Uniform boundedness Principle for unbounded Operators

Abstract

A uniform boundedness principle for unbounded operators is derived. A particular case is: Suppose $\{T\}_{i\in I}$ be a family of linear mappings of a Banach space $X$ into a normed space $Y$ such that $\{T_ix : i \in I\}$ is bounded for each $x \in X$;then there exists a dense subset $A$ of the open unit ball in $X$ such that $\{T_ix : i \in I, x\in A\}$  is bounded. A closed graph theorem and a bounded inverse theorem are obtained for families of linear mappings as consequences of this principle. Someapplications of this principle are also obtained.