Absorbing set

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 balanced 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 balanced hull of while the largest balanced set contained within denoted by is called the balanced core 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 absorb if it satisfies any of the following equivalent conditions:

  1. 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 eventually (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.
  2. There exists a real such that for every non-zero[note 1] 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.
  3. There exists a real such that for every non-zero[note 1] 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:

  1. There exists a non-zero scalar such that
    • If then the requirement may be dropped.
  2. There exists a non-zero[note 1] 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:

  1. There exists such that for every scalar satisfying Or stated more succinctly,
  2. 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.
  3. There exists such that
  4. There exists such that
  5. 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
  6. 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
  7. There exists such that
  8. There exists a scalar such that In words, can be scaled to contain the balanced hull of
  9. There exists a scalar such that
  10. There exists a scalar such that In words, can be scaled so that its balanced core contains
  11. There exists a scalar such that
  12. There exists a non-zero[note 1] scalar such that In words, the balanced core of contains some non-zero scalar multiple of

If or then this list can be extended to include:

  1. 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 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 absorbing in 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 bounded if it is absorbed by every neighborhood of the origin. A set is called bornivorous 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 not exist any 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 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):

  1. 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.
  2. absorbs every finite subset of
  3. For every there exists a real such that for any scalar satisfying
  4. For every there exists a real such that for any scalar satisfying
  5. 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.
  6. 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[note 2] 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 non-zero and if this 1-dimensional space is endowed with the (unique) Hausdorff vector topology, 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).
  7. 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.
    • Connection to vector/TVS topologies: 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[note 2] 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).[note 3] Thus is absorbing.
      • The reason why the Euclidean topology is distinguished in this characterization ultimately stems from the defining requirement on TVS topologies[note 2] that scalar multiplication be continuous when the scalar field is given this (Euclidean) topology.
  8. 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 characterization shows that the property of being absorbing in depends only 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 barrel and it will be a neighborhood of the origin in because like every finite-dimensional Euclidean space, is a barrelled space).

If then to this list can be appended:

  1. The algebraic interior of contains the origin (that is, ).

If is balanced then to this list can be appended:

  1. For every there exists a scalar such that [1] (or equivalently, such that ).
  2. For every there exists a scalar such that

If is convex or balanced then to this list can be appended:

  1. 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[proof 1] and a summary is given below.
      • Summary of proof: By assumption, for any 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.
  2. 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
  3. 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 [2]

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

For a set to be absorbing

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 [3] 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 absolutely convex set or a disk) 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.[4] If this normed space is a Banach space then is called a Banach disk.

Properties

Every absorbing set contains the origin. If is an absorbing disk in a vector space then there exists an absorbing disk in such that [5]

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

See also

Notes

  1. The requirement that be scalar be non-zero cannot be dropped from this characterization.
  2. A topology on a vector space is called a vector topology or a TVS-topology 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.
  3. 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 some 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 all 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.

Proofs

  1. Proof: Let be a vector space over the field with being or and endow the field with its usual normed Euclidean topology. Let be a convex set such that for every there exists a positive real such that Because if then the proof is complete so assume Clearly, every non-empty convex subset of the real line is an interval (possibly open, closed, or half-closed; possibly degenerate (that is, a singleton set); possibly bounded or unbounded). Recall that the intersection of convex sets is convex so that for every the sets and are convex, where now the convexity of (which contains the origin and is contained in the line ) implies that is an interval contained in the line Lemma: If then the interval contains an open sub-interval that contains the origin. Proof of lemma: By assumption, since we can pick some such that and (because ) we can also pick some such that where and (since ). Because is convex and contains the distinct points and it contains the convex hull of the points which (in particular) contains the open sub-interval where this open sub-interval contains the origin (to see why, take which satisfies ), which proves the lemma. Now fix let Because was arbitrary, to prove that is absorbing in it is necessary and sufficient to show that is a neighborhood of the origin in when is given its usual Hausdorff Euclidean topology, where recall that this topology makes the map defined by into a TVS-isomorphism. If then the fact that the interval contains an open sub-interval around the origin means exactly that is a neighborhood of the origin in which completes the proof. So assume that Write so that and (naively, is the "-axis" and is the "-axis" of ). The set is contained in the convex set so that the convex hull of is contained in By the lemma, each of and are line segments (intervals) with each segment containing the origin in an open sub-interval; moreover, they clearly intersect at the origin. Pick a real such that and Let denote the convex hull of which is contained in the convex hull of and thus also contained in the convex set To finish the proof, it suffices to show that is a neighborhood of in Viewed as a subset of the complex plane is shaped like an open square with its four corners on the positive and negative and -axes (that is, in and ). So it is readily verified that contains the open ball of radius centered at the origin of Thus is a neighborhood of the origin in as desired.

Citations

  1. Narici & Beckenstein 2011, pp. 107–110.
  2. Narici & Beckenstein 2011, pp. 441–457.
  3. Narici & Beckenstein 2011, pp. 67–113.
  4. Narici & Beckenstein 2011, pp. 115–154.
  5. Narici & Beckenstein 2011, pp. 149–153.

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