barrelled set

Let $X$ be a topological vector space (TVS).

A subset of $X$ is called a {{em|barrel}} if it is closed convex balanced and absorbing in $X.$

A subset of $X$ is called {{em|bornivorous}}{{sfn|Narici|Beckenstein|2011|pp=441-457}} and a {{em|bornivore}} if it absorbs every bounded subset of $X.$ Every bornivorous subset of $X$ is necessarily an absorbing subset of $X.$

Let $B\_0\; \backslash subseteq\; X$ be a subset of a topological vector space $X.$ If $B\_0$ is a balanced absorbing subset of $X$ and if there exists a sequence $\backslash left(B\_i\backslash right)\_\{i=1\}^\{\backslash infty\}$ of balanced absorbing subsets of $X$ such that $B\_\{i+1\}\; +\; B\_\{i+1\}\; \backslash subseteq\; B\_i$ for all $i\; =\; 0,\; 1,\; \backslash ldots,$ then $B\_0$ is called a {{em|suprabarrel}}{{sfn|Khaleelulla|1982|p=65}} in $X,$ where moreover, $B\_0$ is said to be a(n):

• {{em|bornivorous suprabarrel}} if in addition every $B\_i$ is a closed and bornivorous subset of $X$ for every $i\; \backslash geq\; 0.${{sfn|Khaleelulla|1982|p=65}}
• {{em|ultrabarrel}} if in addition every $B\_i$ is a closed subset of $X$ for every $i\; \backslash geq\; 0.${{sfn|Khaleelulla|1982|p=65}}
• {{em|bornivorous ultrabarrel}} if in addition every $B\_i$ is a closed and bornivorous subset of $X$ for every $i\; \backslash geq\; 0.${{sfn|Khaleelulla|1982|p=65}}

In this case, $\backslash left(B\_i\backslash right)\_\{i=1\}^\{\backslash infty\}$ is called a {{em|defining sequence}} for $B\_0.${{sfn|Khaleelulla|1982|p=65}}

Note that every bornivorous ultrabarrel is an ultrabarrel and that every bornivorous suprabarrel is a suprabarrel.

• In a semi normed vector space the closed unit ball is a barrel.
• Every locally convex topological vector space has a neighbourhood basis consisting of barrelled sets, although the space itself need not be a barreled space.

• {{annotated link|Barrelled space}}
• {{annotated link|Space of linear maps}}
• {{annotated link|Ultrabarrelled space}}

{{reflist}}

• {{cite book|last=Hogbe-Nlend|first=Henri|title=Bornologies and functional analysis|publisher=North-Holland Publishing Co.|location=Amsterdam|year=1977|pages=xii+144|isbn=0-7204-0712-5|mr=0500064}}
• {{Khaleelulla Counterexamples in Topological Vector Spaces}}
• {{Narici Beckenstein Topological Vector Spaces|edition=2}}
• {{cite book|author=H.H. Schaefer|title=Topological Vector Spaces|publisher=Springer-Verlag|series=GTM|volume=3|year=1970|isbn=0-387-05380-8}}
• {{Cite book|isbn=9783540115656|title=Counterexamples in Topological Vector Spaces|last1=Khaleelulla|first1=S.M.|year=1982|publisher=Springer-Verlag|location=Berlin Heidelberg|series=GTM|volume=936 |pages=29–33, 49, 104}}
• {{Cite book|isbn=9780821807804|title=The Convenient Setting of Global Analysis|last1=Kriegl|first1=Andreas|year=1997|publisher=American Mathematical Society|last2=Michor|first2=Peter W.|series=Mathematical Surveys and Monographs}}

{{Functional Analysis}}

{{BoundednessAndBornology}}

{{TopologicalVectorSpaces}}

Category:Topological vector spaces