Barreled 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 (for example, Inner product space#Definition, inner product, Norm (mathematics ...
and related areas of
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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 (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neighbourh ...
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 id ...
. 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 polytop ...
, balanced, 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
subset In mathematics, a 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 a ...
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 \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 cl ...
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 (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neigh ...
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 has a neighborhood basis 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 (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neigh ...
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

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 polar of some \sigma(Y, X, b)-bounded subset of Y.
  • Suppose M is a vector subspace of finite codimension in a locally convex space X and B \subseteq M. If B is a barrel (resp. bornivorous barrel, bornivorous disk) in M then there exists a barrel (resp. bornivorous barrel, bornivorous disk) C in X such that B = C \cap M.


    • Characterizations of barreled spaces

    Denote by L(X; Y) the space of continuous linear maps from X into Y. If (X, \tau) is a Hausdorff space">Hausdorff
    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) with continuous dual space X^ then the following are equivalent:
    1. X is barrelled.
    2. : Every barrel in X is a neighborhood of the origin. * This definition is similar to a characterization of Baire TVSs proved by Saxon 974 who proved that a TVS Y with a topology that is not the
      indiscrete topology In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the conseque ...
      is a
      Baire space In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior. According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are ...
      if and only if every absorbing balanced subset is a neighborhood of point of Y (not necessarily the origin).
    3. For any Hausdorff TVS Y every pointwise bounded subset of L(X; Y) is equicontinuous.
    4. For any F-space Y every pointwise bounded subset of L(X; Y) is equicontinuous. * An F-space is a complete metrizable TVS.
    5. Every closed linear operator from X into a complete metrizable TVS is continuous. * A linear map F : X \to Y is called closed if its graph is a closed subset of X \times Y.
    6. Every Hausdorff TVS topology \nu on X that has a neighborhood basis of the origin consisting of \tau-closed set is course than \tau.
    If (X, \tau) is locally convex space then this list may be extended by appending:
    1. There exists a TVS Y not carrying the
      indiscrete topology In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the conseque ...
      (so in particular, Y \neq \) such that every pointwise bounded subset of L(X; Y) is equicontinuous.
    2. For any locally convex TVS Y, every pointwise bounded subset of L(X; Y) is equicontinuous. * It follows from the above two characterizations that in the class of locally convex TVS, barrelled spaces are exactly those for which the uniform boundedness principle holds.
    3. Every \sigma\left(X^, X\right)-bounded subset of the continuous dual space X is equicontinuous (this provides a partial converse to the Banach-Steinhaus theorem)..
    4. X carries the strong dual topology \beta\left(X, X^\right).
    5. Every lower semicontinuous
      seminorm In mathematics, particularly in functional analysis, a seminorm is like a Norm (mathematics), norm but need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some Absorbing ...
      on X is continuous.
    6. Every linear map F : X \to Y into a locally convex space Y is almost continuous. * A linear map F : X \to Y is called if for every neighborhood V of the origin in Y, the closure of F^(V) is a neighborhood of the origin in X.
    7. Every surjective linear map F : Y \to X from a locally convex space Y is almost open. * This means that for every neighborhood V of 0 in Y, the closure of F(V) is a neighborhood of 0 in X.
    8. If \omega is a locally convex topology on X such that (X, \omega) has a neighborhood basis at the origin consisting of \tau-closed sets, then \omega is weaker than \tau.
    If X is a Hausdorff locally convex space then this list may be extended by appending:
    1. Closed graph theorem: Every closed linear operator F : X \to Y into a
      Banach space In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...
      Y is continuous. * The linear operator is called if its graph is a closed subset of X \times Y.
    2. For every subset A of the continuous dual space of X, the following properties are equivalent: A is
      1. equicontinuous;
      2. relatively weakly compact;
      3. strongly bounded;
      4. weakly bounded.
    3. The 0-neighborhood bases in X and the fundamental families of bounded sets in X_^ correspond to each other by polarity.
    If X is
    metrizable topological vector space In functional analysis and related areas of mathematics, a metrizable (resp. pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric). An LM-space is an inductive limit of a sequence of ...
    then this list may be extended by appending:
    1. For any complete metrizable TVS Y every pointwise bounded in L(X; Y) is equicontinuous.
    If X is a locally convex
    metrizable topological vector space In functional analysis and related areas of mathematics, a metrizable (resp. pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric). An LM-space is an inductive limit of a sequence of ...
    then this list may be extended by appending:
    1. (): The weak* topology on X^ is sequentially complete.
    2. (): Every weak* bounded subset of X^ is \sigma\left(X^, X\right)-relatively countably compact.
    3. (): Every countable weak* bounded subset of X^ is equicontinuous.
    4. (): X is not the union of an increase sequence of nowhere dense disks.


    Examples and sufficient conditions

    Each of the following topological vector spaces is barreled:
    1. TVSs that are
      Baire space In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior. According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are ...
      . * Consequently, every topological vector space that is of the second category in itself is barrelled.
    2. F-spaces, Fréchet spaces,
      Banach space In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...
      s, and
      Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
      s. * However, there exist normed vector spaces that are barrelled. For example, if the L^p-space L^2(
      , 1 The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
      is topologized as a subspace of L^1(
      , 1 The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
      , then it is not barrelled.
    3. Complete pseudometrizable TVSs. * Consequently, every finite-dimensional TVS is barrelled.
    4. Montel spaces.
    5. Strong dual spaces of Montel spaces (since they are necessarily Montel spaces).
    6. A locally convex quasi-barrelled space that is also a σ-barrelled space.
    7. A sequentially complete quasibarrelled space.
    8. A quasi-complete Hausdorff locally convex infrabarrelled space. * A TVS is called quasi-complete if every closed and bounded subset is complete.
    9. A TVS with a dense barrelled vector subspace. * Thus the completion of a barreled space is barrelled.
    10. A Hausdorff locally convex TVS with a dense infrabarrelled vector subspace. * Thus the completion of an infrabarrelled Hausdorff locally convex space is barrelled.
    11. A vector subspace of a barrelled space that has countable codimensional. * In particular, a finite codimensional vector subspace of a barrelled space is barreled.
    12. A locally convex ultrabarelled TVS.
    13. A Hausdorff locally convex TVS X such that every weakly bounded subset of its continuous dual space is equicontinuous.
    14. A locally convex TVS X such that for every Banach space B, a closed linear map of X into B is necessarily continuous.
    15. A product of a family of barreled spaces.
    16. A locally convex direct sum and the inductive limit of a family of barrelled spaces.
    17. A quotient of a barrelled space.
    18. A Hausdorff sequentially complete quasibarrelled boundedly summing TVS.
    19. A locally convex Hausdorff
      reflexive space In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space for which the canonical evaluation map from X into its bidual (which is the strong dual of the strong dual of X) is a homeomo ...
      is barrelled.


    Counterexamples


    Properties of barreled spaces


    Banach–Steinhaus generalization

    The importance of barrelled spaces is due mainly to the following results. The Banach-Steinhaus theorem is a corollary of the above result. When the vector space Y consists of the complex numbers then the following generalization also holds. Recall that a linear map F : X \to Y is called closed if its graph is a closed subset of X \times Y.


    Other properties


    See also

    * * * * * * * *


    References


    Bibliography

    * * * * * * * * * * * * * * * * * * * * * {{Topological vector spaces Topological vector spaces