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.
the smallest balanced set containing
is a balanced set then this list can be extended to include: If
to be an absorbing set, or to be a neighborhood of the origin in a topology) then this list can be extended to include: If
then this list can be extended to include: A set absorbing a point A set is said to absorb a point
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.
containing the origin is the one and only singleton subset that absorbs itself.
is the unit circle (centered at the origin
In contrast, every neighborhood of the origin absorbs every bounded subset of
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): If
to be absorbing) then it suffices to check any of the above conditions for all non-zero
be a linear map between vector spaces and let
is a topological vector space (TVS) then any neighborhood of the origin in
This fact is one of the primary motivations for defining the property "absorbing in
is a non-convex balanced set that is not absorbing in
absorbing then the same is true of the symmetric set
into a seminormed space that carries its canonical pseduometrizable topology.
as a limit point) forms a neighborhood basis of absorbing disks at the origin for this locally convex topology.
is a topological vector space and if this convex absorbing subset
does not contain any non-trivial vector subspace then
will form what is known as an auxiliary normed space.
Every absorbing set contains the origin.
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
necessarily contains a non-empty open subset of