# Ultra-barrelled space

From Encyclopedia of Mathematics

2010 Mathematics Subject Classification: *Primary:* 46A [MSN][ZBL]

A topological vector space $E$ with a topology $\mathfrak{T}$ for which any topology $\mathfrak{T}'$ having a neighbourhood base of zero consisting of $\mathfrak{T}$-closed sets is weaker than $\mathfrak{T}$. Every topological vector space which is not a set of the first category is ultra-barrelled. If a locally convex space is ultra-barrelled, it is also barrelled, but a barrelled space need not be ultra-barrelled.

#### References

[1] | R.E. Edwards, "Functional analysis: theory and applications" , Holt, Rinehart & Winston (1965) |

[2] | W.S. Robertson, "Completions of topological vector spaces" Proc. London Math. Soc. , 8 : 30 (1958) pp. 242–257 |

**How to Cite This Entry:**

Ultra-barrelled space.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Ultra-barrelled_space&oldid=41970

This article was adapted from an original article by V.I. Lomonosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article