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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

Every barrel must contain the origin. If and if is any subset of then is a convex, balanced, and absorbing set of if and only if this is all true of in for every -dimensional vector subspace thus if then the requirement that a barrel be a closed subset of is the only defining property that does not depend *solely* on (or lower)-dimensional vector subspaces of

If is any TVS then every closed convex and balanced neighborhood of the origin is necessarily a barrel in (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 is equal to (if considered as a complex vector space) or equal to (if considered as a real vector space). Regardless of whether is a real or complex vector space, every barrel in is necessarily a neighborhood of the origin (so is an example of a barrelled space). Let be any function and for every angle let denote the closed line segment from the origin to the point Let Then is always an absorbing subset of (a real vector space) but it is an absorbing subset of (a complex vector space) if and only if it is a neighborhood of the origin. Moreover, is a balanced subset of if and only if for every (if this is the case then and are completely determined by 's values on ) but is a balanced subset of if and only it is an open or closed ball centered at the origin (of radius ). In particular, barrels in are exactly those closed balls centered at the origin with radius in If then is a closed subset that is absorbing in but not absorbing in and that is neither convex, balanced, nor a neighborhood of the origin in By an appropriate choice of the function it is also possible to have be a balanced and absorbing subset of that is neither closed nor convex. To have be a balanced, absorbing, and closed subset of that is *neither* convex nor a neighborhood of the origin, define on as follows: for let (alternatively, it can be any positive function on that is continuously differentiable, which guarantees that and that is closed, and that also satisfies which prevents from being a neighborhood of the origin) and then extend to by defining which guarantees that is balanced in

- In any topological vector space (TVS) every barrel in absorbs every compact convex subset of
^{[1]} - In any locally convex Hausdorff TVS every barrel in absorbs every convex bounded complete subset of
^{[1]} - If is locally convex then a subset of is -bounded if and only if there exists a barrel in such that
^{[1]} - Let be a pairing and let be a locally convex topology on consistent with duality. Then a subset of is a barrel in if and only if is the polar of some -bounded subset of
^{[1]} - Suppose is a vector subspace of finite codimension in a locally convex space and If is a barrel (resp. bornivorous barrel, bornivorous disk) in then there exists a barrel (resp. bornivorous barrel, bornivorous disk) in such that
^{[2]}

Denote by the space of continuous linear maps from into

If is a Hausdorff topological vector space (TVS) with continuous dual space then the following are equivalent:

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

- This definition is similar to a characterization of Baire TVSs proved by Saxon [1974], who proved that a TVS 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
- For any Hausdorff TVS every pointwise bounded subset of is equicontinuous.
^{[3]} - For any F-space every pointwise bounded subset of is equicontinuous.
^{[3]}- An F-space is a complete metrizable TVS.

- Every closed linear operator from into a complete metrizable TVS is continuous.
^{[4]}- A linear map is called
**closed**if its graph is a closed subset of

- A linear map is called
- Every Hausdorff TVS topology on that has a neighborhood basis of the origin consisting of -closed set is course than
^{[5]}

If is locally convex space then this list may be extended by appending:

- There exists a TVS not carrying the indiscrete topology (so in particular, ) such that every pointwise bounded subset of is equicontinuous.
^{[2]} - For any locally convex TVS every pointwise bounded subset of 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 -bounded subset of the continuous dual space is equicontinuous (this provides a partial converse to the Banach-Steinhaus theorem).
^{[2]}^{[6]} - carries the strong dual topology
^{[2]} - Every lower semicontinuous seminorm on is continuous.
^{[2]} - Every linear map into a locally convex space is almost continuous.
^{[2]}- A linear map is called
if for every neighborhood of the origin in the closure of is a neighborhood of the origin in*almost continuous*

- A linear map is called
- Every surjective linear map from a locally convex space is almost open.
^{[2]}- This means that for every neighborhood of 0 in the closure of is a neighborhood of 0 in

- If is a locally convex topology on such that has a neighborhood basis at the origin consisting of -closed sets, then is weaker than
^{[2]}

If is a Hausdorff locally convex space then this list may be extended by appending:

**Closed graph theorem**: Every closed linear operator into a Banach space is continuous.^{[7]}- The linear operator is called
*closed*if its graph is a closed subset of

- The linear operator is called
- For every subset of the continuous dual space of the following properties are equivalent: is
^{[6]}- equicontinuous;
- relatively weakly compact;
- strongly bounded;
- weakly bounded.

- The 0-neighborhood bases in and the fundamental families of bounded sets in correspond to each other by polarity.
^{[6]}

If is metrizable topological vector space then this list may be extended by appending:

- For any complete metrizable TVS every pointwise bounded
*sequence*in is equicontinuous.^{[3]}

If is a locally convex metrizable topological vector space then this list may be extended by appending:

- (Property S): The weak* topology on is sequentially complete.
^{[8]} - (Property C): Every weak* bounded subset of is -relatively countably compact.
^{[8]} - (𝜎-barrelled): Every countable weak* bounded subset of is equicontinuous.
^{[8]} - (Baire-like): is not the union of an increase sequence of nowhere dense disks.
^{[8]}

Each of the following topological vector spaces is barreled:

- TVSs that are Baire space.
- Consequently, 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 exist normed vector spaces that are
*not*barrelled. For example, if the -space is topologized as a subspace of then it is not barrelled.

- However, there exist normed vector spaces that are
- Complete pseudometrizable TVSs.
^{[9]}- Consequently, every finite-dimensional TVS is barrelled.

- Montel spaces.
- Strong dual spaces of Montel spaces (since they are necessarily Montel spaces).
- A locally convex quasi-barrelled 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 ultrabarelled TVS.
^{[11]} - A Hausdorff locally convex TVS such that every weakly bounded subset of its continuous dual space is equicontinuous.
^{[12]} - A locally convex TVS such that for every Banach space a closed linear map of into 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.

- 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 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.

**Theorem ^{[19]}** — Let be a barrelled TVS and be a locally convex TVS.
Let be a subset of the space of continuous linear maps from into .
The following are equivalent:

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

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

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

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

Recall that a linear map is called **closed** if its graph is a closed subset of

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

- 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 -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

- ^
^{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. - ^
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^{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.

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