Ultra-barrelled space

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