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Difference between revisions of "Ultra-barrelled space"

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A [[Topological vector space|topological vector space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095000/u0950001.png" /> with a topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095000/u0950002.png" /> for which any topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095000/u0950003.png" /> having a neighbourhood base of zero consisting of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095000/u0950004.png" />-closed sets is weaker than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095000/u0950005.png" />. 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|barrelled space]] need not be ultra-barrelled.
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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.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R.E. Edwards,  "Functional analysis: theory and applications" , Holt, Rinehart &amp; Winston  (1965)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W.S. Robertson,  "Completions of topological vector spaces"  ''Proc. London Math. Soc.'' , '''8''' :  30  (1958)  pp. 242–257</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  R.E. Edwards,  "Functional analysis: theory and applications" , Holt, Rinehart &amp; Winston  (1965)</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  W.S. Robertson,  "Completions of topological vector spaces"  ''Proc. London Math. Soc.'' , '''8''' :  30  (1958)  pp. 242–257</TD></TR>
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</table>

Revision as of 20:25, 2 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=41789
This article was adapted from an original article by V.I. Lomonosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article