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The uniform bounded ness principle is of great importance as throughout analysis as there are many instances of results related to this theorem. The development of this theorem greatly centres on Baire’s category theorem which is contained in section one of this work. Section two is the detail of the theorem in full while section three centres on its applications. Finally, the major objective of this work is to classically present the Uniform Bounded ness Principle as a vital ingredient in the theory of Hilbert Space Analysis.
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