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.
Definition
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
Preliminaries
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
One set absorbing another
If and are subsets of then is said to 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 ), which thus ties this definition to the usual Euclidean topology 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-zeroThe requirement that the 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 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 (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: There exists such that for every scalar satisfying Or stated more succinctly, There exists such that for every scalar satisfying Or stated more succinctly, The 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 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 non-zero scalar such that In words, the balanced core of contains some non-zero scalar multiple of There exists a scalar such that In words, can be scaled to contain the balanced hull 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 scalar such that In words, the balanced core of can be scaled to contain the balanced hull of The balanced core of absorbs the balanced hull (according to any defining condition of "absorbs" other than this one).
If or then 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 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).
Absorbing set
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): 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. Because the inclusion holds if and only if This proves the next statement. For every there exists a real such that where Connection to topology: If is given its usual Hausdorff Euclidean topology then the set is a neighborhood of the origin in thus, there exists a real such that if and only if is a neighborhood of the origin in Consequently, satisfies this condition if and only if for every is a neighborhood of in when is given the Euclidean topology. This gives the next characterization. The only TVS topologies on a 1-dimensional vector space are the (non-Hausdorff) trivial topology and the Hausdorff Euclidean topology. Every 1-dimensional vector subspace of is of the form for some and if this 1-dimensional space is endowed with the (unique) , then the map defined by is necessarily a TVS-isomorphism (where as usual, is endowed with its standard Euclidean topology induced by the Euclidean metric). 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). 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. -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 topologyA topology on a vector space is called a or a if its makes vector addition and scalar multiplication continuous when the scalar field is given its usual norm-induced Euclidean topology (that norm being the absolute value ). Since restrictions of continuous functions are continuous, if is a vector subspace of a TVS then 's vector addition and scalar multiplication operations will also be continuous. Thus the subspace topology that any vector subspace inherits from a TVS will once again be 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).If is a neighborhood of the origin in a TVS 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. Thus is absorbing. 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). This characterization 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 every finite-dimensional Euclidean space, including is a barrelled space).
If then to this list can be appended: The algebraic interior of contains the origin (that is, ).
If is balanced 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 or balanced 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 follows immediately from condition (10) above and the fact that for all scalars (where is real). 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.