# Barrelled space

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).

Definition: 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.
Definition: A barrel or a barrelled set in a TVS is a subset that is a closed absorbing disk.

Note that 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 TVS X, every barrel in X absorbs every compact convex subset of X.
• In any locally convex Hausdorff TVS X, every barrel in X absorbs every convex bounded complete subset of X.
• If X is locally convex then a subset H of X' is 𝜎(X', X)-bounded if and only if there exists a barrel B in X such that HB°.
• 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.
• Suppose M is a vector subspace of finite codimension in a locally convex space X and BM. 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 = CM.

## 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:

1. X is barrelled;
2. (definition) Every barrel in X is a neighborhood of the origin;
• This definition is similar to a characterization of Baire TVSs proved by Saxon , 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).

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

1. For any Hausdorff TVS Y, every pointwise bounded subset of L(X; Y) is equicontinuous;
2. For any F-space Y, every pointwise bounded subset of L(X; Y) is equicontinuous;
3. Every closed linear operator from X into a complete metrizable TVS is continuous.
• Recall that a linear map F : XY is called closed if its graph is a closed subset of X × Y.
4. Every Hausdorff TVS topology 𝜐 on X that has a neighborhood basis of 0 consisting of 𝜏-closed set is course than 𝜏.

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

1. 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;
• 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.
3. Every σ(X', X)-bounded subset of the continuous dual space X is equicontinuous (this provides a partial converse to the Banach-Steinhaus theorem);
4. X carries the strong topology β(X, X');
5. Every lower semicontinuous seminorm on X is continuous;
6. Every linear map F : XY into a locally convex space Y is almost continuous;
• this means that for every neighborhood V of 0 in Y, the closure of F-1(V) is a neighborhood of 0 in X;
7. Every surjective linear map F : YX from a locally convex space Y is almost open;
• this means that for every neighborhood V of 0 in Y, the closure of F(V) is a neighborhood of 0 in X;
8. 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;

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

1. Closed graph theorem: Every closed linear operator F : XY into a Banach space Y is continuous;
• a closed linear operator is a linear operator whose graph is closed in X × Y.
2. for all subsets A of the continuous dual space of X, the following properties are equivalent: A is 
1. equicontinuous;
2. relatively weakly compact;
3. strongly bounded;
4. weakly bounded;
3. the 0-neighborhood bases in X and the fundamental families of bounded sets in Eβ' correspond to each other by polarity;

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

1. For any complete metrizable TVS Y, every pointwise bounded sequence in L(X; Y) is equicontinuous;

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

1. (property S): the weak* topology on X' is sequentially complete;
2. (property C): every weak* bounded subset of X' is 𝜎(X', X)-relatively countably compact;
3. (𝜎-barrelled): every countable weak* bounded subset of X' is equicontinuous;
4. (Baire-like): X is not the union of an increase sequence of nowhere dense disks.

## Examples and sufficient conditions

Each of the following topological vector spaces is barreled:

1. TVSs that are Baire space.
• thus, also 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 are normed vector spaces that are not barrelled. For instance, if L2([0, 1]) is topologized as a subspace of L1([0, 1]), then it is not barrelled.
3. Complete pseudometrizable TVSs.
4. Montel spaces.
5. Strong duals of Montel spaces (since they are Montel spaces).
6. A locally convex quasi-barreled space that is also a 𝜎-barrelled space.
7. A sequentially complete quasibarrelled space.
8. A quasi-complete Hausdorff locally convex infrabarrelled space.
• A TVS is called quasi-complete if every closed and bounded subset is complete.
9. A TVS with a dense barrelled vector subspace.
• Thus the completion of a barreled space is barrelled.
10. A Hausdorff locally convex TVS with a dense infrabarrelled vector subspace.
• Thus the completion of an infrabarrelled Hausdorff locally convex space is barrelled.
11. A vector subspace of a barrelled space that has countable codimensional.
• In particular, a finite codimensional vector subspace of a barrelled space is barreled.
12. A locally convex ultrabelled TVS.
13. A Hausdorff locally convex TVS X such that every weakly bounded subset of its continuous dual space is equicontinuous.
14. A locally convex TVS X such that for every Banach space B, a closed linear map of X into B is necessarily continuous.
15. A product of a family of barreled spaces.
16. A locally convex direct sum and the inductive limit of a family of barrelled spaces.
17. A quotient of a barrelled space.
18. A Hausdorff sequentially complete quasibarrelled boundedly summing TVS.
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.
• A closed subspace of a barreled space is not necessarily countably quasi-barreled (and thus not necessarily barrelled).
• There exists a dense vector subspace of the Fréchet barrelled space that is not barrelled.
• There exist complete locally convex TVSs that are not barrelled.
• 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).

## Properties of barreled spaces

### Banach-Steinhaus Generalization

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

Theorem — 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:

1. H is bounded for the topology of pointwise convergence;
2. H is bounded for the topology of bounded convergence;
3. H is equicontinuous.

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

Theorem — 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:

1. H is weakly bounded;
2. H is strongly bounded;
3. H is equicontinuous;
4. H is relatively compact in the weak dual topology.

Recall that a linear map F : XY is called closed if its graph is a closed subset of X × Y.

Closed Graph Theorem — 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.
• 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.
• A linear map with a closed graph from a barreled TVS into a Br-complete TVS is necessarily continuous.

## History

Barrelled spaces were introduced by Bourbaki (1950).