TITLE: Original proof of Uniform Boundedness Principle (Banach Steinhaus) and related questions
QUESTION [7 upvotes]: Can someone please provide me with any of the things listed below :

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?

REPLY [2 votes]: 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, http://dx.doi.org/10.4169/amer.math.monthly.118.05.450