
05898742
j
2011f.00591
Sokal, Alan D.
A really simple elementary proof of the uniform boundedness theorem.
Am. Math. Mon. 118, No. 5, 450452 (2011).
2011
Mathematical Association of America (MAA), Washington, DC
EN
I95
uniform boundedness principle
BanachSteinhaus theorem
gliding hump argument
doi:10.4169/amer.math.monthly.118.05.450
arxiv:1005.1585
The paper lives up to its title: based on the easy lemma that $\sup_{\yx\0$, 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.
Dirk Werner (Berlin)