View source for Ultra-barrelled space

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.

====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>