Barrelled space#barrel

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 {{harvs|last=Bourbaki|authorlink=Nicolas Bourbaki|year=1950|txt}}.

Barrels

A convex and balanced subset of a real or complex vector space is called a {{em|disk}} and it is said to be {{em|disked}}, {{em|absolutely convex}}, or {{em|convex balanced}}.

A {{em|{{visible anchor|barrel}}}} or a {{em|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 $\dim X \geq 2$ and if $S$ is any subset of $X,$ then $S$ is a convex, balanced, and absorbing set of $X$ if and only if this is all true of $S \cap Y$ in $Y$ for every $2$ -dimensional vector subspace $Y;$ thus if $\dim X > 2$ then the requirement that a barrel be a closed subset of $X$ is the only defining property that does not depend {{em|solely}} on $2$ (or lower)-dimensional vector subspaces of $X.$

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

=Properties of barrels=

In any topological vector space (TVS) $X,$ every barrel in $X$ absorbs every compact convex subset of $X.$ {{sfn|Narici|Beckenstein|2011|pp=225-273}}

In any locally convex Hausdorff TVS $X,$ every barrel in $X$ absorbs every convex bounded complete subset of $X.$ {{sfn|Narici|Beckenstein|2011|pp=225-273}}

If $X$ is locally convex then a subset $H$ of $X^{\prime}$ is $\sigma\left(X^{\prime}, X\right)$ -bounded if and only if there exists a barrel $B$ in $X$ such that $H \subseteq B^{\circ}.$ {{sfn|Narici|Beckenstein|2011|pp=225-273}}

Let $(X, Y, b)$ be a pairing and let $\nu$ be a locally convex topology on $X$ consistent with duality. Then a subset $B$ of $X$ is a barrel in $(X, \nu)$ if and only if $B$ is the polar of some $\sigma(Y, X, b)$ -bounded subset of $Y.$ {{sfn|Narici|Beckenstein|2011|pp=225-273}}

Suppose $M$ is a vector subspace of finite codimension in a locally convex space $X$ and $B \subseteq M.$ 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 $B = C \cap M.$ {{sfn|Narici|Beckenstein|2011|pp=371-423}}

Characterizations of barreled spaces

Denote by $L(X; Y)$ the space of continuous linear maps from $X$ into $Y.$

If $(X, \tau)$ is a Hausdorff topological vector space (TVS) with continuous dual space $X^{\prime}$ then the following are equivalent:

$X$ is barrelled.

{{em|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 proved 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 {{em|some}} point of $Y$ (not necessarily the origin).{{sfn|Narici|Beckenstein|2011|pp=371-423}}
For any Hausdorff TVS $Y$ every pointwise bounded subset of $L(X; Y)$ is equicontinuous.{{sfn|Adasch|Ernst|Keim|1978|p=39}}

For any F-space Y every pointwise bounded subset of L(X; Y) is equicontinuous.{{sfn|Adasch|Ernst|Keim|1978|p=39}}
- An F-space is a complete metrizable TVS.
Every closed linear operator from X into a complete metrizable TVS is continuous.{{sfn|Adasch|Ernst|Keim|1978|p=43}}
- A linear map $F : X \to Y$ is called closed if its graph is a closed subset of $X \times Y.$
Every Hausdorff TVS topology $\nu$ on $X$ that has a neighborhood basis of the origin consisting of $\tau$ -closed set is course than $\tau.$ {{sfn|Adasch|Ernst|Keim|1978|p=32}}

If $(X, \tau)$ is locally convex space then this list may be extended by appending:

There exists a TVS $Y$ not carrying the indiscrete topology (so in particular, $Y \neq \{0\}$ ) such that every pointwise bounded subset of $L(X; Y)$ is equicontinuous.{{sfn|Narici|Beckenstein|2011|pp=371-423}}

For any locally convex TVS Y, every pointwise bounded subset of L(X; Y) is equicontinuous.{{sfn|Narici|Beckenstein|2011|pp=371-423}}
- 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.
Every $\sigma\left(X^{\prime}, X\right)$ -bounded subset of the continuous dual space $X$ is equicontinuous (this provides a partial converse to the Banach-Steinhaus theorem).{{sfn|Narici|Beckenstein|2011|pp=371-423}}{{harvnb|Schaefer|Wolff|1999|pp=127, 141}}{{harvnb|Trèves|2006|p=350}}.

$X$ carries the strong dual topology $\beta\left(X, X^{\prime}\right).$ {{sfn|Narici|Beckenstein|2011|pp=371-423}}

Every lower semicontinuous seminorm on $X$ is continuous.{{sfn|Narici|Beckenstein|2011|pp=371-423}}

Every linear map F : X \to Y into a locally convex space Y is almost continuous.{{sfn|Narici|Beckenstein|2011|pp=371-423}}
- A linear map $F : X \to Y$ is called {{em|{{visible anchor|almost continuous}}}} if for every neighborhood $V$ of the origin in $Y,$ the closure of $F^{-1}(V)$ is a neighborhood of the origin in $X.$
Every surjective linear map F : Y \to X from a locally convex space Y is almost open.{{sfn|Narici|Beckenstein|2011|pp=371-423}}
- This means that for every neighborhood $V$ of 0 in $Y,$ the closure of $F(V)$ is a neighborhood of 0 in $X.$
If $\omega$ is a locally convex topology on $X$ such that $(X, \omega)$ has a neighborhood basis at the origin consisting of $\tau$ -closed sets, then $\omega$ is weaker than $\tau.$ {{sfn|Narici|Beckenstein|2011|pp=371-423}}

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

Closed graph theorem: Every closed linear operator F : X \to Y into a Banach space Y is continuous.{{sfn|Narici|Beckenstein|2011|p=477}}
- The linear operator is called {{em|closed}} if its graph is a closed subset of $X \times Y.$
For every subset 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.

The 0-neighborhood bases in $X$ and the fundamental families of bounded sets in $X_{\beta}^{\prime}$ correspond to each other by polarity.

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

For any complete metrizable TVS $Y$ every pointwise bounded {{em|sequence}} in $L(X; Y)$ is equicontinuous.{{sfn|Adasch|Ernst|Keim|1978|p=39}}

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

({{visible anchor|Property S}}): The weak* topology on $X^{\prime}$ is sequentially complete.{{sfn|Narici|Beckenstein|2011|p=399}}

({{visible anchor|Property C}}): Every weak* bounded subset of $X^{\prime}$ is $\sigma\left(X^{\prime}, X\right)$ -relatively countably compact.{{sfn|Narici|Beckenstein|2011|p=399}}

({{visible anchor|𝜎-barrelled}}): Every countable weak* bounded subset of $X^{\prime}$ is equicontinuous.{{sfn|Narici|Beckenstein|2011|p=399}}

({{visible anchor|Baire-like}}): $X$ is not the union of an increase sequence of nowhere dense disks.{{sfn|Narici|Beckenstein|2011|p=399}}

Examples and sufficient conditions

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 {{em|not}} barrelled. For example, if the $L^p$ -space $L^2([0, 1])$ is topologized as a subspace of $L^1([0, 1]),$ then it is not barrelled.
Complete pseudometrizable TVSs.{{sfn|Narici|Beckenstein|2011|p=383}}
- 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.{{sfn|Khaleelulla|1982|pp=28-63}}

A sequentially complete quasibarrelled space.

A quasi-complete Hausdorff locally convex infrabarrelled space.{{sfn|Narici|Beckenstein|2011|pp=371-423}}
- A TVS is called quasi-complete if every closed and bounded subset is complete.
A TVS with a dense barrelled vector subspace.{{sfn|Narici|Beckenstein|2011|pp=371-423}}
- Thus the completion of a barreled space is barrelled.
A Hausdorff locally convex TVS with a dense infrabarrelled vector subspace.{{sfn|Narici|Beckenstein|2011|pp=371-423}}
- Thus the completion of an infrabarrelled Hausdorff locally convex space is barrelled.{{sfn|Narici|Beckenstein|2011|pp=371-423}}
A vector subspace of a barrelled space that has countable codimensional.{{sfn|Narici|Beckenstein|2011|pp=371-423}}
- In particular, a finite codimensional vector subspace of a barrelled space is barreled.
A locally convex ultrabarelled TVS.{{sfn|Narici|Beckenstein|2011|pp=418-419}}

A Hausdorff locally convex TVS $X$ such that every weakly bounded subset of its continuous dual space is equicontinuous.{{sfn|Trèves|2006|p=350}}

A locally convex TVS $X$ such that for every Banach space $B,$ a closed linear map of $X$ into $B$ is necessarily continuous.{{sfn|Schaefer|Wolff|1999|p=166}}

A product of a family of barreled spaces.{{sfn|Schaefer|Wolff|1999|p=138}}

A locally convex direct sum and the inductive limit of a family of barrelled spaces.{{sfn|Schaefer|Wolff|1999|p=61}}

A quotient of a barrelled space.{{sfn|Trèves|2006|p=346}}{{sfn|Schaefer|Wolff|1999|p=61}}

A Hausdorff sequentially complete quasibarrelled boundedly summing TVS.{{sfn|Adasch|Ernst|Keim|1978|p=77}}

A locally convex Hausdorff reflexive space is barrelled.

=Counterexamples=

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.{{sfn|Narici|Beckenstein|2011|pp=371-423}}

A closed subspace of a barreled space is not necessarily countably quasi-barreled (and thus not necessarily barrelled).{{sfn|Schaefer|Wolff|1999|pp=103-110}}

There exists a dense vector subspace of the Fréchet barrelled space $\R^{\N}$ that is not barrelled.{{sfn|Narici|Beckenstein|2011|pp=371-423}}

There exist complete locally convex TVSs that are not barrelled.{{sfn|Narici|Beckenstein|2011|pp=371-423}}

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).{{sfn|Narici|Beckenstein|2011|pp=371-423}}

Properties of barreled spaces

=Banach–Steinhaus generalization=

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

{{Math theorem|name=Theorem{{sfn|Trèves|2006|p=347}}|math_statement=

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.{{sfn|Trèves|2006|p=348}} When the vector space $Y$ consists of the complex numbers then the following generalization also holds.

{{Math theorem|name=Theorem{{sfn|Trèves|2006|p=349}}|math_statement=

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 $F : X \to Y$ is called closed if its graph is a closed subset of $X \times Y.$

{{Math theorem|name=Closed Graph Theorem{{sfn|Adasch|Ernst|Keim|1978|p=41}}|math_statement=

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.{{sfn|Adasch|Ernst|Keim|1978|pp=70-73}}

A linear map from a barrelled space into a locally convex space is almost continuous.

A linear map from a locally convex space {{em|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.{{sfn|Trèves|2006|p=424}}

A linear map with a closed graph from a barreled TVS into a $B_r$ -complete TVS is necessarily continuous.{{sfn|Schaefer|Wolff|1999|p=166}}

References

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Category:Topological vector spaces