# barrel

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

A topological vector space is said to be a *barrelled space* if it is locally convex (http://planetmath.org/LocallyConvexTopologicalVectorSpace), 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 $\U0001d517$ be the set of all those topologies^{} on $V$ making $V$ a TVS. In other words, if $T\in \U0001d517$, then $(V,T)$ is a topological vector space.

Let $V$ and $T\in \U0001d517$ be defined as above. Then $(V,T)$ being barrelled has an equivalent^{} characterization below:

(*) for any ${T}_{1}\in \U0001d517$ such that there is a neighborhood base (http://planetmath.org/LocalBase) 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).

Title | barrel |

Canonical name | Barrel |

Date of creation | 2013-03-22 16:41:26 |

Last modified on | 2013-03-22 16:41:26 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 5 |

Author | CWoo (3771) |

Entry type | Definition |

Classification | msc 46A08 |

Synonym | barreled space |

Synonym | infrabarreled space |

Synonym | ultrabarreled space |

Synonym | barrelled |

Synonym | infrabarrelled |

Synonym | ultrabarrelled |

Defines | barrelled space |

Defines | infrabarrelled space |

Defines | ultrabarrelled space |