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.

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.

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

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 showed 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 0 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 principal 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 ;^[2]
    • This means that for every neighborhood ${\displaystyle V}$ of 0 in ${\displaystyle Y}$, the closure of ${\displaystyle F^{-1}(V)}$ is a neighborhood of 0 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]
    • A closed linear operator is a linear operator whose graph is closed in ${\displaystyle X\times Y}$
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 TVS 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 TVS 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. (