A Course in Functional Analysis and Measure Theory by Vladimir Kadets

Author:Vladimir Kadets
Language: eng
Format: epub, pdf
Publisher: Springer International Publishing, Cham

Theorem 1

(Uniform boundedness principle). Any pointwise bounded family of continuous linear operators from a Banach space X into a Banach space Y is uniformly bounded.

Proof.

Let be a pointwise bounded family. For each , define and . The sets are closed and their union is the whole space X. Therefore, by Baire’s theorem, at least one of the sets is not nowhere dense, and consequently contains some ball. That is, there exist a number and a ball of the form such that for any the inequality holds for all . Then for any and any ,

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