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 ${\displaystyle \dim X\geq 2}$ 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.

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 ${\displaystyle X}$ 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 ${\displaystyle \mathbb {R} ^{2}.}$

Properties of barrels

• 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]

Characterizations of barreled 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 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]

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 ${\displaystyle L^{p}}$-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.

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

Properties of barreled spaces

Banach–Steinhaus generalization

The importance of barrelled spaces is due mainly to the following results.

Theorem ^[19]  — Let ${\displaystyle X}$ be a barrelled TVS and ${\displaystyle Y}$ be a locally convex TVS. Let ${\displaystyle H}$ be a subset of the space ${\displaystyle L(X;Y)}$ of continuous linear maps from ${\displaystyle X}$ into ${\displaystyle Y}$. The following are equivalent:

1. ${\displaystyle H}$ is bounded for the topology of pointwise convergence;
2. ${\displaystyle H}$ is bounded for the topology of bounded convergence;
3. ${\displaystyle H}$ is equicontinuous.

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.

Theorem ^[21]  — If ${\displaystyle X}$ is a barrelled TVS over the complex numbers and ${\displaystyle H}$ is a subset of the continuous dual space of ${\displaystyle X}$, then the following are equivalent:

1. ${\displaystyle H}$ is weakly bounded;
2. ${\displaystyle H}$ is strongly bounded;
3. ${\displaystyle H}$ is equicontinuous;
4. ${\displaystyle H}$ is relatively compact in the weak dual topology.

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

Closed Graph Theorem ^[22]  — Every closed linear operator from a Hausdorff barrelled TVS into a complete metrizable TVS is continuous.

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 ${\displaystyle B_{r}}$-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
• 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

References

1. Narici & Beckenstein 2011, pp. 225–273.
2. Narici & Beckenstein 2011, pp. 371–423.
3. Adasch, Ernst & Keim 1978, p. 39.
4. Adasch, Ernst & Keim 1978, p. 43.
5. Adasch, Ernst & Keim 1978, p. 32.
6. Schaefer & Wolff 1999 , pp. 127, 141 Trèves 2006 , p. 350.
7. Narici & Beckenstein 2011, p. 477.
8. Narici & Beckenstein 2011, p. 399.
9. Narici & Beckenstein 2011, p. 383.
10. Khaleelulla 1982, pp. 28–63.
11. Narici & Beckenstein 2011, pp. 418–419.
12. Trèves 2006, p. 350.
13. Schaefer & Wolff 1999, p. 166.
14. Schaefer & Wolff 1999, p. 138.
15. Schaefer & Wolff 1999, p. 61.
16. Trèves 2006, p. 346.
17. Adasch, Ernst & Keim 1978, p. 77.
18. Schaefer & Wolff 1999, pp. 103–110.
19. Trèves 2006, p. 347.
20. Trèves 2006, p. 348.
21. Trèves 2006, p. 349.
22. Adasch, Ernst & Keim 1978, p. 41.
23. Adasch, Ernst & Keim 1978, pp. 70–73.
24. Trèves 2006, p. 424.

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