Barrelled space

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

If ${\displaystyle X}$ is any TVS then every closed convex and balanced neighborhood of the origin is necessarily a barrel in ${\displaystyle X}$ (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 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.

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

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

1. ${\displaystyle X}$ is barrelled.
2. Definition: Every barrel in ${\displaystyle X}$ 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 ${\displaystyle Y}$ 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 ${\displaystyle Y}$ (not necessarily the origin).[2]
3. For any Hausdorff TVS ${\displaystyle Y.}$ every pointwise bounded subset of ${\displaystyle L(X;Y)}$ is equicontinuous.[3]
4. For any F-space ${\displaystyle Y.}$ every pointwise bounded subset of ${\displaystyle L(X;Y)}$ is equicontinuous.[3]
   • An F-space is a complete metrizable TVS.
5. Every closed linear operator from ${\displaystyle X}$ into a complete metrizable TVS is continuous.[4]
   • A linear map ${\displaystyle F:X\to Y}$ is called closed if its graph is a closed subset of ${\displaystyle X\times Y.}$
6. Every Hausdorff TVS topology ${\displaystyle \nu }$ on ${\displaystyle X}$ that has a neighborhood basis of the origin consisting of ${\displaystyle \tau }$-closed set is course than ${\displaystyle \tau .}$[5]

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

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

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

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

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

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

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

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

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 ${\displaystyle L^{p}}$-space ${\displaystyle L^{2}([0,1])}$ is topologized as a subspace of ${\displaystyle L^{1}([0,1]),}$ 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 ${\displaystyle X}$ such that every weakly bounded subset of its continuous dual space is equicontinuous.[12]
14. A locally convex TVS ${\displaystyle X}$ such that for every Banach space ${\displaystyle B,}$ a closed linear map of ${\displaystyle X}$ into ${\displaystyle B}$ 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.

• Barrelled set
• Countably barrelled space
• Distinguished space – TVS whose strong dual is barralled
• Quasibarrelled space
• Ultrabarrelled space
• Uniform boundedness principle#Generalisations – A theorem stating that pointwise boundedness implies uniform boundedness
• Ursescu theorem – Generalization of closed graph, open mapping, and uniform boundedness theorem
• Webbed space – Space where open mapping and closed graph theorems hold

• Voigt, Jürgen (2020). A Course on Topological Vector Spaces. Compact Textbooks in Mathematics. Cham: Birkhäuser Basel. ISBN 978-3-030-32945-7. OCLC 1145563701.
• Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.