title: Quasi-ultrabarrelled space
text: # Quasi-ultrabarrelled space
  
  In functional analysis and related areas of mathematics, a quasi-ultrabarrelled space is a topological vector spaces (TVS) for which every bornivorous ultrabarrel is a neighbourhood of the origin.
  
  ## Definition
  
  A subset $B_0$ of a TVS X is called a bornivorous ultrabarrel if it is a closed, balanced, and bornivorous subset of X and if there exists a sequence $\left(B_{i}\right)_{i=1}^{\infty }$ of closed balanced and bornivorous subsets of X such that $B_{i+1}$ + $B_{i+1}$ ⊆ $B_i$ for all i = 0, 1, .... In this case, $\left(B_{i}\right)_{i=1}^{\infty }$ is called a defining sequence for $B_0.$ A TVS X is called quasi-ultrabarrelled if every bornivorous ultrabarrel in X is a neighbourhood of the origin.
  
  ## Properties
  
  A locally convex quasi-ultrabarrelled space is quasi-barrelled.
  
  ## Examples and sufficient conditions
  
  Ultrabarrelled spaces and ultrabornological spaces are quasi-ultrabarrelled. Complete and metrizable TVSs are quasi-ultrabarrelled.
  
  