In
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
, a subset of a
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
(TVS) is called a barrel or a barrelled set if it is closed
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytope ...
balanced
In telecommunications and professional audio, a balanced line or balanced signal pair is a circuit consisting of two conductors of the same type, both of which have equal impedances along their lengths and equal impedances to ground and to other ...
and
absorbing.
Barrelled sets play an important role in the definitions of several classes of topological vector spaces, such as
barrelled space
In functional analysis and related areas of mathematics, a barrelled space (also written barreled space) is a topological vector space (TVS) for which every barrelled set in the space is a neighbourhood for the zero vector.
A barrelled set or a b ...
s.
Definitions
Let
be a
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
(TVS).
A subset of
is called a if it is closed
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytope ...
balanced
In telecommunications and professional audio, a balanced line or balanced signal pair is a circuit consisting of two conductors of the same type, both of which have equal impedances along their lengths and equal impedances to ground and to other ...
and
absorbing in
A subset of
is called and a if it
absorbs every
bounded subset
:''"Bounded" and "boundary" are distinct concepts; for the latter see boundary (topology). A circle in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary.
In mathematical analysis and related areas of mat ...
of
Every bornivorous subset of
is necessarily an absorbing subset of
Let
be a subset of a topological vector space
If
is a
balanced
In telecommunications and professional audio, a balanced line or balanced signal pair is a circuit consisting of two conductors of the same type, both of which have equal impedances along their lengths and equal impedances to ground and to other ...
absorbing subset of
and if there exists a sequence
of balanced absorbing subsets of
such that
for all
then
is called a in
where moreover,
is said to be a(n):
* if in addition every
is a closed and
bornivorous subset of
for every
* if in addition every
is a
closed subset
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a clo ...
of
for every
* if in addition every
is a closed and bornivorous subset of
for every
In this case,
is called a for
Properties
Note that every bornivorous ultrabarrel is an ultrabarrel and that every bornivorous suprabarrel is a suprabarrel.
Examples
* In a
semi normed vector space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" ...
the closed
unit ball
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* ''Unit'' (alb ...
is a barrel.
* Every
locally convex topological vector space
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vec ...
has a
neighbourhood basis In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point x in a topological space is the collection of all neighbourhoods of x.
Definitions
Neighbour ...
consisting of barrelled sets, although the space itself need not be a barreled space.
See also
*
*
*
References
Bibliography
*
*
*
*
*
*
{{TopologicalVectorSpaces
Topological vector spaces