In functional analysis and related areas of mathematics, a **barrelled space** (also written **barreled space**) is a topological vector spaces (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.

## Barrels

Let X be a topological vector space (TVS).

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 TVS is a subset that is a closed absorbing disk.

The only topological requirement on a barrel is that it be a closed subset of the TVS; all other requirements (i.e. being a disk and being absorbing) are purely algebraic properties.

### Properties of barrels

- In any topological vector space (TVS) X, every barrel in X absorbs every compact convex subset of X.
^{[1]} - In any locally convex Hausdorff TVS X, every barrel in X absorbs every convex bounded complete subset of X.
^{[1]} - If X is locally convex then a subset H of is -bounded if and only if there exists a barrel B in X such that
*H*⊆*B*°.^{[1]} - Let (
*X*,*Y*,*b*) be a pairing and let 𝜏 be a locally convex topology on X consistent with duality. Then a subset B of X is a barrel in (*X*, 𝜏) if and only if B is the polar of some 𝜎(*Y*,*X*,*b*)-bounded subset of Y.^{[1]} - Suppose M is a vector subspace of finite codimension in a locally convex space X and
*B*⊆*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*∩*M*.^{[2]}

## Characterizations of barreled spaces

**Notation**: Let L(*X*;*Y*) denote the space of continuous linear maps from X into Y.

If (*X*, 𝜏) is a topological vector space (TVS) with continuous dual *X*' then the following are equivalent:

- X is barrelled;
- (definition) Every barrel in X is a neighborhood of the origin;
- This definition is similar to a characterization of Baire TVSs proved by Saxon [1974], who showed that a TVS 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 Y (not necessarily the origin).^{[2]}

- This definition is similar to a characterization of Baire TVSs proved by Saxon [1974], who showed that a TVS 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

If (*X*, 𝜏) is Hausdorff then we may add to this list:

- For any Hausdorff TVS Y, every pointwise bounded subset of L(
*X*;*Y*) is equicontinuous;^{[3]} - For any F-space Y, every pointwise bounded subset of L(
*X*;*Y*) is equicontinuous;^{[3]}- An F-space is a complete metrizable TVS.

- Every closed linear operator from X into a complete metrizable TVS is continuous.
^{[4]}- Recall that a linear map
*F*:*X*→*Y*is called**closed**if its graph is a closed subset of*X*×*Y*.

- Recall that a linear map
- Every Hausdorff TVS topology 𝜐 on X that has a neighborhood basis of 0 consisting of 𝜏-closed set is course than 𝜏.
^{[5]}

If (*X*, 𝜏) is locally convex space then we may add to this list:

- There exists a TVS Y not carrying the indiscrete topology (so in particular,
*Y*≠ { 0 }) 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;^{[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.

- Every σ(
*X*',*X*)-bounded subset of the continuous dual space X is equicontinuous (this provides a partial converse to the Banach-Steinhaus theorem);^{[2]}^{[6]} - X carries the strong topology β(
*X*,*X*');^{[2]} - Every lower semicontinuous seminorm on X is continuous;
^{[2]} - Every linear map
*F*:*X*→*Y*into a locally convex space Y is almost continuous;^{[2]}- this means that for every neighborhood V of 0 in Y, the closure of
*F*^{−1}(*V*) is a neighborhood of 0 in X;

- this means that for every neighborhood V of 0 in Y, the closure of
- Every surjective linear map
*F*:*Y*→*X*from a locally convex space Y is almost open;^{[2]}- this means that for every neighborhood V of 0 in Y, the closure of
*F*(*V*) is a neighborhood of 0 in X;

- this means that for every neighborhood V of 0 in Y, the closure of
- If ϖ is a locally convex topology on X such that (
*X*, ϖ) has a neighborhood basis at the origin consisting of 𝜏-closed sets, then ϖ is weaker than 𝜏;^{[2]}

If X is a Hausdorff locally convex space then we may add to this list:

**Closed graph theorem**: Every closed linear operator*F*:*X*→*Y*into a Banach space Y is continuous;^{[7]}- a closed linear operator is a linear operator whose graph is closed in
*X*×*Y*.

- a closed linear operator is a linear operator whose graph is closed in
- for all subsets A of the continuous dual space of X, the following properties are equivalent: A is
^{[6]}- equicontinuous;
- relatively weakly compact;
- strongly bounded;
- weakly bounded;

- the 0-neighborhood bases in X and the fundamental families of bounded sets in
*E*_{β}' correspond to each other by polarity;^{[6]}

If X is metrizable TVS then we may add to this list:

- For any complete metrizable TVS Y, every pointwise bounded
*sequence*in L(*X*;*Y*) is equicontinuous;^{[3]}

If X is a locally convex metrizable TVS then we may add to this list:

- (property S): the weak* topology on
*X*' is sequentially complete;^{[8]} - (property C): every weak* bounded subset of
*X*' is 𝜎(*X*',*X*)-relatively countably compact;^{[8]} - (𝜎-barrelled): every countable weak* bounded subset of
*X*' is equicontinuous;^{[8]} - (Baire-like): 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:

- TVSs that are Baire space.
- thus, also every topological vector space that is of the second category in itself is barrelled.

- F-spaces, Fréchet spaces, Banach spaces, and Hilbert spaces.
- However, there are normed vector spaces that are
*not*barrelled. For instance, if L^{2}([0, 1]) is topologized as a subspace of L^{1}([0, 1]), then it is not barrelled.

- However, there are normed vector spaces that are
- Complete pseudometrizable TVSs.
^{[9]} - Montel spaces.
- Strong duals of Montel spaces (since they are Montel spaces).
- A locally convex quasi-barreled space that is also a 𝜎-barrelled space.
^{[10]} - A sequentially complete quasibarrelled space.
- A quasi-complete Hausdorff locally convex infrabarrelled space.
^{[2]}- A TVS is called
**quasi-complete**if every closed and bounded subset is complete.

- A TVS is called
- A TVS with a dense barrelled vector subspace.
^{[2]}- Thus the completion of a barreled space is barrelled.

- 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]}

- Thus the completion of an infrabarrelled Hausdorff locally convex space is barrelled.
- 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.

- A locally convex ultrabelled TVS.
^{[11]} - A Hausdorff locally convex TVS X such that every weakly bounded subset of its continuous dual space is equicontinuous.
^{[12]} - A locally convex TVS X such that for every Banach space B, a closed linear map of X into B is necessarily continuous.
^{[13]} - A product of a family of barreled spaces.
^{[14]} - A locally convex direct sum and the inductive limit of a family of barrelled spaces.
^{[15]} - A quotient of a barrelled space.
^{[16]}^{[15]} - A Hausdorff sequentially complete quasibarrelled boundedly summing TVS.
^{[17]} - 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 ℝ
^{ℕ}that is not barrelled.^{[2]} - There exist complete locally convex TVSs that are not barrelled.
^{[2]} - The finest locally convex topology on a vector space is 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 X be a barrelled TVS and Y be a locally convex TVS.
Let H be a subset of the space

*L*(

*X*;

*Y*) of continuous linear maps from X into Y. The following are equivalent:

- H is bounded for the topology of pointwise convergence;
- H is bounded for the topology of bounded convergence;
- H is equicontinuous.

The Banach-Steinhaus theorem is a corollary of the above result.^{[20]} When the vector space Y consists of the complex numbers then the following generalization also holds.

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

- H is weakly bounded;
- H is strongly bounded;
- H is equicontinuous;
- H is relatively compact in the weak dual topology.

Recall that a linear map *F* : *X* → *Y* is called **closed** if its graph is a closed subset of *X* × *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
*on*to 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
*B*_{r}-complete TVS is necessarily continuous.^{[13]}

## History

Barrelled spaces were introduced by Bourbaki (1950).

## See also

- Countably barrelled space
- Infrabarrelled space
- Quasibarrelled space
- Ultrabarrelled space
- Uniform boundedness principle#Generalisations
- Ursescu theorem – A theorem that simultaneously generalizes the closed graph, open mapping, and Banach–Steinhaus theorems.
- Webbed space – Topological vector spaces for which the open mapping and closed graphs theorems hold

## References

- ^
^{a}^{b}^{c}^{d}Narici & Beckenstein 2011, pp. 225-273. - ^
^{a}^{b}^{c}^{d}^{e}^{f}^{g}^{h}^{i}^{j}^{k}^{l}^{m}^{n}^{o}^{p}^{q}^{r}^{s}Narici & Beckenstein 2011, pp. 371-423. - ^
^{a}^{b}^{c}Adasch, Ernst & Keim 1978, p. 39. **^**Adasch, Ernst & Keim 1978, p. 43.**^**Adasch, Ernst & Keim 1978, p. 32.- ^
^{a}^{b}^{c}Schaefer & Wolff 1999, pp. 127, 141Trèves 2006, p. 350. **^**Narici & Beckenstein 2011, p. 477.- ^
^{a}^{b}^{c}^{d}Narici & Beckenstein 2011, p. 399. **^**Narici & Beckenstein 2011, p. 383.**^**Khaleelulla 1982, pp. 28-63.**^**Narici & Beckenstein 2011, pp. 418-419.**^**Trèves 2006, p. 350.- ^
^{a}^{b}Schaefer & Wolff 1999, p. 166. **^**Schaefer & Wolff 1999, p. 138.- ^
^{a}^{b}Schaefer & Wolff 1999, p. 61. **^**Trèves 2006, p. 346.**^**Adasch, Ernst & Keim 1978, p. 77.**^**Schaefer & Wolff 1999, pp. 103-110.**^**Trèves 2006, p. 347.**^**Trèves 2006, p. 348.**^**Trèves 2006, p. 349.**^**Adasch, Ernst & Keim 1978, p. 41.**^**Adasch, Ernst & Keim 1978, pp. 70-73.**^**Trèves 2006, p. 424.

## Bibliography

- Adasch, Norbert; Ernst, Bruno; Keim, Dieter (1978).
*Topological Vector Spaces: The Theory Without Convexity Conditions*. Lecture Notes in Mathematics.**639**. Berlin New York: Springer-Verlag. ISBN 978-3-540-08662-8. OCLC 297140003. - Berberian, Sterling K. (1974).
*Lectures in Functional Analysis and Operator Theory*. Graduate Texts in Mathematics.**15**. New York: Springer. ISBN 978-0-387-90081-0. OCLC 878109401. - Bourbaki, Nicolas (1950). "Sur certains espaces vectoriels topologiques".
*Annales de l'Institut Fourier*(in French).**2**: 5–16 (1951). doi:10.5802/aif.16. MR 0042609. - Bourbaki, Nicolas (1987) [1981].
*Topological Vector Spaces: Chapters 1–5*[*Sur certains espaces vectoriels topologiques*].*Annales de l'Institut Fourier*. Éléments de mathématique.**2**. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag. ISBN 978-3-540-42338-6. OCLC 17499190. - Conway, John (1990).
*A course in functional analysis*. Graduate Texts in Mathematics.**96**(2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908. - Edwards, Robert E. (1995).
*Functional Analysis: Theory and Applications*. New York: Dover Publications. ISBN 978-0-486-68143-6. OCLC 30593138. - Grothendieck, Alexander (1973).
*Topological Vector Spaces*. Translated by Chaljub, Orlando. New York: Gordon and Breach Science Publishers. ISBN 978-0-677-30020-7. OCLC 886098. - Husain, Taqdir (1978).
*Barrelledness in topological and ordered vector spaces*. Berlin New York: Springer-Verlag. ISBN 3-540-09096-7. OCLC 4493665. - Jarchow, Hans (1981).
*Locally convex spaces*. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342. - Köthe, Gottfried (1969).
*Topological Vector Spaces I*. Grundlehren der mathematischen Wissenschaften.**159**. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704. - Khaleelulla, S. M. (1982). Written at Berlin Heidelberg.
*Counterexamples in Topological Vector Spaces*. Lecture Notes in Mathematics.**936**. Berlin New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370. - Narici, Lawrence; Beckenstein, Edward (2011).
*Topological Vector Spaces*. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. - Robertson, Alex P.; Robertson, Wendy J. (1964).
*Topological vector spaces*. Cambridge Tracts in Mathematics.**53**. Cambridge University Press. pp. 65–75. - Schaefer, Helmut H.; Wolff, Manfred P. (1999).
*Topological Vector Spaces*. GTM.**8**(Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. - Swartz, Charles (1992).
*An introduction to Functional Analysis*. New York: M. Dekker. ISBN 978-0-8247-8643-4. OCLC 24909067. - Trèves, François (2006) [1967].
*Topological Vector Spaces, Distributions and Kernels*. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.