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A really simple elementary proof of the uniform boundedness theorem. (English)
Am. Math. Mon. 118, No. 5, 450-452 (2011).
The paper lives up to its title: based on the easy lemma that $\sup_{\|y-x\|0$, the author’s proof constructs in just five lines some $x_0\in X$ with $\sup_n \|T_nx_0\|=\infty$ provided that $\sup_n \|T_n\|=\infty$. A wealth of information on other elementary arguments, i.e., those avoiding the Baire category theorem, complements the author’s proof.
Reviewer: Dirk Werner (Berlin)
Classification: I95
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