Difference between revisions of "Ultra-barrelled space"
From Encyclopedia of Mathematics
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| − | 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 [[Category of a set|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. | + | 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 [[Category of a set|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==== | ====References==== | ||
Latest revision as of 06:21, 26 September 2017
2020 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
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