Ultrabarrelled space
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 of a TVS is called an ultrabarrel if it is a closed and balanced subset of and if there exists a sequence of closed balanced and absorbing subsets of such that for all In this case, is called a defining sequence for A TVS is called ultrabarrelled if every ultrabarrel in 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 is a complete locally bounded non-locally convex TVS and if is a closed balanced and bounded neighborhood of the origin, then 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 – A theorem stating that pointwise boundedness implies uniform boundedness