Uniform Boundedness principle for unbounded operators
C. Ganesa Moorthy and CT. Ramasamy Received: November 7, 2013;
Accepted: February 17, 2014
Abstract.
A uniform boundedness principle for unbounded operators is derived. A particular case is:
Suppose {T}_{i}_{i in I}
be a family of linear mappings of a Banach space X into
a normed space Y such that {T : _{i}xi 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 : _{i}xi 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.
Some applications of this principle are also obtained.
Keywords:
Uniform boundedness principle; closed graph theorem.
AMS Subject classification:
Primary: 46A32, 47L60
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