On the uniform boundedness theorem in fuzzy quasi-normed spaces

Abstract

We prove that a family of continuous linear operators from a fuzzy quasi-normed space of the half second category to a fuzzy quasi-normed space is uniformly fuzzy bounded if and only if it is pointwise fuzzy bounded. This result generalizes and unifies several well-known results; in fact, the classical uniform boundedness principle, or Banach–Steinhauss theorem, is deduced as a particular case. Furthermore, we establish the relationship between uniform fuzzy boundedness and equicontinuity which allows us to give a uniform boundedness theorem in the class of paratopological vector spaces. The classical result for topological vector spaces is deduced as a corollary.