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 barrel in a topological vector space is a set that is convex, 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 Bourbaki (1950).

Barrels

A convex and balanced subset of a real or complex vector space is called a disk and it is said to be disked, absolutely convex, or convex balanced.

A barrel or a barrelled set in a topological vector space (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 [math]\displaystyle{ \dim X \geq 2 }[/math] and if [math]\displaystyle{ S }[/math] is any subset of [math]\displaystyle{ X, }[/math] then [math]\displaystyle{ S }[/math] is a convex, balanced, and absorbing set of [math]\displaystyle{ X }[/math] if and only if this is all true of [math]\displaystyle{ S \cap Y }[/math] in [math]\displaystyle{ Y }[/math] for every [math]\displaystyle{ 2 }[/math]-dimensional vector subspace [math]\displaystyle{ Y; }[/math] thus if [math]\displaystyle{ \dim X \gt 2 }[/math] then the requirement that a barrel be a closed subset of [math]\displaystyle{ X }[/math] is the only defining property that does not depend solely on [math]\displaystyle{ 2 }[/math] (or lower)-dimensional vector subspaces of [math]\displaystyle{ X. }[/math]

If [math]\displaystyle{ X }[/math] is any TVS then every closed convex and balanced neighborhood of the origin is necessarily a barrel in [math]\displaystyle{ X }[/math] (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 might 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 [math]\displaystyle{ X }[/math] is equal to [math]\displaystyle{ \Complex }[/math] (if considered as a complex vector space) or equal to [math]\displaystyle{ \R^2 }[/math] (if considered as a real vector space). Regardless of whether [math]\displaystyle{ X }[/math] is a real or complex vector space, every barrel in [math]\displaystyle{ X }[/math] is necessarily a neighborhood of the origin (so [math]\displaystyle{ X }[/math] is an example of a barrelled space). Let [math]\displaystyle{ R : [0, 2\pi) \to (0, \infty] }[/math] be any function and for every angle [math]\displaystyle{ \theta \in [0, 2 \pi), }[/math] let [math]\displaystyle{ S_{\theta} }[/math] denote the closed line segment from the origin to the point [math]\displaystyle{ R(\theta) e^{i \theta} \in \Complex. }[/math] Let [math]\displaystyle{ S := \bigcup_{\theta \in [0, 2 \pi)} S_{\theta}. }[/math] Then [math]\displaystyle{ S }[/math] is always an absorbing subset of [math]\displaystyle{ \R^2 }[/math] (a real vector space) but it is an absorbing subset of [math]\displaystyle{ \Complex }[/math] (a complex vector space) if and only if it is a neighborhood of the origin. Moreover, [math]\displaystyle{ S }[/math] is a balanced subset of [math]\displaystyle{ \R^2 }[/math] if and only if [math]\displaystyle{ R(\theta) = R(\pi + \theta) }[/math] for every [math]\displaystyle{ 0 \leq \theta \lt \pi }[/math] (if this is the case then [math]\displaystyle{ R }[/math] and [math]\displaystyle{ S }[/math] are completely determined by [math]\displaystyle{ R }[/math]'s values on [math]\displaystyle{ [0, \pi) }[/math]) but [math]\displaystyle{ S }[/math] is a balanced subset of [math]\displaystyle{ \Complex }[/math] if and only it is an open or closed ball centered at the origin (of radius [math]\displaystyle{ 0 \lt r \leq \infty }[/math]). In particular, barrels in [math]\displaystyle{ \Complex }[/math] are exactly those closed balls centered at the origin with radius in [math]\displaystyle{ (0, \infty]. }[/math] If [math]\displaystyle{ R(\theta) := 2 \pi - \theta }[/math] then [math]\displaystyle{ S }[/math] is a closed subset that is absorbing in [math]\displaystyle{ \R^2 }[/math] but not absorbing in [math]\displaystyle{ \Complex, }[/math] and that is neither convex, balanced, nor a neighborhood of the origin in [math]\displaystyle{ X. }[/math] By an appropriate choice of the function [math]\displaystyle{ R, }[/math] it is also possible to have [math]\displaystyle{ S }[/math] be a balanced and absorbing subset of [math]\displaystyle{ \R^2 }[/math] that is neither closed nor convex. To have [math]\displaystyle{ S }[/math] be a balanced, absorbing, and closed subset of [math]\displaystyle{ \R^2 }[/math] that is neither convex nor a neighborhood of the origin, define [math]\displaystyle{ R }[/math] on [math]\displaystyle{ [0, \pi) }[/math] as follows: for [math]\displaystyle{ 0 \leq \theta \lt \pi, }[/math] let [math]\displaystyle{ R(\theta) := \pi - \theta }[/math] (alternatively, it can be any positive function on [math]\displaystyle{ [0, \pi) }[/math] that is continuously differentiable, which guarantees that [math]\displaystyle{ \lim_{\theta \searrow 0} R(\theta) = R(0) \gt 0 }[/math] and that [math]\displaystyle{ S }[/math] is closed, and that also satisfies [math]\displaystyle{ \lim_{\theta \nearrow \pi} R(\theta) = 0, }[/math] which prevents [math]\displaystyle{ S }[/math] from being a neighborhood of the origin) and then extend [math]\displaystyle{ R }[/math] to [math]\displaystyle{ [\pi, 2 \pi) }[/math] by defining [math]\displaystyle{ R(\theta) := R(\theta - \pi), }[/math] which guarantees that [math]\displaystyle{ S }[/math] is balanced in [math]\displaystyle{ \R^2. }[/math]

Properties of barrels

• In any topological vector space (TVS) [math]\displaystyle{ X, }[/math] every barrel in [math]\displaystyle{ X }[/math] absorbs every compact convex subset of [math]\displaystyle{ X. }[/math]^[1]
• In any locally convex Hausdorff TVS [math]\displaystyle{ X, }[/math] every barrel in [math]\displaystyle{ X }[/math] absorbs every convex bounded complete subset of [math]\displaystyle{ X. }[/math]^[1]
• If [math]\displaystyle{ X }[/math] is locally convex then a subset [math]\displaystyle{ H }[/math] of [math]\displaystyle{ X^{\prime} }[/math] is [math]\displaystyle{ \sigma\left(X^{\prime}, X\right) }[/math]-bounded if and only if there exists a barrel [math]\displaystyle{ B }[/math] in [math]\displaystyle{ X }[/math] such that [math]\displaystyle{ H \subseteq B^{\circ}. }[/math]^[1]
• Let [math]\displaystyle{ (X, Y, b) }[/math] be a pairing and let [math]\displaystyle{ \nu }[/math] be a locally convex topology on [math]\displaystyle{ X }[/math] consistent with duality. Then a subset [math]\displaystyle{ B }[/math] of [math]\displaystyle{ X }[/math] is a barrel in [math]\displaystyle{ (X, \nu) }[/math] if and only if [math]\displaystyle{ B }[/math] is the polar of some [math]\displaystyle{ \sigma(Y, X, b) }[/math]-bounded subset of [math]\displaystyle{ Y. }[/math]^[1]
• Suppose [math]\displaystyle{ M }[/math] is a vector subspace of finite codimension in a locally convex space [math]\displaystyle{ X }[/math] and [math]\displaystyle{ B \subseteq M. }[/math] If [math]\displaystyle{ B }[/math] is a barrel (resp. bornivorous barrel, bornivorous disk) in [math]\displaystyle{ M }[/math] then there exists a barrel (resp. bornivorous barrel, bornivorous disk) [math]\displaystyle{ C }[/math] in [math]\displaystyle{ X }[/math] such that [math]\displaystyle{ B = C \cap M. }[/math]^[2]

Characterizations of barreled spaces

Denote by [math]\displaystyle{ L(X; Y) }[/math] the space of continuous linear maps from [math]\displaystyle{ X }[/math] into [math]\displaystyle{ Y. }[/math]

If [math]\displaystyle{ (X, \tau) }[/math] is a Hausdorff topological vector space (TVS) with continuous dual space [math]\displaystyle{ X^{\prime} }[/math] then the following are equivalent:

1. [math]\displaystyle{ X }[/math] is barrelled.
2. Definition: Every barrel in [math]\displaystyle{ X }[/math] is a neighborhood of the origin.
   • This definition is similar to a characterization of Baire TVSs proved by Saxon [1974], who proved that a TVS [math]\displaystyle{ Y }[/math] with a topology that is not the indiscrete topology is a Baire space if and only if every absorbing balanced subset is a neighborhood of some point of [math]\displaystyle{ Y }[/math] (not necessarily the origin).^[2]
3. For any Hausdorff TVS [math]\displaystyle{ Y }[/math] every pointwise bounded subset of [math]\displaystyle{ L(X; Y) }[/math] is equicontinuous.^[3]
4. For any F-space [math]\displaystyle{ Y }[/math] every pointwise bounded subset of [math]\displaystyle{ L(X; Y) }[/math] is equicontinuous.^[3]
   • An F-space is a complete metrizable TVS.
5. Every closed linear operator from [math]\displaystyle{ X }[/math] into a complete metrizable TVS is continuous.^[4]
   • A linear map [math]\displaystyle{ F : X \to Y }[/math] is called closed if its graph is a closed subset of [math]\displaystyle{ X \times Y. }[/math]
6. Every Hausdorff TVS topology [math]\displaystyle{ \nu }[/math] on [math]\displaystyle{ X }[/math] that has a neighborhood basis of the origin consisting of [math]\displaystyle{ \tau }[/math]-closed set is course than [math]\displaystyle{ \tau. }[/math]^[5]

If [math]\displaystyle{ (X, \tau) }[/math] is locally convex space then this list may be extended by appending:

7. There exists a TVS [math]\displaystyle{ Y }[/math] not carrying the indiscrete topology (so in particular, [math]\displaystyle{ Y \neq \{0\} }[/math]) such that every pointwise bounded subset of [math]\displaystyle{ L(X; Y) }[/math] is equicontinuous.^[2]
8. For any locally convex TVS [math]\displaystyle{ Y, }[/math] every pointwise bounded subset of [math]\displaystyle{ L(X; Y) }[/math] is equicontinuous.^[2]
   • 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 principal holds.
9. Every [math]\displaystyle{ \sigma\left(X^{\prime}, X\right) }[/math]-bounded subset of the continuous dual space [math]\displaystyle{ X }[/math] is equicontinuous (this provides a partial converse to the Banach-Steinhaus theorem).^[2][6]
10. [math]\displaystyle{ X }[/math] carries the strong dual topology [math]\displaystyle{ \beta\left(X, X^{\prime}\right). }[/math]^[2]
11. Every lower semicontinuous seminorm on [math]\displaystyle{ X }[/math] is continuous.^[2]
12. Every linear map [math]\displaystyle{ F : X \to Y }[/math] into a locally convex space [math]\displaystyle{ Y }[/math] is almost continuous.^[2]
    • A linear map [math]\displaystyle{ F : X \to Y }[/math] is called almost continuous if for every neighborhood [math]\displaystyle{ V }[/math] of the origin in [math]\displaystyle{ Y, }[/math] the closure of [math]\displaystyle{ F^{-1}(V) }[/math] is a neighborhood of the origin in [math]\displaystyle{ X. }[/math]
13. Every surjective linear map [math]\displaystyle{ F : Y \to X }[/math] from a locally convex space [math]\displaystyle{ Y }[/math] is almost open.^[2]
    • This means that for every neighborhood [math]\displaystyle{ V }[/math] of 0 in [math]\displaystyle{ Y, }[/math] the closure of [math]\displaystyle{ F(V) }[/math] is a neighborhood of 0 in [math]\displaystyle{ X. }[/math]
14. If [math]\displaystyle{ \omega }[/math] is a locally convex topology on [math]\displaystyle{ X }[/math] such that [math]\displaystyle{ (X, \omega) }[/math] has a neighborhood basis at the origin consisting of [math]\displaystyle{ \tau }[/math]-closed sets, then [math]\displaystyle{ \omega }[/math] is weaker than [math]\displaystyle{ \tau. }[/math]^[2]

If [math]\displaystyle{ X }[/math] is a Hausdorff locally convex space then this list may be extended by appending:

15. Closed graph theorem: Every closed linear operator [math]\displaystyle{ F : X \to Y }[/math] into a Banach space [math]\displaystyle{ Y }[/math] is continuous.^[7]
    • The linear operator is called closed if its graph is a closed subset of [math]\displaystyle{ X \times Y. }[/math]
16. For every subset [math]\displaystyle{ A }[/math] of the continuous dual space of [math]\displaystyle{ X, }[/math] the following properties are equivalent: [math]\displaystyle{ A }[/math] is^[6]
    1. equicontinuous;
    2. relatively weakly compact;
    3. strongly bounded;
    4. weakly bounded.
17. The 0-neighborhood bases in [math]\displaystyle{ X }[/math] and the fundamental families of bounded sets in [math]\displaystyle{ X_{\beta}^{\prime} }[/math] correspond to each other by polarity.^[6]

If [math]\displaystyle{ X }[/math] is metrizable topological vector space then this list may be extended by appending:

18. For any complete metrizable TVS [math]\displaystyle{ Y }[/math] every pointwise bounded sequence in [math]\displaystyle{ L(X; Y) }[/math] is equicontinuous.^[3]

If [math]\displaystyle{ X }[/math] is a locally convex metrizable topological vector space then this list may be extended by appending:

19. (Property S): The weak* topology on [math]\displaystyle{ X^{\prime} }[/math] is sequentially complete.^[8]
20. (Property C): Every weak* bounded subset of [math]\displaystyle{ X^{\prime} }[/math] is [math]\displaystyle{ \sigma\left(X^{\prime}, X\right) }[/math]-relatively countably compact.^[8]
21. (𝜎-barrelled): Every countable weak* bounded subset of [math]\displaystyle{ X^{\prime} }[/math] is equicontinuous.^[8]
22. (Baire-like): [math]\displaystyle{ X }[/math] is not the union of an increase sequence of nowhere dense disks.^[8]

Examples and sufficient conditions

Each of the following topological vector spaces is barreled:

1. TVSs that are Baire space.
   • Consequently, every topological vector space that is of the second category in itself is barrelled.
2. F-spaces, Fréchet spaces, Banach spaces, and Hilbert spaces.
   • However, there exist normed vector spaces that are not barrelled. For example, if the [math]\displaystyle{ L^p }[/math]-space [math]\displaystyle{ L^2([0, 1]) }[/math] is topologized as a subspace of [math]\displaystyle{ L^1([0, 1]), }[/math] then it is not barrelled.
3. Complete pseudometrizable TVSs.^[9]
   • 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.^[10]
7. A sequentially complete quasibarrelled space.
8. A quasi-complete Hausdorff locally convex infrabarrelled space.^[2]
   • A TVS is called quasi-complete if every closed and bounded subset is complete.
9. A TVS with a dense barrelled vector subspace.^[2]
   • Thus the completion of a barreled space is barrelled.
10. A Hausdorff locally convex TVS with a dense infrabarrelled vector subspace.^[2]
    • Thus the completion of an infrabarrelled Hausdorff locally convex space is barrelled.^[2]
11. A vector subspace of a barrelled space that has countable codimensional.^[2]
    • In particular, a finite codimensional vector subspace of a barrelled space is barreled.
12. A locally convex ultrabarelled TVS.^[11]
13. A Hausdorff locally convex TVS [math]\displaystyle{ X }[/math] such that every weakly bounded subset of its continuous dual space is equicontinuous.^[12]
14. A locally convex TVS [math]\displaystyle{ X }[/math] such that for every Banach space [math]\displaystyle{ B, }[/math] a closed linear map of [math]\displaystyle{ X }[/math] into [math]\displaystyle{ B }[/math] is necessarily continuous.^[13]
15. A product of a family of barreled spaces.^[14]
16. A locally convex direct sum and the inductive limit of a family of barrelled spaces.^[15]
17. A quotient of a barrelled space.^[16][15]
18. A Hausdorff sequentially complete quasibarrelled boundedly summing TVS.^[17]
19. A locally convex Hausdorff reflexive space is barrelled.

Counter examples

• A barrelled space need not be Montel, complete, metrizable, unordered Baire-like, nor the inductive limit of Banach spaces.
• Not all normed spaces are barrelled. However, they are all infrabarrelled.^[2]
• A closed subspace of a barreled space is not necessarily countably quasi-barreled (and thus not necessarily barrelled).^[18]
• There exists a dense vector subspace of the Fréchet barrelled space [math]\displaystyle{ \R^{\N} }[/math] that is not barrelled.^[2]
• There exist complete locally convex TVSs that are not barrelled.^[2]
• The finest locally convex topology on an infinite-dimensional vector space is a Hausdorff barrelled space that is a meagre subset of itself (and thus not a Baire space).^[2]

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.^[20] When the vector space [math]\displaystyle{ Y }[/math] consists of the complex numbers then the following generalization also holds.

Recall that a linear map [math]\displaystyle{ F : X \to Y }[/math] is called closed if its graph is a closed subset of [math]\displaystyle{ X \times Y. }[/math]

Other properties

• Every Hausdorff barrelled space is quasi-barrelled.^[23]
• A linear map from a barrelled space into a locally convex space is almost continuous.
• A linear map from a locally convex space onto a barrelled space is almost open.
• A separately continuous bilinear map from a product of barrelled spaces into a locally convex space is hypocontinuous.^[24]
• A linear map with a closed graph from a barreled TVS into a [math]\displaystyle{ B_r }[/math]-complete TVS is necessarily continuous.^[13]

See also

• Barrelled set
• Countably barrelled space
• Distinguished space – TVS whose strong dual is barralled
• Quasibarrelled space
• Ultrabarrelled space
• Ursescu theorem – Generalization of closed graph, open mapping, and uniform boundedness theorem
• Webbed space – Space where open mapping and closed graph theorems hold

References

Bibliography

• Template:Voigt A Course on Topological Vector Spaces
• Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114. 

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