A really simple elementary proof of the uniform boundedness theorem
arXiv:1005.1585 · doi:10.4169/amer.math.monthly.118.05.450
Abstract
I give a proof of the uniform boundedness theorem that is elementary (i.e. does not use any version of the Baire category theorem) and also extremely simple.
LaTex2e, 5 pages. Version 2 improves the exposition by isolating the key lemma. To appear in the American Mathematical Monthly