International Journal of Functional Analysis, Operator Theory and Applications
Volume 2, Issue 1, Pages 51 - 55
(June 2010)
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OPTIMALITY IN THE BANACH-STEINHAUS THEOREM
Charles Swartz
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Abstract:
The Banach-Steinhaus Theorem asserts that the pointwise limit of a sequence of continuous linear operators defined on a barrelled or complete metric linear space is continuous and the convergence is uniform on precompact subsets. If the domain of such a sequence of continuous linear operators is a complete metric linear AK-space, then we show that the family of precompact subsets is the optimal family for which the uniform convergence holds. We show that a similar result holds for a version of the Hahn-Schur Theorem. |
| Keywords and phrases: Banach-Steinhaus, precompact, AK-space, Hahn-Schur. |
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