In functional analysis and related areas of mathematics, an ultrabarrelled space is a topological vector spaces (TVS) for which every ultrabarrel is a neighbourhood of the origin.
Definition A subset B 0 {\displaystyle B_{0}} of a TVS X {\displaystyle X} is called an ultrabarrel if it is a closed and balanced subset of X {\displaystyle X} and if there exists a sequence ( B i ) i = 1 ∞ {\displaystyle \left(B_{i}\right)_{i=1}^{\infty }} of closed balanced and absorbing subsets of X {\displaystyle X} such that B i + 1 + B i + 1 ⊆ B i {\displaystyle B_{i+1}+B_{i+1}\subseteq B_{i}} for all i = 0 , 1 , … . {\displaystyle i=0,1,\ldots .} In this case, ( B i ) i = 1 ∞ {\displaystyle \left(B_{i}\right)_{i=1}^{\infty }} is called a defining sequence for B 0 . {\displaystyle B_{0}.} A TVS X {\displaystyle X} is called ultrabarrelled if every ultrabarrel in X {\displaystyle X} is a neighbourhood of the origin.
Properties A locally convex ultrabarrelled space is a barrelled space. Every ultrabarrelled space is a quasi-ultrabarrelled space.
Examples and sufficient conditions Complete and metrizable TVSs are ultrabarrelled. If X {\displaystyle X} is a complete locally bounded non-locally convex TVS and if B 0 {\displaystyle B_{0}} is a closed balanced and bounded neighborhood of the origin, then B 0 {\displaystyle B_{0}} is an ultrabarrel that is not convex and has a defining sequence consisting of non-convex sets.
Counter-examples There exist barrelled spaces that are not ultrabarrelled. There exist TVSs that are complete and metrizable (and thus ultrabarrelled) but not barrelled.
See also Barrelled space – Type of topological vector space Countably barrelled space Countably quasi-barrelled space Infrabarreled space Uniform boundedness principle#Generalisations – Theorem stating that pointwise boundedness implies uniform boundedness
Citations
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