In functional analysis and related areas of mathematics an absorbing set in a vector space is a set S 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 X is a vector space over the field \mathbb{K} of real numbers \R or complex numbers \Complex, and for any let denote the open ball (respectively, the closed ball) of radius r in \mathbb{K} centered at 0. Define the product of a set of scalars with a set A of vectors as and define the product of with a single vector x as

Preliminaries

Balanced core and balanced hull A subset S of X is said to be Balanced set if a s \in S for all s \in S and all scalars a satisfying |a| \leq 1; this condition may be written more succinctly as and it holds if and only if Given a set T, the smallest balanced set containing T, denoted by is called the of T while the largest balanced set contained within T, denoted by is called the of T. 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 T 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 c is any scalar then while if c \neq 0 is non-zero or if 0 \in T then also

One set absorbing another

If**** S and A are subsets of**** X,**** then**** A is**** said**** to**** **** S if**** it**** satisfies any of**** the following equivalent**** conditions****:**** If A is a balanced set then this list can be extended to include:

<ol> There exists a non-zero scalar c \neq 0 such that There exists a non-zero scalar c \neq 0 such that </li></ol> If 0 \in A (a necessary condition for A to be an [absorbing set](https://bliptext.com/articles/), or to be a neighborhood of the origin in a topology) then this list can be extended to include: <ol> There exists r > 0 such that for every scalar c satisfying |c| < r. Or stated more succinctly, </li> <li>There exists r > 0 such that for every scalar c satisfying |c| \leq r. Or stated more succinctly, <li>There exists r > 0 such that </li> <li>There exists r > 0 such that </li> <li>There exists r > 0 such that <li>There exists r > 0 such that In words, a set is absorbed by A if it is contained in some positive scalar multiple of the [balanced core](https://bliptext.com/articles/) of A.</li> <li>There exists r > 0 such that </li> <li>There exists a non-zero scalar c \neq 0 such that In words, the [balanced core](https://bliptext.com/articles/) of A contains some non-zero scalar multiple of S.</li> <li>There exists a scalar c such that In words, A can be scaled to contain the [balanced hull](https://bliptext.com/articles/) of S.</li> <li>There exists a scalar c such that </li> <li>There exists a scalar c such that In words, A can be scaled so that its [balanced core](https://bliptext.com/articles/) contains S.</li> <li>There exists a scalar c such that </li> <li>There exists a scalar c such that In words, the [balanced core](https://bliptext.com/articles/) of A can be scaled to contain the [balanced hull](https://bliptext.com/articles/) of S.</li> <li>The balanced core of A absorbs the balanced hull S (according to any defining condition of "absorbs" other than this one).</li> </ol> If 0 \not\in S or 0 \in A then this list can be extended to include: <ol> absorbs S (according to any defining condition of "absorbs" other than this one). </ol> **A set absorbing a point** ****A**** ****s****e****t**** ****i****s**** ****s****a****i****d**** ****t****o**** **** ****x**** ****i****f**** ****i****t**** ****a****b****s****o****r****b****s**** ****t****h****e**** ****s****i****n****g****l****e****t****o****n**** ****s****e****t**** ****\****{****x****\****}****.**** A set A absorbs the origin if and only if it contains the origin; that is, if and only if 0 \in A. As detailed below, a set is said to be if it absorbs every point of X. This notion of one set absorbing another is also used in other definitions: A subset of a [topological vector space](https://bliptext.com/articles/topological-vector-space) X 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 X is equal to either \R^2 or \Complex. If is the [unit circle](https://bliptext.com/articles/unit-circle) (centered at the origin \mathbf{0}) together with the origin, then is the one and only non-empty set that A absorbs. Moreover, there does exist non-empty subset of X that is absorbed by the unit circle S^1. In contrast, every [neighborhood](https://bliptext.com/articles/neighbourhood-mathematics) of the origin absorbs every [bounded subset](https://bliptext.com/articles/bounded-set-topological-vector-space) of X (and so in particular, absorbs every singleton subset/point).

Absorbing set

A subset A of a vector space** X** over a field \mathbb{K} is called an of** X** and is said to be ** X** 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: A [absorbs](https://bliptext.com/articles/) every point of X; that is, for every x \in X, A [absorbs](https://bliptext.com/articles/) \{x\}. </li> <li>A absorbs every finite subset of X.</li> <li>For every x \in X, there exists a real r > 0 such that x \in c A for any scalar satisfying |c| \geq r.</li> <li>For every x \in X, there exists a real r > 0 such that c x \in A for any scalar satisfying |c| \leq r.</li> <li>For every x \in X, there exists a real r > 0 such that </li> <li>For every x \in X, there exists a real r > 0 such that where </li> <li>A contains the origin and for every 1-dimensional vector subspace Y of X, A \cap Y is a neighborhood of the origin in Y when Y is given [its unique Hausdorff vector topology](https://bliptext.com/articles/) (i.e. the [Euclidean topology](https://bliptext.com/articles/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 \mathbb{K} is given this (Euclidean) topology.</li> <li>**0-Neighborhoods are absorbing**: This condition gives insight as to why every neighborhood of the origin in every [topological vector space](https://bliptext.com/articles/topological-vector-space) (TVS) is necessarily absorbing: If U is a neighborhood of the origin in a TVS X then for every 1-dimensional vector subspace Y, U \cap Y is a neighborhood of the origin in Y when Y is endowed with the [subspace topology](https://bliptext.com/articles/subspace-topology) induced on it by X. This [subspace topology](https://bliptext.com/articles/subspace-topology) is always a vector topology and because Y is 1-dimensional, the only vector topologies on it are [the Hausdorff Euclidean topology](https://bliptext.com/articles/) and the [trivial topology](https://bliptext.com/articles/trivial-topology), which is a subset of the Euclidean topology. So regardless of which of these vector topologies is on Y, the set U \cap Y will be a neighborhood of the origin in Y with respect to its unique Hausdorff vector topology (the Euclidean topology). Thus U is absorbing.</li> </ul> </li> <li>A contains the origin and for every 1-dimensional vector subspace Y of X, A \cap Y is absorbing in Y (according to any defining condition of "absorbing" other than this one). </li> </ol> If then to this list can be appended: If A is [balanced](https://bliptext.com/articles/balanced-set) then to this list can be appended: If A is [convex](https://bliptext.com/articles/convex-set) **or** [balanced](https://bliptext.com/articles/balanced-set) then to this list can be appended: If 0 \in A (which is necessary for A to be absorbing) then it suffices to check any of the above conditions for all non-zero x \in X, rather than all x \in X.

Examples and sufficient conditions

For one set to absorb another

Let F : X \to Y be a linear map between vector spaces and let and be balanced sets. Then C absorbs F(B) if and only if F^{-1}(C) absorbs B. If a set A absorbs another set B then any superset of A also absorbs B. A set A absorbs the origin if and only if the origin is an element of A. A set A absorbs a finite union of sets if and only it absorbs each set individuality (that is, if and only if A absorbs B_i for every ). In particular, a set A is an absorbing subset of X if and only if it absorbs every finite subset of X.

For a set to be absorbing

The unit ball of any normed vector space (or seminormed vector space) is absorbing. More generally, if X is a topological vector space (TVS) then any neighborhood of the origin in X is absorbing in X. This fact is one of the primary motivations for defining the property "absorbing in X." 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 X := \Reals 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 D is a disk in X, then D is absorbing in X if and only if This conclusion is not guaranteed if the set is balanced but not convex; for example, the union D of the x and y 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 A absorbing then the same is true of the symmetric set Auxiliary normed spaces If W is convex and absorbing in X then the symmetric set will be convex and balanced (also known as an or a ) in addition to being absorbing in X. This guarantees that the Minkowski functional of D will be a seminorm on X, thereby making into a seminormed space that carries its canonical pseduometrizable topology. The set of scalar multiples r D as r ranges over (or over any other set of non-zero scalars having 0 as a limit point) forms a neighborhood basis of absorbing disks at the origin for this locally convex topology. If X is a topological vector space and if this convex absorbing subset W is also a bounded subset of X, then all this will also be true of the absorbing disk if in addition D does not contain any non-trivial vector subspace then p_D will be a norm and will form what is known as an auxiliary normed space. If this normed space is a Banach space then D is called a.

Properties

Every absorbing set contains the origin. If D is an absorbing disk in a vector space X then there exists an absorbing disk E in X such that If A is an absorbing subset of X then and more generally, for any sequence of scalars such that Consequently, if a topological vector space X is a non-meager subset of itself (or equivalently for TVSs, if it is a Baire space) and if A is a closed absorbing subset of X then A necessarily contains a non-empty open subset of X (in other words, A's topological interior will not be empty), which guarantees that A - A is a neighborhood of the origin in X. Every absorbing set is a total set, meaning that every absorbing subspace is dense.

Citations