Barrelled space
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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 ...
and related areas of
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural are ...
for the
zero vector In mathematics, a zero element is one of several generalizations of 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 identi ...
. A barrelled set or a barrel in a topological vector space is a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
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 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 ...
, absorbing, and
closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
. Barrelled spaces are studied because a form of the
Banach–Steinhaus theorem In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerst ...
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 polytope ...
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 ...
subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
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 Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
absorbing disk; that is, a barrel is a convex, balanced, closed, and absorbing subset. Every barrel must contain the origin. If \dim X \geq 2 and if S is any subset of X, then S is a convex, balanced, and absorbing set of X if and only if this is all true of S \cap Y in Y for every 2-dimensional vector subspace Y; thus if \dim X > 2 then the requirement that a barrel be 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 X is the only defining property that does not depend on 2 (or lower)-dimensional vector subspaces of X. If X is any TVS then every closed convex and balanced
neighborhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
of the origin is necessarily a barrel in X (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 vec ...
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 X is equal to \Complex (if considered as a complex vector space) or equal to \R^2 (if considered as a real vector space). Regardless of whether X is a real or complex vector space, every barrel in X is necessarily a neighborhood of the origin (so X is an example of a barrelled space). Let R : , 2\pi) \to (0, \infty/math> be any function and for every angle \theta \in [0, 2 \pi), let S_ denote the closed line segment from the origin to the point R(\theta) e^ \in \Complex. Let S := \bigcup_ S_. Then S is always an absorbing subset of \R^2 (a real vector space) but it is an absorbing subset of \Complex (a complex vector space) if and only if it is a
neighborhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
of the origin. Moreover, S is a balanced subset of \R^2 if and only if R(\theta) = R(\pi + \theta) for every 0 \leq \theta < \pi (if this is the case then R and S are completely determined by R's values on [0, \pi)) but S is a balanced subset of \Complex if and only it is an open or closed ball centered at the origin (of radius 0 < r \leq \infty). In particular, barrels in \Complex are exactly those closed balls centered at the origin with radius in (0, \infty]. If R(\theta) := 2 \pi - \theta then S is a closed subset that is absorbing in \R^2 but not absorbing in \Complex, and that is neither convex, balanced, nor a neighborhood of the origin in X. By an appropriate choice of the function R, it is also possible to have S be a balanced and absorbing subset of \R^2 that is neither closed nor convex. To have S be a balanced, absorbing, and closed subset of \R^2 that is convex nor a neighborhood of the origin, define R on \lim__R(\theta)_=_R(0)_>_0_and_that_S_is_closed,_and_that_also_satisfies_\lim__R(\theta)_=_0,_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

_(TVS)_X,_every_barrel_in_X_
\lim__R(\theta)_=_R(0)_>_0_and_that_S_is_closed,_and_that_also_satisfies_\lim__R(\theta)_=_0,_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

_(TVS)_X,_every_barrel_in_X_Absorbing_set">absorbs_every_compact_convex_subset_of_X.
  • In_any_
    \lim__R(\theta)_=_R(0)_>_0_and_that_S_is_closed,_and_that_also_satisfies_\lim__R(\theta)_=_0,_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

    _(TVS)_X,_every_barrel_in_X_Absorbing_set">absorbs_every_compact_convex_subset_of_X.
  • In_any_Locally_convex_topological_vector_space">locally_convex_ 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_ve_...
    _Hausdorff_TVS_X,_every_barrel_in_X_absorbs_every_convex_bounded_complete_subset_of_X.
  • If_X_is_locally_convex_then_a_subset_H_of_X^_is_\sigma\left(X^,_X\right)-bounded_if_and_only_if_there_exists_a_
    \lim__R(\theta)_=_R(0)_>_0_and_that_S_is_closed,_and_that_also_satisfies_\lim__R(\theta)_=_0,_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

    _(TVS)_X,_every_barrel_in_X_Absorbing_set">absorbs_every_compact_convex_subset_of_X.
  • In_any_Locally_convex_topological_vector_space">locally_convex_ 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_ve_...
    _Hausdorff_TVS_X,_every_barrel_in_X_absorbs_every_convex_bounded_complete_subset_of_X.
  • If_X_is_locally_convex_then_a_subset_H_of_X^_is_\sigma\left(X^,_X\right)-bounded_if_and_only_if_there_exists_a_#barrel">barrel_ A_barrel_or_cask_is_a_hollow__cylindrical_container_with_a_bulging_center,_longer_than_it_is_wide._They_are_traditionally_made_of_wooden__staves_and_bound_by_wooden_or_metal_hoops._The_word_vat_is_often_used_for_large_containers_for_liquids,__...
    _B_in_X_such_that_H_\subseteq_B^.
  • Let_(X,_Y,_b)_be_a_
    \lim__R(\theta)_=_R(0)_>_0_and_that_S_is_closed,_and_that_also_satisfies_\lim__R(\theta)_=_0,_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

    _(TVS)_X,_every_barrel_in_X_Absorbing_set">absorbs_every_compact_convex_subset_of_X.
  • In_any_Locally_convex_topological_vector_space">locally_convex_ 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_ve_...
    _Hausdorff_TVS_X,_every_barrel_in_X_absorbs_every_convex_bounded_complete_subset_of_X.
  • If_X_is_locally_convex_then_a_subset_H_of_X^_is_\sigma\left(X^,_X\right)-bounded_if_and_only_if_there_exists_a_#barrel">barrel_ A_barrel_or_cask_is_a_hollow__cylindrical_container_with_a_bulging_center,_longer_than_it_is_wide._They_are_traditionally_made_of_wooden__staves_and_bound_by_wooden_or_metal_hoops._The_word_vat_is_often_used_for_large_containers_for_liquids,__...
    _B_in_X_such_that_H_\subseteq_B^.
  • Let_(X,_Y,_b)_be_a_Dual_system">pairing_ In_mathematics,_a_pairing_is_an_''R''-bilinear_map_from_the_Cartesian_product_of_two_''R''-modules,_where_the_underlying_ring_''R''_is_commutative. _Definition Let_''R''_be_a_commutative_ring_with_unit,_and_let_''M'',_''N''_and_''L''_be__''R''-modu_...
    _and_let_\nu_be_a_locally_convex_topology_on_X_consistent_with_duality._Then_a_subset_B_of_X_is_a_barrel_in_(X,_\nu)_if_and_only_if_B_is_the_ \lim__R(\theta)_=_R(0)_>_0_and_that_S_is_closed,_and_that_also_satisfies_\lim__R(\theta)_=_0,_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