closed ball) of radius in centered at Define the product of a set of scalars with a set of vectors as and define the product of with a single vector as
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
Given a set the smallest balanced set containing denoted by is called the of while the largest balanced set contained within denoted by is called the of These sets are given by the formulas
and
(these formulas show that the balanced hull and the balanced core always exist and are unique). 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
while if is non-zero or if then also
If and are subsets of then is said to if it satisfies any of the following equivalent conditions:
If is a balanced set then this list can be extended to include: <ol> <li value="4">There exists a non-zero scalar such that
<li value="5">There exists a non-zero scalar such that </li></ol>
If (a necessary condition for to be an absorbing set, or to be a neighborhood of the origin in a topology) then this list can be extended to include: <ol> <li value="6">There exists such that for every scalar satisfying Or stated more succinctly, </li> <li>There exists such that for every scalar satisfying Or stated more succinctly,
<li>There exists such that </li> <li>There exists such that </li> <li>There exists such that
<li>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 </li> <li>There exists such that </li> <li>There exists a non-zero scalar such that In words, the balanced core of contains some non-zero scalar multiple of </li> <li>There exists a scalar such that In words, can be scaled to contain the balanced hull of </li> <li>There exists a scalar such that </li> <li>There exists a scalar such that In words, can be scaled so that its balanced core contains </li> <li>There exists a scalar such that </li> <li>There exists a scalar such that In words, the balanced core of can be scaled to contain the balanced hull of </li> <li>The balanced core of absorbs the balanced hull (according to any defining condition of "absorbs" other than this one).</li> </ol>
If or then this list can be extended to include: <ol><li value="20"> absorbs (according to any defining condition of "absorbs" other than this one).
</ol>
A set absorbing a point
A set is said to if it absorbs the singleton set 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 is called if it is absorbed by every neighborhood of the origin. A set is called if it absorbs every bounded subset.
First examples
Every set absorbs the empty set but the empty set does not absorb 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 (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 of the origin absorbs every bounded subset of (and so in particular, absorbs every singleton subset/point).
A subset of a vector space over a field is called an of and is said to be 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): <ol> <li>Definition: absorbs every point of that is, for every absorbs
</li> <li> absorbs every finite subset of </li> <li>For every there exists a real such that for any scalar satisfying </li> <li>For every there exists a real such that for any scalar satisfying </li> <li>For every there exists a real such that
</li> <li>For every there exists a real such that where
</li> <li> 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 (i.e. the Euclidean topology). <ul> <li>The reason why the Euclidean topology is distinguished in this characterization ultimately stems from the defining requirement on TVS topologies that scalar multiplication be continuous when the scalar field is given this (Euclidean) topology.</li> <li>-Neighborhoods are absorbing: This condition gives insight as to why every neighborhood of the origin in every topological vector space (TVS) is necessarily absorbing: If is a neighborhood of the origin in a TVS then for every 1-dimensional vector subspace is a neighborhood of the origin in when is endowed with the subspace topology induced on it by This subspace topology is always a vector topology and because is 1-dimensional, the only vector topologies on it are the Hausdorff Euclidean topology and the trivial topology, which is a subset of the Euclidean topology. So regardless of which of these vector topologies is on the set will be a neighborhood of the origin in with respect to its unique Hausdorff vector topology (the Euclidean topology). Thus is absorbing.</li> </ul> </li> <li> contains the origin and for every 1-dimensional vector subspace of is absorbing in (according to any defining condition of "absorbing" other than this one).
</li> </ol>
If then to this list can be appended:
If is balanced then to this list can be appended:
If is convex or balanced then to this list can be appended:
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
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 of sets 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
The unit ball of any normed vector space (or seminormed vector space) is absorbing. More generally, if is a topological vector space (TVS) then any neighborhood of the origin in is absorbing in This fact is one of the primary motivations for defining the property "absorbing in "
Every superset of an absorbing set is absorbing. Consequently, the union of any family of (one or more) absorbing sets is absorbing. The intersection of finitely many absorbing subsets is once again an absorbing subset. However, the open balls of radius are all absorbing in although their intersection is not absorbing.
If is a disk (a convex and balanced subset) then and so in particular, a disk is always 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
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). If absorbing then the same is true of the symmetric set
Auxiliary normed spaces
If is convex and absorbing in then the symmetric set will be convex and balanced (also known as an or a ) in addition to being absorbing in This guarantees that the Minkowski functional of will be a seminorm on thereby making into a seminormed space 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 topology. If is a topological vector space and if this convex absorbing subset is also a bounded subset of then all this will also be true of the absorbing disk if in addition does not contain any non-trivial vector subspace then will be a norm and will form what is known as an auxiliary normed space. If this normed space is a Banach space then is called a .
Every absorbing set contains the origin. If is an absorbing disk 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 is a non-meager subset of itself (or equivalently for TVSs, if it is a Baire space) and if is a closed absorbing subset of then necessarily contains a non-empty open subset of (in other words, 's topological interior will not be empty), which guarantees that is a neighborhood of the origin in
Every absorbing set is a total set, meaning that every absorbing subspace is dense.
Proofs