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, norm, topology, etc.) and the linear functions defined o ...
and related areas of
mathematics, a barrelled space (also written
barreled space) is 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) for which every barrelled set in the space is a
neighbourhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; American and British English spelling differences, see spelling differences) is a geographically localised community ...
for the
zero vector
In mathematics, a zero element is one of several generalizations of 0, the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context.
Additive identities
An additive iden ...
.
A barrelled set or a barrel in a topological vector space is a
set that is
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 polytop ...
,
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 ci ...
,
absorbing, and
closed.
Barrelled spaces are studied because a form of the
Banach–Steinhaus theorem still holds for them.
Barrelled spaces were introduced by .
Barrels
A
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 polytop ...
and
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 ci ...
subset of a real or complex vector space is called a and it is said to be , , or .
A or a in 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 a subset that is a
closed absorbing disk; that is, a barrel is a convex, balanced, closed, and absorbing subset.
Every barrel must contain the origin. If
and if
is any subset of
then
is a convex, balanced, and absorbing set of
if and only if this is all true of
in
for every
-dimensional vector subspace
thus if
then the requirement that a barrel be a
closed subset of
is the only defining property that does not depend on
(or lower)-dimensional vector subspaces of
If
is any TVS then every closed convex and balanced
neighborhood of the origin is necessarily a barrel in
(because every neighborhood of the origin is necessarily an absorbing subset). In fact, 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 ...
has a
neighborhood 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 ...
at its origin consisting entirely of barrels. However, in general, there exist barrels that are not neighborhoods of the origin; "barrelled spaces" are exactly those TVSs in which every barrel is necessarily a neighborhood of the origin. Every finite dimensional topological vector space is a barrelled space so examples of barrels that are not neighborhoods of the origin can only be found in infinite dimensional spaces.
Examples of barrels and non-barrels
The closure of any convex, balanced, and absorbing subset is a barrel. This is because the closure of any convex (respectively, any balanced, any absorbing) subset has this same property.
''A family of examples'': Suppose that
is equal to
(if considered as a complex vector space) or equal to
(if considered as a real vector space). Regardless of whether
is a real or complex vector space, every barrel in
is necessarily a neighborhood of the origin (so
is an example of a barrelled space). Let
_which_prevents_
S_from_being_a_neighborhood_of_the_origin)_and_then_extend_
R_to_
[\pi,_2_\pi)_by_defining_
R(\theta)_:=_R(\theta_-_\pi),_which_guarantees_that_
S_is_balanced_in_
\R^2.
_Properties_of_barrels
- In_any_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_...