ultrabarrelled space

A subset $B\_0$ of a TVS $X$ is called an ultrabarrel if it is a closed and balanced subset of $X$ and if there exists a sequence $\backslash left(B\_i\backslash right)\_\{i=1\}^\{\backslash infty\}$ of closed balanced and absorbing subsets of $X$ such that $B\_\{i+1\}\; +\; B\_\{i+1\}\; \backslash subseteq\; B\_i$ for all $i\; =\; 0,\; 1,\; \backslash ldots.$

In this case, $\backslash left(B\_i\backslash right)\_\{i=1\}^\{\backslash infty\}$ is called a defining sequence for $B\_0.$

A TVS $X$ is called ultrabarrelled if every ultrabarrel in $X$ is a neighbourhood of the origin.{{sfn|Khaleelulla|1982|pp=65-76}}

A locally convex ultrabarrelled space is a barrelled space.{{sfn|Khaleelulla|1982|pp=65-76}}

Every ultrabarrelled space is a quasi-ultrabarrelled space.{{sfn|Khaleelulla|1982|pp=65-76}}

Complete and metrizable TVSs are ultrabarrelled.{{sfn|Khaleelulla|1982|pp=65-76}}

If $X$ is a complete locally bounded non-locally convex TVS and if $B\_0$ is a closed balanced and bounded neighborhood of the origin, then $B\_0$ is an ultrabarrel that is not convex and has a defining sequence consisting of non-convex sets.{{sfn|Khaleelulla|1982|pp=65-76}}

There exist barrelled spaces that are not ultrabarrelled.{{sfn|Khaleelulla|1982|pp=65-76}}

There exist TVSs that are complete and metrizable (and thus ultrabarrelled) but not barrelled.{{sfn|Khaleelulla|1982|pp=65-76}}

• {{annotated link|Barrelled space}}
• {{annotated link|Countably barrelled space}}
• {{annotated link|Countably quasi-barrelled space}}
• {{annotated link|Infrabarreled space}}
• {{annotated link|Uniform boundedness principle#Generalisations}}

{{reflist}}

• {{cite journal

|last=Bourbaki|first=Nicolas|authorlink=Nicolas Bourbaki|journal=Annales de l'Institut Fourier|language=French|mr=0042609|pages=5–16 (1951)|title=Sur certains espaces vectoriels topologiques|url=http://www.numdam.org/item?id=AIF_1950__2__5_0|volume=2|year=1950| doi=10.5802/aif.16|doi-access=free}}

• {{Husain Khaleelulla Barrelledness in Topological and Ordered Vector Spaces}}
• {{Jarchow Locally Convex Spaces}}
• {{Khaleelulla Counterexamples in Topological Vector Spaces}}
• {{Narici Beckenstein Topological Vector Spaces|edition=2}}
• {{cite book|last1=Robertson|first1=Alex P.|first2= Wendy J.|last2=Robertson |title=Topological vector spaces|series=Cambridge Tracts in Mathematics|volume=53|year=1964|publisher=Cambridge University Press|pages=65–75}}
• {{Schaefer Wolff Topological Vector Spaces|edition=2}}
• {{Trèves François Topological vector spaces, distributions and kernels}}

{{Functional analysis}}

{{Boundedness and bornology}}

{{Topological vector spaces}}

Category:Topological vector spaces