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# barrel

Let $V$ be a topological vector space (TVS). A *barrel* $B$ is a subset of $V$ that is closed, convex, balanced, and absorbing. For example, in a Banach space $A$, any ball $\{v\in A\mid||v||\leq r\}$ for some $r>0$ is a barrel.

A topological vector space is said to be a *barrelled space* if it is locally convex, and every barrel is a neighborhood of $0$. Every Banach space is a barrelled space.

A weaker form of a barrelled space is that of an *infrabarrelled space*. A TVS is said to be *infrabarrelled* if it is locally convex, and every barrel that absorbs every bounded set is a neighborhood of $0$.

Let $V$ be a vector space and $\mathfrak{T}$ be the set of all those topologies on $V$ making $V$ a TVS. In other words, if $T\in\mathfrak{T}$, then $(V,T)$ is a topological vector space.

Let $V$ and $T\in\mathfrak{T}$ be defined as above. Then $(V,T)$ being barrelled has an equivalent characterization below:

(*) for any $T_{1}\in\mathfrak{T}$ such that there is a neighborhood base of $0$ consisting of $T$-closed sets, then $T_{1}$ is coarser than $T$.

A variation of a barrelled space is that of an *ultrabarrelled space*. A topological vector space is said to be *ultrabarrelled* if it satisfies (*) above. A locally convex ultrabarrelled space is barrelled.

# References

- 1 H. H. Schaefer, Topological Vector Spaces, Springer-Verlag, New York (1970).
- 2 R. E. Edwards, Functional Analysis, Theory and Applications, Holt, Reinhart and Winston, New York (1965).

## Mathematics Subject Classification

46A08*no label found*

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