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

is a vector space over the field of real numbers or complex numbers denote the open ball (respectively, the closed ball

Preliminaries

Balanced core and balanced hull

is said to be balanced

the smallest balanced set is called the is called the

One set absorbing another

is said to if it satisfies any of the following equivalent conditions:

Definition

• ) then 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.

for every non-zero

• which is the next characterization.

for every non-zero

is a balanced set then this list can be extended to include:

• may be dropped.

There exists a non-zero

to be an absorbing set, or to be a neighborhood of the origin in a topology) then this list can be extended to include:

• which gives the next statement.

• the balanced hull) its balanced core

if it is contained in some positive scalar multiple of the balanced core There exists a non-zero In words, the balanced core can be scaled to contain the balanced hull can be scaled so that its balanced core In words, the balanced core can be scaled to contain the balanced hull (according to any defining condition of "absorbs" other than this one).

then this list can be extended to include: (according to any defining condition of "absorbs" other than this one).

• ).

A set absorbing a point

A set is said to if it absorbs the singleton set As detailed below, a set is said to be

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

containing the origin is the one and only singleton subset that absorbs itself.

is the unit circle absorbs. Moreover, there does exist In contrast, every neighborhood of the origin absorbs every bounded subset (and so in particular, absorbs every singleton subset/point).

Absorbing set

is called an 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 absorbs

• Every absorbing set must contain the origin.

• The closed ball can be used in place of the open ball.

• This proves the next statement.

• Connection to topology is given its usual Hausdorff Euclidean topology is a neighborhood 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 for some is endowed with the (unique) is necessarily a TVS-isomorphism is endowed with its standard Euclidean topology induced by the Euclidean metric).

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 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 is endowed with the subspace topology This subspace topology is always a vector topology is 1-dimensional, the only vector topologies on it are the Hausdorff Euclidean topology and the trivial topology with respect to its unique Hausdorff vector topology (the Euclidean topology). is absorbing. (according to any defining condition of "absorbing" other than this one).

• depends to be an absorbing set that is also convex, balanced, and closed (such a set is called a is a barrelled space).

then to this list can be appended:

1. The algebraic interior).

is balanced then to this list can be appended:

1. ).

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

• The proof that a balanced is real).

• The proof that a convex 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 is called an interval its unique Hausdorff vector topology 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 quadrilateral

  which gives the next characterization.

Examples and sufficient conditions

For one set to absorb another

For a set to be absorbing

The unit ball of any normed vector space (or seminormed vector space is a topological vector space"

is not absorbing.

is a disk

absorbing then the same is true of the symmetric set

Auxiliary normed spaces

is convex then the symmetric set will be convex and balanced (also known as an or a This guarantees that the Minkowski functional will be a seminorm into a seminormed space that carries its canonical pseduometrizable as a limit point) forms a neighborhood basis of absorbing disks at the origin for this locally convex is a topological vector space is also a bounded subset will be a norm will form what is known as an auxiliary normed space. If this normed space is a Banach space is called a .

Properties

is an absorbing disk

Consequently, if a topological vector space is a non-meager subset of itself (or equivalently for TVSs, if it is a Baire space's topological interior is a neighborhood of the origin

Every absorbing set is a total set, meaning that every absorbing subspace is dense.

See also

Notes

Proofs

Citations

References