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).

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.

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

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

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 principle 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.

• 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 ${\displaystyle \mathbb {R} ^{\mathbb {N} }}$ 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]

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 ${\displaystyle Y}$ consists of the complex numbers then the following generalization also holds.

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

• 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 ${\displaystyle B_{r}}$-complete TVS is necessarily continuous.^[13]

• 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.