Original proof of Uniform Boundedness Principle (Banach Steinhaus) and related questions Can someone please provide me with any of the things listed below :


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*a list of different proofs of (some version of) the Uniform Boundedness Principle (also known as the Banach Steinhaus theorem), I already know Rudin's proof that seems quite general, a proof in the case of Banach spaces found in Haïm Brezis' book on functional analysis (also based on Baire Category) and a different proof altogether (making no use of Baire Category) found here https://pantherfile.uwm.edu/kevinm/www/qtbook/notes/pub.pdf.

*if possible the original proof and formulation of the theorem.


The reason I ask is that I don't understand how people realised completeness was a necessary condition, also I wonder if using the Baire Category Theorem was standard in Banach's time. Finally, what motivating examples did Banach or Steinhaus consider?
 A: Well, it is a good idea to look at the Wikipedia page first. There you'll find a link to the original article by Banach-Steinhaus1 which was motivated by the theory of Fourier series.
Concerning your question whether it was standard to apply the Baire category theorem at those times, very much so. The Polish school (e.g. Sierpinski, Lusin, Souslin, and of course Banach) applied it routinely, just look at their papers in the early volumes in Fundamenta Mathematicae.
Lastly, the proof in that paper you link to is quite a common gliding hump argument, which was recently published by Sokal in the Monthly. You'll find further historical remarks and discussion in his article.
1Banach, S.; Steinhaus, H., Sur le principe de la condensation de singularités., Fundamenta 9, 50-61 (1927); eudml. ZBL53.0243.02.
A: See here: Alan D. Sokal, A really simple elementary proof of the uniform boundedness theorem, Amer. Math. Monthly 118, 450-452 (2011)
https://arxiv.org/abs/1005.1585, https://doi.org/10.4169/amer.math.monthly.118.05.450
