A subset ${\displaystyle B_{0}}$  of a TVS ${\displaystyle X}$  is called an ultrabarrel if it is a closed and balanced subset of ${\displaystyle X}$  and if there exists a sequence ${\displaystyle \left(B_{i}\right)_{i=1}^{\infty }}$  of closed balanced and absorbing subsets of ${\displaystyle X}$  such that ${\displaystyle B_{i+1}+B_{i+1}\subseteq B_{i}}$  for all ${\displaystyle i=0,1,\ldots .}$  In this case, ${\displaystyle \left(B_{i}\right)_{i=1}^{\infty }}$  is called a defining sequence for ${\displaystyle B_{0}.}$  A TVS ${\displaystyle X}$  is called ultrabarrelled if every ultrabarrel in ${\displaystyle X}$  is a neighbourhood of the origin.^[1]

A locally convex ultrabarrelled space is a barrelled space.^[1] Every ultrabarrelled space is a quasi-ultrabarrelled space.^[1]

Complete and metrizable TVSs are ultrabarrelled.^[1] If ${\displaystyle X}$  is a complete locally bounded non-locally convex TVS and if ${\displaystyle B_{0}}$  is a closed balanced and bounded neighborhood of the origin, then ${\displaystyle B_{0}}$  is an ultrabarrel that is not convex and has a defining sequence consisting of non-convex sets.^[1]

Counter-examples edit

There exist barrelled spaces that are not ultrabarrelled.^[1] There exist TVSs that are complete and metrizable (and thus ultrabarrelled) but not barrelled.^[1]

• Barrelled space – Type of topological vector space
• Countably barrelled space
• Countably quasi-barrelled space
• Infrabarreled space
• Uniform boundedness principle#Generalisations – A theorem stating that pointwise boundedness implies uniform boundedness

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